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P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> C H A P T E R 22 Describing the distribution of a single variable Objectives PL E P1: FXS/ABE To introduce the two main types of data—categorical and numerical To use bar charts to display frequency distributions of categorical data To use histograms and frequency polygons to display frequency distributions of SA M numerical data To use cumulative frequency polygons and cumulative relative frequency polygons to display cumulative frequency distributions To use the stem-and-leaf plot to display numerical data To use the histogram to display numerical data To use these plots to describe the distribution of a numerical variable in terms of symmetry, centre, spread and outliers To define and calculate the summary statistics mean, median, range, interquartile range, variance and standard deviation To understand the properties of these summary statistics and when each is appropriate To construct and interpret boxplots, and use them to compare data sets 22.1 Types of variables A characteristic about which information is recorded is called a variable, because its value is not always the same. Several types of variable can be identified. Consider the following situations. 500 Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> Chapter 22 — Describing the distribution of a single variable 501 PL E Students answer a question by selecting ‘yes’, ‘no’ or ‘don’t know’. Students say how they feel about a particular statement by ticking one of ‘strongly agree’, ‘agree’, ‘no opinion’, ‘disagree’ or ‘strongly disagree’. Students write down the size shoe that they take. Students write down their height. These situations give rise to two different types of data. The data arising from the first two situations are called categorical data, because the data can only be classified by the name of the category from which they come; there is no quantity associated with each category. The data arising from the third and fourth examples is called numerical data. These examples differ slightly from each other in the type of numerical data they each generate. Shoe sizes are of the form . . . , 6, 6.5, 7, 7.5, . . . . These are called discrete data, because the data can only take particular values. Discrete data often arise in situations where counting is involved. The other type of numerical data is continuous data where the variable may take any value (sometimes within a specified interval). Such data arise when students measure height. In fact, continuous data often arise when measuring is involved. Exercise 22A 1 Classify the data which arise from the following situations into categorical, or numerical. SA M a Kindergarten pupils bring along their favourite toy, and they are grouped together under the headings: ‘dolls’, ‘soft toys’, ‘games’, ‘cars’, and ‘other’. b The number of students on each of twenty school buses are counted. c A group of people each write down their favourite colour. d Each student in a class is weighed in kilograms. e Each student in a class is weighed and then classified as ‘light’, ‘average’ or ‘heavy’. f People rate their enthusiasm for a certain rock group as ‘low’, ‘medium’, or ‘high’. 2 Classify the data which arise from the following situations as categorical or numerical. a The intelligence quotient (IQ) of a group of students is measured using a test. b A group of people are asked to indicate their attitude to capital punishment by selecting a number from 1 to 5 where 1 = strongly disagree, 2 = disagree, 3 = undecided, 4 = agree, and 5 = strongly agree. 3 Classify the following numerical data as either discrete or continuous. a b c d e The number of pages in a book. The price paid to fill the tank of a car with petrol. The volume of petrol used to fill the tank of a car. The time between the arrival of successive customers at an autobank teller. The number of tosses of a die required before a six is thrown. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> 502 22.2 Essential Advanced General Mathematics Displaying categorical data—the bar chart Suppose a group of 130 students were asked to nominate their favourite kind of music under the categories ‘hard rock’, ‘oldies’, ‘classical’, ‘rap’, ‘country’ or ‘other’. The table shows the data for the first few students. Favourite music hard rock classical country hard rock PL E Student’s name Daniel Karina John Jodie The table gives data for individual students. To consider the group as a whole the data should be collected into a table called a frequency distribution by counting how many of each of the different values of the variable have been observed. Counting the number of students who responded to the question on favourite kinds of music gave the following results in each category. Hard rock 62 Other 27 Oldies 20 Classical 15 Rap 3 Country 3 Number of students SA M While a clear indication of the group’s preferences can be seen from the table, a visual display may be constructed to illustrate this. When the data are categorical, the appropriate display is a bar chart. The categories are indicated on the horizontal axis and the corresponding numbers in each category shown on the vertical axis. 70 60 50 40 30 20 10 0 Hard rock Other Oldies Classical Type of music Rap Country The order in which the categories are listed on the horizontal axis is not important, as no order is inherent in the category labels. In this particular bar chart, the categories are listed in decreasing order by number. From the bar chart the music preferences for the group of students may be easily compared. The value which occurs most frequently is called the mode of the variable. Here it can be seen that the mode is hard rock. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> Chapter 22 — Describing the distribution of a single variable 503 Exercise 22B 1 A group of students were asked to select their favourite type of fast food, with the following results. Food type hamburgers chicken fish and chips Chinese pizza other Number of students 23 7 6 7 18 8 PL E a Draw a bar chart for these data. b Which is the most popular food type? 2 The following responses were received to a question regarding the return of capital punishment. a Draw a bar chart for these data. b How many respondents either agree or strongly agree? SA M 3 A video shop proprietor took note of the type of films borrowed during a particular day with the following results. a Construct a bar chart to illustrate these data. b Which is the least popular film type? 4 A survey of secondary school students’ preferred ways of spending their leisure time at home gave the following results. a Construct a bar chart to illustrate these data. b What is the most common leisure activity? 22.3 strongly agree agree don’t know disagree strongly disagree comedy drama horror music other watch TV read listen to music watch a video phone friends other 21 11 42 53 129 53 89 42 15 33 42% 13% 23% 12% 4% 6% Displaying numerical data—the histogram In previous studies you have been introduced to various ways of summarising and displaying numerical data, including dotplots, stem-and-leaf plots, histograms and boxplots. Constructing a histogram for discrete numerical data is demonstrated in Example 1. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> 504 Essential Advanced General Mathematics Example 1 The numbers of siblings reported by each student in Year 11 at a local school is as follows: 2 0 2 3 2 3 4 1 4 0 1 1 3 4 1 2 5 0 3 3 9 0 2 0 4 5 1 1 6 1 0 1 1 0 1 1 1 1 1 2 0 0 3 2 1 Construct a frequency distribution of the number of siblings. Solution PL E P1: FXS/ABE To construct the frequency distribution count the numbers of students corresponding to each of the numbers of siblings, as shown. Number Frequency 0 9 1 15 2 7 3 6 4 4 5 2 6 1 7 0 8 0 9 1 A histogram looks similar to a bar chart, but because the data are numeric there is a natural order to the plot which may not occur with a bar chart. Usually for discrete data the actual data values are located at the middle of the appropriate column, as shown. 10 SA M Frequency 15 5 0 0 1 2 3 4 5 6 Number of siblings 7 8 9 An alternative display for a frequency distribution is a frequency polygon. It is formed by plotting the values in the frequency histogram with points, which are then joined by straight lines. A frequency polygon for the data in Example 1 is shown by the red line in this diagram. Frequency 15 10 5 0 0 1 2 3 4 5 6 Number of siblings 7 8 9 When the range of responses is large it is usual to gather the data together into sub-groups or class intervals. The number of data values corresponding to each class interval is called the class frequency. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> Chapter 22 — Describing the distribution of a single variable 505 Class intervals should be chosen according to the following principles: Every data value should be in an interval The intervals should not overlap There should be no gaps between the intervals. The choice of intervals can vary, but generally a division which results in about 5 to 15 groups is preferred. It is also usual to choose an interval width which is easy for the reader to interpret, such as 10 units, 100 units, 1000 units etc (depending on the data). By convention, the beginning of the interval is given the appropriate exact value, rather than the end. For example, intervals of 0–49, 50–99, 100–149 would be preferred over the intervals 1–50, 51–100, 101–150 etc. PL E P1: FXS/ABE Example 2 A researcher asked a group of people to record how many cups of coffee they drank in a particular week. Here are her results. 0 5 8 0 9 10 23 25 0 17 14 3 6 19 25 25 0 0 0 0 34 32 0 0 30 0 4 0 33 23 0 32 13 21 22 6 0 2 28 25 14 20 12 17 16 Construct a frequency distribution and hence a histogram of these data. Solution Frequency SA M Because there are so many different results and they are spread over a wide range, the data are summarised into class intervals. As the minimum value is 0 and the Number of Frequency maximum is 34, intervals of width 5 cups of coffee would be appropriate, giving the 0–4 16 frequency distribution shown in the table. 5–9 5 10–14 5 The corresponding histogram 15–19 4 may then be drawn. 20–24 5 25–29 5 20 30–34 5 15 10 5 0 5 15 20 25 30 10 Number of cups of coffee 35 Example 2 was concerned with a discrete numerical variable. When constructing a frequency distribution of continuous data, the data are again grouped, as shown in Example 3. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> 506 Essential Advanced General Mathematics Example 3 The following are the heights of the players in a basketball club, measured to the nearest millimetre. 178.1 183.3 192.4 196.3 185.6 180.3 203.7 189.6 173.3 182.0 191.1 183.9 193.4 183.6 189.7 177.7 183.1 184.5 191.1 184.1 193.0 185.8 180.4 183.8 188.3 189.1 180.0 174.7 189.5 184.6 202.4 170.9 178.6 194.7 185.3 188.7 180.1 170.5 179.3 193.8 178.9 PL E P1: FXS/ABE Construct a frequency distribution and hence a histogram of these data. Solution From the data it seems that intervals of width 5 will be suitable. All values of the variable which are 170 or more, but less than 175, have been included in the first interval. The second interval includes values from 175 to less than 180, and so on for the rest of the table. Frequency SA M The histogram of these data is shown here. Player heights 170 – 175 – 180 – 185 – 190 – 195 – 200 – Frequency 4 5 13 9 7 1 2 15 10 5 0 170 175 180 185 190 195 Player heights 200 205 The interval in a frequency distribution which has the highest class frequency is called the modal class. Here the modal class is 180.0–184.9. Using the TI-Nspire The calculator can be used to construct a histogram for numerical data. This will be illustrated using the basketball player height data from Example 3. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> Chapter 22 — Describing the distribution of a single variable 507 PL E The data is easiest entered in a Lists & 3). Spreadsheet application ( ) to Firstly, use the up/down arrows ( name the first column height. Then enter each of the 41 numbers as shown. Open a Data & Statistics application ( 5 ) to graph the data. At first the data displays as shown. SA M Specify the x variable by selecting Add X Variable from the Plot Properties (b 2 4) and selecting height. The data now displays as shown. (Note: It is also possible to use the NavPad to move down below the x-axis and click to add the x variable.) Select Histogram from the Plot Type menu (b 1 3). The data now displays as shown. Select Bin Settings from the Histogram Properties submenu of Plot Properties menu (b 2 2 2). Let width = 5 and Alignment = 170. Finally, select Zoom, Data from the Window/Zoom menu (b 5 2) to display the data as shown. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> 508 Essential Advanced General Mathematics Using the Casio ClassPad The calculator can be used to construct a histogram for numerical data. This will be illustrated using the basketball player height data from Example 3. SA M PL E enter the data into list1, tapping EXE to enter and move down the column. In Tap SetGraph, Setting . . . and the tab for Graph 1, enter the settings shown and tap SET. Tap SetGraph, StatGraph1 and then tap the box to tick and select the graph. to produce the graph selecting HStart Tap = 4 (the left bound of the histogram) and HStep = 4 (the desired interval width) when prompted. The histogram is produced as shown. With the graph window selected (bold border) tap 6 to adjust the viewing window for the graph. Tap Analysis, Trace and use the navigator key to move from column to column and display the count for that column. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> Chapter 22 — Describing the distribution of a single variable 509 Relative and percentage frequencies When frequencies are expressed as a proportion of the total number they are called relative frequencies. By expressing the frequencies as relative frequencies more information is obtained about the data set. Multiplying the relative frequencies by 100 readily converts them to percentage frequencies, which are easier to interpret. An example of the calculation of relative and percentage frequencies is shown in Example 4. Example 4 PL E P1: FXS/ABE Construct a relative frequency distribution and a percentage frequency distribution for the player height data. Player heights (cm) Frequency 170 – 4 175 – 5 180 – 13 185 – 9 190 – 7 195 – 1 200 – 2 SA M Solution From this table it can be seen, for example, that nine out of forty-one, or 22% of players, have heights from 185 cm to less than 190 cm. Relative frequency 4 41 5 41 13 41 9 41 7 41 1 41 2 41 Percentage frequency = 0.10 10% = 0.12 12% = 0.32 32% = 0.22 22% = 0.17 17% = 0.02 2% = 0.05 5% Both the relative frequency histogram and the percentage frequency histogram are identical to the frequency histogram—only the vertical scale is changed. To construct either of these histograms from a list of data use a graphics calculator to construct the frequency histogram, and then convert the individual frequencies to either relative frequencies or percentage frequencies one by one as required. Cumulative frequency distribution To answer questions concerning the number or proportion of the data values which are less than a given value a cumulative frequency distribution, or a cumulative relative frequency distribution can be constructed. In both a cumulative frequency distribution and a cumulative relative frequency distribution, the number of observations in each class are accumulated from low to high values of the variable. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> 510 Essential Advanced General Mathematics Example 5 Construct a cumulative frequency distribution and a cumulative relative frequency distribution for the data in Example 4. Solution <170 <175 <180 <185 <190 <195 <200 <205 Frequency Cumulative frequency Cumulative relative frequency 0 4 5 13 9 7 1 2 0 4 9 22 31 38 39 41 0 0.10 0.22 0.54 0.76 0.93 0.95 1.00 PL E Player heights (cm) SA M Each cumulative frequency was obtained by adding preceding values of the frequency. In the same way the cumulative relative frequencies were obtained by adding preceding relative frequencies. Thus it can be said that a proportion of 0.54, or 54%, of players are less than 185 cm tall. Cumulative frequency A graphical representation of a cumulative frequency distribution is called a cumulative frequency 40 polygon and has a distinctive appearance, as it 30 always starts at zero and is non-decreasing. This graph shows, on the vertical axis, the 20 number of players shorter than any height 10 given on the horizontal axis. The cumulative relative frequency distribution could also be 0 170 175 180 185 190 195 200 plotted as a cumulative relative frequency Player heights polygon, which would differ from the cumulative frequency polygon only in the scale on the vertical axis, which would run from 0 to 1. 205 Exercise 22C Example 1 1 The number of pets reported by each student in a class is given in the following table: 2 0 3 2 4 1 0 1 3 4 2 5 3 3 0 2 4 5 1 6 0 1 Construct a frequency distribution of the numbers of pets reported by each student. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> Chapter 22 — Describing the distribution of a single variable 511 Number of students 2 The number of children in the family for each student in a class is shown in this histogram. 10 5 0 a b c d PL E P1: FXS/ABE 1 2 3 4 5 6 7 Size of family 8 9 10 How many students are the only child in a family? What is the most common number of children in the family? How many students come from families with six or more children? How many students are there in the class? 10 SA M Number of students 3 The following histogram gives the scores on a general knowledge quiz for a class of Year 11 students. 5 0 10 20 30 40 50 60 Marks 70 80 90 100 a How many students scored from 10–19 marks? b How many students attempted the quiz? c What is the modal class? d If a mark of 50 or more is designated as a pass, how many students passed the quiz? 4 The maximum temperatures for several capital cities around the world on a particular day, in degrees Celsius, were: 17 16 17 31 Example 2 Example 4 a b c d 26 15 23 19 36 18 28 25 32 25 36 22 17 30 45 24 12 23 17 29 32 33 19 32 2 33 37 38 Use a class interval of 5 to construct a frequency distribution for these data. Construct the corresponding relative frequency distribution. Draw a histogram from the frequency distribution. What percentage of cities had a maximum temperature of less than 25◦ C? Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> 512 Essential Advanced General Mathematics 5 A student purchases 21 new text books from a school book supplier with the following prices (in dollars). 21.65 7.80 8.90 Example 3 Example 5 14.95 3.50 17.15 12.80 7.99 4.55 7.95 42.98 21.95 32.50 18.50 7.60 23.99 19.95 5.99 23.99 3.20 14.50 a Draw a histogram of these data using appropriate class intervals. b What is the modal class? c Construct a cumulative frequency distribution for these data and draw the cumulative frequency polygon. PL E P1: FXS/ABE 6 A group of students were asked to draw a line which they estimated to be the same length as a 30 cm ruler. The lines were then measured (in cm) with the following results. 30.3 32.2 32.1 30.9 30.1 31.2 31.2 31.6 30.7 32.3 32.1 32.1 31.3 31.4 30.8 30.7 31.8 29.7 32.8 32.9 30.1 31.0 31.9 28.9 33.3 29.4 30.7 31.6 a Construct a histogram of the frequency distribution. b Construct a cumulative frequency distribution for these data and draw the cumulative frequency polygon. c Write a sentence to describe the students’ performance on this task. SA M 7 The following are the marks obtained by a group of Year 11 Chemistry students on the end of year exam. 21 33 47 49 52 52 58 59 63 68 68 71 72 82 92 31 47 48 49 52 53 59 59 65 68 70 71 72 91 99 a Using a graphics calculator, or otherwise, construct a histogram of the frequency distribution. b Construct a cumulative frequency distribution for these data and draw the cumulative frequency polygon. c Write a sentence to describe the students’ performance on this exam. 8 The following 50 values are the lengths (in metres) of some par 4 golf holes from Melbourne golf courses. 302 371 376 366 398 272 334 332 361 407 311 369 338 299 337 351 334 320 321 371 338 320 321 361 266 325 374 364 312 354 314 364 317 305 331 307 353 362 408 409 336 366 310 245 385 310 260 280 279 260 a Construct a histogram of the frequency distribution. b Construct a cumulative frequency distribution for these data and draw the cumulative frequency polygon. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> Chapter 22 — Describing the distribution of a single variable 513 c Use the cumulative frequency polygon to estimate: i the proportion of par 4 holes below 300 m in length ii the proportion of par 4 holes 360 m or more in length iii the length which is exceeded by 90% of the par 4 holes. 22.4 Characteristics of distributions of numerical variables a PL E Distributions of numerical variables are characterised by their shapes and special features such as centre and spread. Two distributions are said to differ in centre if the values of the variable in one distribution are generally larger than the values of the variable in the other distribution. Consider, for example, the following histograms shown on the same scale. b 0 5 10 15 0 5 10 15 SA M It can be seen that plot b is identical to plot a but moved horizontally several units to the right, indicating that these distributions differ in the location of their centres. The next pair of histograms also differ, but not in the same way. While both histograms are centred at about the same place, histogram d is more spread out. Two distributions are said to differ in spread if the values of the variable in one distribution tend to be more spread out than the values of the variable in the other distribution. c 0 5 10 15 d 0 5 10 15 A distribution is said to be symmetric if it forms a mirror image of itself when folded in the ‘middle’ along a vertical axis; otherwise it is said to be skewed. Histogram e is perfectly symmetrical, while f shows a distribution which is approximately symmetric. f e 0 5 10 15 0 5 10 15 Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> 514 Essential Advanced General Mathematics If a histogram has a short tail to the left and a long tail pointing to the right it is said to be positively skewed (because of the many values towards the positive end of the distribution) as shown in the histogram g. If a histogram has a short tail to the right and a long tail pointing to the left it is said to be negatively skewed (because of the many values towards the negative end of the distribution), as shown in histogram h. g h negatively skewed positively skewed 0 5 PL E P1: FXS/ABE 10 15 0 5 10 15 Knowing whether a distribution is skewed or symmetric is important as this gives considerable information concerning the choice of appropriate summary statistics, as will be seen in the next section. Exercise 22D 1 Do the following pairs of distributions differ in centre, spread, both or neither? SA M a b 0 0 0 0 c Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> Chapter 22 — Describing the distribution of a single variable 515 2 Describe the shape of each of the following histograms. a b 0 0 PL E c 0 3 What is the shape of the histogram drawn in 6, Exercise 22C? 4 What is the shape of the histogram drawn in 7, Exercise 22C? 5 What is the shape of the histogram drawn in 8, Exercise 22C? 22.5 Stem-and-leaf plots SA M An informative data display for a small (less than 50 values) numerical data set is the stem-and-leaf plot. The construction of the stem-and-leaf plot is illustrated in Example 6. Example 6 By the end of 2004 the number of test matches played, as captain, by each of the Australian cricket captains was: 3 10 1 16 11 39 2 2 2 1 5 25 8 25 1 3 5 30 6 24 48 4 1 7 8 24 28 21 2 93 2 17 50 15 1 57 10 5 9 6 28 6 Construct a stem-and-leaf plot of these data. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> 516 Essential Advanced General Mathematics Solution To make a stem-and-leaf plot find the smallest and the largest data values. From the table above, the smallest value is 1, which is given a 0 in the ten’s column, and the largest is 93, which has a 9 in the ten’s column. This means that the stems are chosen to be from 0–9. These are written in a column with a vertical line to their right, as shown. 0 1 2 3 4 5 6 7 8 9 PL E P1: FXS/ABE The units for each data point are then entered to the right of the dividing line. They are entered initially in the order in which they appear in the data. When all data points are entered in the table, the stem-and-leaf plot looks like this. 3 6 1 9 8 0 2 1 8 3 6 4 8 2 6 2 5 5 1 2 1 5 1 2 1 7 9 6 5 0 0 1 7 5 4 4 8 5 8 0 7 SA M 0 1 2 3 4 5 6 7 8 9 3 To complete the plot the leaves are ordered, and a key added to specify the place value of the stem and the leaves. 0 1 2 3 4 5 6 7 8 9 1 0 1 0 8 0 1 1 1 1 2 2 2 2 2 3 3 4 5 5 5 6 6 6 7 8 8 9 0 1 5 6 7 4 4 5 5 8 8 9 7 3 | 9 indicates 39 matches 3 It can be seen from this plot that one captain has led Australia in many more test matches than any other (Allan Border, who captained Australia in 93 test matches). When a value sits away from the main body of the data it is called an outlier. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> Chapter 22 — Describing the distribution of a single variable 517 Stem-and-leaf plots have the advantage of retaining all the information in the data set while achieving a display not unlike that of a histogram (turned on its side). In addition, a stem-and-leaf plot clearly shows: the range of values where the values are concentrated the shape of the data set whether there are any gaps in which no values are observed any unusual values (outliers). Grouping the leaves in tens is simplest—other convenient groupings are in fives or twos, as shown in Example 7. Example 7 PL E P1: FXS/ABE The birth weights, in kilograms, of the first 30 babies born at a hospital in a selected month are as follows. 2.9 3.7 2.8 2.7 3.6 3.5 3.5 3.2 3.3 3.6 2.9 3.1 2.8 3.2 3.0 3.6 2.5 4.2 3.7 2.6 3.2 3.6 3.8 2.4 3.6 3.0 4.3 2.9 4.2 3.2 Construct a stem-and-leaf plot of these data. Solution SA M A stem-and-leaf plot of the birth weights, with the stem representing units and the leaves representing one-tenth of a unit, may be constructed. 2 4 5 6 7 8 8 9 9 9 3 0 0 1 2 2 2 2 3 5 5 6 6 6 6 6 7 7 8 3 | 0 indicates 3.0 kilograms 4 2 2 3 The plot, which allows one row for each different stem, appears to be too compact. These data may be better displayed by constructing a stem-and-leaf plot with two rows for each stem. These rows correspond to the digits {0, 1, 2, 3, 4} in the first row and {5, 6, 7, 8, 9} in the second row. 2 2 3 3 4 4 5 0 5 2 6 0 5 2 7 1 6 3 8 2 6 8 2 6 9 2 6 9 2 6 9 3 7 7 8 3 | 0 indicates 3.0 kilograms The only other possibility for a stem-and-leaf plot is one which has five rows per stem. These rows correspond to the digits {0, 1}, {2, 3}, {4, 5}, {6, 7} and {8, 9}. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> 518 Essential Advanced General Mathematics 2 2 2 3 3 3 3 3 4 4 4 6 8 0 2 5 6 8 5 7 8 0 2 5 6 2 2 9 1 2 9 9 2 3 6 6 6 7 7 PL E P1: FXS/ABE 3 | 0 indicates 3.0 kilograms 3 SA M None of the stem-and-leaf displays shown are correct or incorrect. A stem-and-leaf plot is used to explore data and more than one may need to be constructed before the most informative one is obtained. Again, from 5 to 15 rows is generally the most helpful, but this may vary in individual cases. When the data have too many digits for a convenient stem-and-leaf plot they should be rounded or truncated. Truncating a number means simply dropping off the unwanted digits. So, for example, a value of 149.99 would become 149 if truncated to three digits, but 150 if rounded to three digits. Since the object of a stem-and-leaf display is to give a feeling for the shape and patterns in the data set, the decision on whether to round or truncate is not very important; however, generally when constructing a stem-and-leaf display the data is truncated, as this is what commonly used data analysis computer packages will do. Some of the most interesting investigations in statistics involve comparing two or more data sets. Stem-and-leaf plots are useful displays for the comparison of two data sets, as shown in the following example. Example 8 The following table gives the number disposals by members of the Port Adelaide and Brisbane football teams, in the 2004 AFL Grand Final. Port Adelaide 25 12 20 11 19 11 18 11 18 11 17 10 16 10 15 9 14 9 13 7 12 7 Brisbane 25 19 19 18 17 16 15 15 13 13 13 10 10 9 9 8 8 7 6 5 4 0 Construct back to back stem-and-leaf plots of these data. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> Chapter 22 — Describing the distribution of a single variable 519 Solution To compare the two groups, the stem-and-leaf plots are drawn back to back, using two rows per stem. Port Adelaide Brisbane 0 0 1 1 2 2 9 9 7 7 4 3 2 2 1 1 1 1 0 0 9 8 8 7 6 5 0 5 0 | 2 represents 20 disposals 0 5 0 5 4 6 7 8 8 9 9 0 3 3 3 5 6 7 8 9 9 PL E P1: FXS/ABE 5 2 | 0 represents 20 disposals The leaves on the left of the stem are centred slightly higher than the leaves on the right, which suggests that, overall, Port Adelaide recorded more disposals. The spread of disposals for Port Adelaide appears narrower than that of the Brisbane players. Exercise 22E Example 6 1 The monthly rainfall for Melbourne, in a particular year, is given in the following table (in millimetres). J F M A M SA M Month Rainfall (mm) J J A S O N D 48 57 52 57 58 49 49 50 59 67 60 59 a Construct a stem-and-leaf plot of the rainfall, using the following stems. 4 5 6 b In how many months is the rainfall 60 mm or more? Example 7 2 An investigator recorded the amount of time 24 similar batteries lasted in a toy. Her results in hours were: 25.5 4.2 39.7 25.6 29.9 16.9 23.6 18.9 26.9 46.0 31.3 33.8 21.4 36.8 27.4 27.5 19.5 25.1 29.8 31.3 33.4 41.2 21.8 32.9 a Make a stem-and-leaf plot of these times with two rows per stem. b How many of the batteries lasted for more than 30 hours? 3 The amount of time (in minutes) that a class of students spent on homework on one particular night was: 10 39 27 70 46 19 63 37 20 67 33 20 15 28 21 23 16 0 14 29 15 10 Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> 520 Essential Advanced General Mathematics a Make a stem-and-leaf plot of these times. b How many students spent more than 60 minutes on homework? c What is the shape of the distribution? 4 The cost of various brands of track shoes at a retail outlet are as follows. $49.99 $75.49 $68.99 $164.99 $75.99 $210.00 $84.99 $36.98 $95.49 $28.99 $46.99 $76.99 $82.99 $79.99 $149.99 a Construct a stem-and-leaf plot of these data. b What is the shape of the distribution? Example 8 $39.99 $25.49 $35.99 $78.99 52.99 $45.99 PL E P1: FXS/ABE 5 The students in a class were asked to write down the ages of their mothers and fathers. Mother’s age 49 50 43 44 Father’s age 50 51 43 46 43 40 50 39 47 40 50 41 40 43 46 45 49 48 49 38 42 43 44 37 38 43 41 44 55 44 51 48 48 43 47 48 47 43 52 46 54 48 41 49 44 45 40 46 SA M a Construct a back to back stem-and-leaf plot of these data sets. b How do the ages of the students’ mothers and fathers compare in terms of shape, centre and spread? 6 The results of a mathematics test for two different classes of students are given in the table. Class A 22 19 85 79 48 45 39 82 68 81 47 80 58 91 77 99 76 55 89 65 85 79 82 71 Class B 12 13 74 76 80 80 81 81 83 82 98 84 70 84 70 88 71 69 72 73 72 88 73 91 a Construct a back to back stem-and-leaf plot to compare the data sets. b How many students in each class scored less than 50%? c Which class do you think performed better overall on the test? Give reasons for your answer. 22.6 Summarising data A statistic is a number that can be computed from data. Certain special statistics are called summary statistics, because they numerically summarise special features of the data set under consideration. Of course, whenever any set of numbers is summarised into just one or two figures much information is lost, but if the summary statistics are well chosen they will also help to reveal the message which may be hidden in the data set. Summary statistics are generally either measures of centre or measures of spread. There are many different examples for each of these measures and there are situations when one of the measures is more appropriate than another. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> Chapter 22 — Describing the distribution of a single variable 521 Measures of centre Mean The most commonly used measure of centre of a distribution of a numerical variable is the mean. This is calculated by summing all the data values and dividing by the number of values in the data set. PL E Example 9 The following data set shows the number of premierships won by each of the current AFL teams, up until the end of 2004. Find the mean of the number of premiership wins. Premierships 16 16 14 12 11 10 9 6 4 3 2 2 1 1 1 0 SA M Team Carlton Essendon Collingwood Melbourne Fitzroy/Lions Richmond Hawthorn Geelong Kangaroos Sydney West Coast Adelaide Port Adelaide W Bulldogs St Kilda Fremantle Solution mean = 16 + 16 + 14 + 12 + 11 + 10 + 9 + 6 + 4 + 3 + 2 + 2 + 1 + 1 + 1 + 0 = 6.8 16 The mean of a sample is always denoted by the symbol x̄, which is called ‘x bar’. In general, if n observations are denoted by x1 , x2 , . . . ., xn the mean is x1 + x2 + · · · · · · + xn n or, in a more compact version x̄ = n 1 xi n i=1 where the symbol is the upper case Greek sigma, which in mathematics means ‘the sum of the terms’. x̄ = Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> 522 Essential Advanced General Mathematics Note: The subscripts on the x’s are used to identify all of the n different values of x. They do not mean that the x’s have to be written in any special order. The values of x in the example are in order only because they were listed in that way in the table. Median Another useful measure of the centre of a distribution of a numerical variable is the middle value, or median. To find the value of the median, all the observations are listed in order and the middle one is the median. The median of 2 3 4 PL E P1: FXS/ABE 5 median 6 7 5 7 8 8 11 is 6, as there are five observations on either side of this value when the data are listed in order. Example 10 Find the median number of premierships in the AFL ladder using the data in Example 9. Solution As the data are already given in order, it only remains to decide which is the middle observation. 1 1 1 2 2 3 4 6 9 10 11 12 14 16 16 SA M 0 Since there are 16 entries in the table there is no actual middle observation, so the median is chosen as the value half way between the two middle observations, in this 1 case the eighth and ninth (6 and 4). Thus the median is equal to (6 + 4) = 5. The 2 interpretation here is that of the teams currently playing in the AFL, half (or 50%) have won the premiership 5 or more times and half (or 50%) have have won the premiership 5 or less times. In general, to compute the median of a distribution: Arrange all the observations in ascending order according to size. n + 1 th If n, the number of observations, is odd, then the median is the 2 observation from the end of the list. If n, the number of observations, is even, then the median is found by averaging the nth and the two middle observations in the list. That is, to find the median the 2 n th + 1 observations are added together, and divided by 2. 2 The median value is easily determined from a stem-and-leaf plot by counting to the required observation or observations from either end. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> Chapter 22 — Describing the distribution of a single variable 523 From Examples 10 and 11, the mean number of times premierships won (6.8) and the median number of premierships won (5) have already been determined. These values are different and the interesting question is: why are they different, and which is the better measure of centre for this example? To help answer this question consider a stem-and-leaf plot of these data. 0 0 1 1 0 6 0 6 1 9 1 6 1 1 2 4 2 2 3 4 PL E P1: FXS/ABE From the stem-and-leaf plot it can be seen that the distribution is positively skewed. This example illustrates a property of the mean. When the distribution is skewed or if there are one or two very extreme values, then the value of the mean may be quite significantly affected. The median is not so affected by unusual observations, however, and is thus often a preferable measure of centre. When this is the case, the median is generally preferred as a measure of centre as it will give a better ‘typical’ value of the variable under consideration. Mode SA M The mode is the observation which occurs most often. It is a useful summary statistic, particularly for categorical data which do not lend themselves to some of the other numerical summary methods. Many texts state that the mode is a third option for a measure of centre but this is generally not true. Sometimes data sets do not have a mode, or they have several modes, or they have a mode which is at one or other end of the range of values. Measures of spread Range A measure of spread is calculated in order to judge the variability of a data set. That is, are most of the values clustered together, or are they rather spread out? The simplest measure of spread can be determined by considering the difference between the smallest and the largest observations. This is called the range. Example 11 Consider the marks, for two different tasks, awarded to a group of students. Task A 2 35 6 38 9 38 10 39 11 42 12 46 13 47 22 47 23 52 24 52 26 56 26 56 27 59 33 91 34 94 16 59 19 63 21 65 23 68 28 71 31 72 31 73 33 75 38 78 41 78 49 78 52 86 53 88 54 91 Task B 11 56 Find the range of each of these data sets. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> 524 Essential Advanced General Mathematics Solution For Task A, the minimum mark is 2 and the maximum mark is 94. Range for Task A = 94 − 2 = 92 For Task B, the minimum mark is 11 and the maximum mark is 91. Range for Task B = 91 − 11 = 80 The range for Task A is greater than the range for Task B. Is the range a useful summary statistic for comparing the spread of the two distributions? To help make this decision, consider the stem-and-leaf plots of the data sets: 7 9 6 8 9 3 6 8 7 6 9 2 4 5 7 6 PL E P1: FXS/ABE Task A 6 2 1 0 3 2 4 3 6 2 2 2 1 0 1 2 3 4 5 6 7 8 9 SA M 4 Task B 1 1 1 1 2 3 1 6 1 6 3 1 9 3 5 2 8 9 8 3 4 8 3 8 6 9 5 8 8 8 From the stem-and-leaf plots of the data it appears that the spread of marks for the two tasks is not well described by the range. The marks for Task A are more concentrated than the marks for Task B, except for the two unusual values for Task A. Another measure of spread is needed, one which is not so influenced by these extreme values. For this the interquartile range is used. Interquartile range To find the interquartile range of a distribution: Arrange all observations in order according to size. Divide the observations into two equal-sized groups. If n, the number of observations, is odd, then the median is omitted from both groups. Locate Q 1 , the first quartile, which is the median of the lower half of the observations, and Q 3 , the third quartile, which is the median of the upper half of the observations. The interquartile range IQR is defined as the difference between the quartiles. That is IQR = Q 3 − Q 1 Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> Chapter 22 — Describing the distribution of a single variable 525 Definitions of the quartiles of a distribution sometimes differ slightly from the one given here. Using different definitions may result in slight differences in the values obtained, but these will be minimal and should not be considered a difficulty. Example 12 Find the interquartile ranges for Task A and Task B data given in Example 11. Solution PL E P1: FXS/ABE For Task A the marks listed in order are: 2 35 6 38 9 38 10 39 11 42 12 46 13 47 22 47 23 52 24 52 26 56 26 56 27 59 33 91 34 94 Since there is an even number of observations, then the lower ‘half’ is: 2 6 9 10 11 12 13 22 23 24 26 26 27 33 34 The median of this lower group is the eighth observation, 22, so Q 1 = 22. The upper half is: 35 38 38 39 42 46 47 47 52 52 56 56 59 91 94 SA M The median of this upper group is 47, so Q 3 = 47 Thus, the interquartile range, IQR = 47 − 22 = 25 Similarly, for Task B data, the lower quartile = 31 and the upper quartile = 73, giving an interquartile range for this data set of 42. Comparing the two values of interquartile range shows the spread of Task A marks to be much smaller than the spread of Task B marks, which seems consistent with the display. The interquartile range is a measure of spread of a distribution which describes the range of the middle 50% of the observations. Since the upper 25% and the lower 25% of the observations are discarded, the interquartile range is generally not affected by the presence of outliers in the data set, which makes it a reliable measure of spread. The median and quartiles of a distribution may also be determined from a cumulative relative frequency polygon. Since the median is the observation which divides the data set in half, this is the data value which corresponds to a cumulative relative frequency of 0.5 or 50%. Similarly, the first quartile corresponds to a cumulative relative frequency of 0.25 or 25%, and the third quartile corresponds to a cumulative relative frequency of 0.75 or 75%. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> 526 Essential Advanced General Mathematics Example 13 Use the cumulative relative frequency polygon to find the median and the interquartile range for the data set shown in the graph. % 100 75 50 25 0 PL E P1: FXS/ABE 2 Solution 4 6 8 10 12 14 16 18 From the plot of the data it can be seen that the median is 10, the first quartile is 8, the third quartile is 12 and hence the interquartile range is 12 − 8 = 4. Standard deviation SA M Another extremely useful measure of spread is the standard deviation. It is derived by considering the distance of each observation from the sample mean. If the average of these distances is used as a measure of spread it will be found that, as some of these distances are positive and some are negative, adding them together results in a total of zero. A more useful measure will result if the distances are squared (which makes them all positive) and are then added together. The variance is defined as a kind of average of these squared distances. When the variance is calculated from a sample, rather than the whole population, the average is calculated by dividing by n − 1, rather than n. For the remainder of this discussion it will be assumed that the data under consideration are from a sample. Since the variance has been calculated by squaring the data values it is sensible to find the square root of the variance, so that the measure reverts to a scale comparable to the original data. This results in measure of spread which is called the standard deviation. Standard deviation calculated from a sample is denoted s. Formally the standard deviation may be defined as follows. If a data set consists of n observations denoted x1 , x2 , . . . , xn , the standard deviation is 1 (x1 − x̄)2 + (x2 − x̄)2 + · · · + (xn − x̄)2 s= n−1 or, in more compact notation, n 1 (xi − x̄)2 s= n − 1 i=1 Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> 527 Chapter 22 — Describing the distribution of a single variable Example 14 Calculate the standard deviation of the following data set. 13 12 14 6 15 12 7 6 7 8 Solution Construct a table as shown. (xi − x̄)2 9 4 16 16 25 4 9 16 9 4 (xi − x̄)2 = 1̄12 112 √ = 12.44 = 3.53 From the table, the standard deviation s is: s = 9 xi − x̄ 3 2 4 −4 5 2 −3 −4 −3 −2 SA M PL E xi 13 12 14 6 15 12 7 6 7 8 xi = 1̄00 Interpreting the standard deviation The standard deviation can be made more meaningful by interpreting it in relation to the data set. The interquartile range gives the spread of the middle 50% of the data. Can similar statements be made about the standard deviation? It can be shown that, for most data sets, about 95% of the observations lie within two standard deviations of the mean. Example 15 The cost of a lettuce at a number of different shops on a particular day is given in the table: $3.85 $3.81 $2.65 $1.69 $1.90 $3.66 $2.95 $2.60 $2.40 $2.70 $2.42 $3.10 $2.63 $2.80 $3.20 $1.80 $4.20 $2.88 $2.33 $1.40 $0.85 Calculate the mean cost, the standard deviation and the interval equivalent to two standard deviations above and below the mean. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> 528 Essential Advanced General Mathematics Solution The mean cost is $2.66 and the standard deviation is $0.84. The interval equivalent to two standard deviations above and below the mean is: [2.66 − 2 × 0.84, 2.66 + 2 × 0.84] = [0.98, 4.34]. In this case, 20 of the 21 observations, or 95% of observations, have values within the interval calculated. Example 16 PL E P1: FXS/ABE The prices of forty secondhand motorbikes listed in a newspaper are as follows: $5442 $2220 $3457 $6469 $5294 $5439 $1356 $4689 $7148 $3847 $2523 $738 $8218 $10 884 $4219 $2358 $656 $11 091 $14 450 $4786 $2363 $715 $11 778 $15 731 $2280 $2244 $1000 $11 637 $13 153 $3019 $1963 $1214 $8770 $10 067 $7645 $2142 $1788 $8450 $9878 $8079 Determine the interval equivalent to two standard deviations above and below the mean. Solution SA M The mean price is $5729 and the standard deviation is $4233 (to the nearest whole dollar). The interval equivalent to two standard deviations above and below the mean is: [5729 − 2 × 4233, 5729 + 2 × 4233] = [−2737, 14 195]. The negative value does not give a sensible solution and should be replaced by 0. 38 of the 40 observations, or 95% of observations, have values within the interval. The exact percentage of observations which lie within two standard deviations of the mean varies from data set to data set, but in general it will be around 95%, particularly for symmetric data sets. It was noted earlier that even a single outlier can have a very marked effect on the value of the mean of a data set, while leaving the median unchanged. The same is true when the effect of an outlier on the standard deviation is considered, in comparison to the interquartile range. The median and interquartile range are called resistant measures, while the mean and standard deviation are not resistant measures. When considering a data set it is necessary to do more than just compute the mean and standard variation. First it is necessary to examine the data, using a histogram or stem-and-leaf plot to determine which set of summary statistics is more suitable. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> Chapter 22 — Describing the distribution of a single variable 529 Using the TI-Nspire PL E The calculator can be used to calculate the values of all of the summary statistics in this section. Consider the data from Example 16. The data is easiest entered in a Lists & 3). Spreadsheet application ( ) to Firstly, use the up/down arrows ( name the first column bike. Then enter each of the 40 numbers as shown. SA M 1) to Open a Calculator application ( calculate the summary statistics. Select the One-Variable Statistics command from the Stat Calculations submenu of the Statistics menu (b 6 1 1), specify in the dialog box that there is only one list, and then complete the final dialog box as shown. Press enter to calculate the values of the summary statistics. Use the up arrow ( ) to view the rest of the summary statistics. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> 530 Essential Advanced General Mathematics The calculator can also be used to determine the summary statistics when the data is given in a frequency table such as: x Fr equency 1 5 2 8 3 7 4 2 PL E The data is easiest entered in a Lists & 3). Spreadsheet application ( ) to Firstly, use the up/down arrows ( name the first column x and the second column freq. Then enter the data as shown. SA M 1) to Open a Calculator application ( calculate the summary statistics. Select the One-Variable Statistics command from the Stat Calculations submenu of the Statistics menu (b 6 1 1), specify in the dialog box that there is only one list, and then complete the final dialog box as shown. Press enter to calculate the values of the summary statistics. Using the Casio ClassPad Consider the following heights in cm of a group of eight women. 176, 160, 163, 157, 168, 172, 173, 169 Enter the data into list1 in the module. Tap Calc, One-Variable and when prompted ensure that the XList is set to list1 and the Freq = 1 (since each score is entered individually). The calculator returns the results as shown and all univariate statistics can be viewed by using the scroll bar. Note that the standard deviation is given by xn−1 . Where data is grouped, the scores are entered in list1 and the frequencies in list2. In this case, in Set Calculation use the drop-down arrow to select list2 as the location for the frequencies. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> Chapter 22 — Describing the distribution of a single variable 531 Exercise 22F 1 Find the mean and the median of the following data sets. Examples 9, 10 a 29 14 11 24 14 14 28 14 18 22 14 b 5 9 11 3 12 13 12 6 13 7 3 15 12 15 5 6 d 1.5 1.0 0.2 3.4 PL E c 8.3 5.6 8.2 6.5 8.2 7.0 7.9 7.1 7.8 7.5 0.7 1.3 0.7 0.9 0.2 1.1 0.2 5.8 0.1 2.7 1.7 3.2 0.5 0.6 1.2 4.6 2.0 0.5 1.7 3.1 2 Find the mean and the median of the following data sets. x 1 2 3 4 5 a Frequency 6 3 10 7 8 −2 5 x Frequency b −1 8 0 11 1 3 2 2 3 The price, in dollars, of houses sold in a particular suburb during a one-week period are given in the following list. $129 500 $135 500 $93 400 $140 000 $400 000 $186 000 SA M $187 500 $133 500 $118 000 $140 000 $168 000 $204 000 $550 000 $122 000 Find the mean and the median of the prices. Which do you think is a better measure of centre of the data set? Explain your answer. 4 Concerned with the level of absence from his classes a teacher decided to investigate the number of days each student had been absent from the classes for the year to date. These are his results. No. of days missed No. of students 0 1 2 3 4 5 6 9 21 4 2 14 10 16 18 10 2 1 Find the mean and the median number of days each student had been absent so far that year. Which is the better measure of centre in this case? Examples 11, 12 5 Find the range and the interquartile range for each of the following data sets. a 718 630 1002 b 0.7 −1.6 c 8.56 8.51 d 20 19 16 715 −1.2 0.2 8.96 18 560 8.39 16 1085 −1.0 8.62 18 750 8.51 21 20 510 3.4 3.7 8.58 8.82 17 15 1112 1093 0.8 8.54 22 19 Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> 532 Essential Advanced General Mathematics 6 The serum cholesterol levels for a sample of twenty people are: 231 190 159 192 203 209 304 161 248 206 238 224 209 276 193 196 225 189 244 199 a Find the range of the serum cholesterol levels. b Find the interquartile range of the serum cholesterol levels. 7 Twenty babies were born at a local hospital on one weekend. Their birth weights, in kg, are given in the stem-and-leaf plot below. 2 2 3 3 4 4 1 5 1 5 1 5 PL E P1: FXS/ABE 7 3 6 2 9 3 7 2 9 4 7 3 4 9 3|6 represent 3.6 kg a Find the range of the birth weights. b Find the interquartile range of the birth weights. Example 14 8 Find the standard deviation for the following data sets. a 30 16 $4.38 $5.65 23 18 $3.60 $6.89 18 $2.30 $1.98 SA M b $2.52 $4.32 22 c 200 300 950 200 200 14 56 $3.45 $4.60 300 13 $5.40 $5.12 840 26 9 $4.43 $3.79 350 31 $2.27 $4.99 200 $4.50 $3.02 200 d 86 74 75 77 79 82 81 75 78 79 80 75 78 78 81 80 76 77 82 9 For each of the following data sets Example 15 a calculate the mean and the standard deviation b determine the percentage of observations falling within two standard deviations of the mean. i 41 16 6 21 1 21 5 31 20 27 17 10 3 32 2 48 8 12 21 44 1 56 5 12 3 1 13 11 15 14 10 12 18 64 3 10 ii 141 152 Example 13 260 141 164 239 235 145 167 134 266 150 150 237 255 254 168 150 245 265 258 140 239 132 10 A group of university students was asked to write down their ages with the following results. 17 17 17 17 17 17 17 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 19 19 19 20 20 20 21 24 25 31 41 44 45 a Construct a cumulative relative frequency polygon and use it to find the median and the interquartile range of this data set. b Find the mean and standard deviation of the ages. c Find the percentage of students whose ages fall within two standard deviations of the mean. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> 533 Chapter 22 — Describing the distribution of a single variable 11 The results of a student’s chemistry experiment are as follows. 7.3 8.3 5.9 7.4 6.2 7.4 5.8 6.0 i Find the mean and the median of the results. ii Find the interquartile range and the standard deviation of the results. b Unfortunately when the student was transcribing his results into his chemistry book he made a small error, and wrote: a 8.3 5.9 7.4 6.2 7.4 5.8 60 PL E 7.3 i Find the mean and the median of these results. ii Find the interquartile range and the standard deviation of these results. c Describe the effect the error had on the summary statistics calculated in parts a and b. Example 17 12 A selection of shares traded on the stock exchange had a mean price of $50 with a standard deviation of $3. Determine an interval which would include approximately 95% of the share prices. 13 A store manager determined the store’s mean daily receipts as $550, with a standard deviation of $200. On what proportion of days were the daily receipts between $150 and $950? The boxplot SA M 22.7 Knowing the median and quartiles of a distribution means that quite a lot is known about the central region of the data set. If something is known about the tails of the distribution then a good picture of the whole data set can be obtained. This can be achieved by knowing the maximum and minimum values of the data. These five important statistics can be derived from a data set: the median, the two quartiles and the two extremes. These values are called the five-figure summary and can be used to provide a succinct pictorial representation of a data set called the box and whisker plot, or boxplot. For this visual display, a box is drawn with the ends at the first and third quartiles. Lines are drawn which join the ends of the box to the minimum and maximum observations. The median is indicated by a vertical line in the box. Example 17 Draw a boxplot to show the number of hours spent on a project by individual students in a particular school. 24 59 9 4 102 3 166 13 48 147 108 27 97 2 264 90 71 86 36 102 9 92 147 40 226 56 146 37 181 19 111 35 76 Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> 534 Essential Advanced General Mathematics Solution First arrange the data in order. 2 3 4 9 37 40 48 56 102 102 108 111 9 59 146 13 71 147 19 76 147 24 86 166 27 90 181 35 92 226 36 97 264 From this ordered list prepare the five-figure summary. median, m = 71 24 + 27 = 25.5 first quartile, Q 1 = 2 108 + 111 = 109.5 third quartile, Q 3 = 2 minimum = 2 maximum = 264 PL E P1: FXS/ABE The boxplot can then be drawn. 0 200 300 m = 71 Q3 = 109.5 max = 264 SA M min = 2 Q1 = 25.5 100 In general, to draw a boxplot: Arrange all the observations in order, according to size. Determine the minimum value, the first quartile, the median, the third quartile, and the maximum value for the data set. Draw a horizontal box with the ends at the first and third quartiles. The height of the box is not important. Join the minimum value to the lower end of the box with a horizontal line. Join the maximum value to the upper end of the box with a horizontal line. Indicate the location of the median with a vertical line. Using a graphics calculator A graphics calculator can be used to construct a boxplot. Consider the data from Example 17. Enter the data into a list named HOURS. To draw the boxplot press 2ND STAT PLOT and select and turn on Plot1, as previously described. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> Chapter 22 — Describing the distribution of a single variable 535 Press the down arrow key and select from the Type menu the boxplot icon as shown, then press ENTER . Use the LIST menu to paste HOURS as the Xlist. Your calculator screen should appear like this. PL E To bring up the boxplot, press ZOOM and then 9:ZoomStat. Your calculator screen should now look like this. To find out values for the five-figure summary, select TRACE . SA M The symmetry of a data set can be determined from a boxplot. If a data set is symmetric, then the median will be located approximately in the centre of the box, and the tails will be of similar length. This is illustrated in the following diagram, which shows the same data set displayed as a histogram and a boxplot. A median placed towards the left of the box, and/or a long tail to the right indicates a positively skewed distribution, as shown in this plot. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> 536 Essential Advanced General Mathematics PL E A median placed towards the right of the box, and/or a long tail to the left indicates a negatively skewed distribution, as illustrated here. A more sophisticated version of a boxplot can be drawn with the outliers in the data set identified. This is very informative, as one cannot tell from the previous boxplot if an extremely long tail is caused by many observations in that region or just one. Before drawing this boxplot the outliers in the data set must be identified. The term outlier is used to indicate an observation which is rather different from other observations. Sometimes it is difficult to decide whether or not an observation should be designated as an outlier. The interquartile range can be used to give a very useful definition of an outlier. SA M An outlier is any number which is more than 1.5 interquartile ranges above the upper quartile, or more than 1.5 interquartile ranges below the lower quartile. When drawing a boxplot, any observation identified as an outlier is indicated by an asterisk, and the whiskers are joined to the smallest and largest values which are not outliers. Example 18 Use the data from Example 17 to draw a boxplot with outliers. Solution median = 71 interquartile range = Q 3 − Q 1 = 109.5 − 25.5 = 84 An outlier will be any observation which is less than 25.5 − 1.5 × 84 = −100.5, which is impossible, or greater than 109.5 + 1.5 × 84 = 235.5. From the data it can be seen that there is only one observation greater than this, 264, which would be denoted with an asterisk. The upper whisker is now drawn from the edge of the box to the largest observation less than 235.5, which is 226. * 0 100 200 300 Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> Chapter 22 — Describing the distribution of a single variable 537 Using the TI-Nspire PL E The calculator can be used to construct a boxplot. Consider the data from Example 17. The data is easiest entered in a Lists & 3). Spreadsheet application ( ) to Firstly, use the up/down arrows ( name the first column hours. Then enter each of the 33 numbers as shown. Open a Data & Statistics application ( 5 ) to graph the data. At first the data displays as shown. SA M Specify the x variable by selecting Add X Variable from the Plot Properties (b 2 4) and selecting hours. The data now displays as shown. (Note: It is also possible to use the NavPad to move down below the x-axis and click to add the x variable.) Select Box Plot from the Plot Type menu (b 1 2). The data now displays as shown. Notice how the calculator, by default, shows any outlier(s). Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> 538 Essential Advanced General Mathematics PL E To not show the outlier(s), select Extend Box Plot Whiskers from the Plot Properties menu (b 2 3). The data now displays as shown. Note: It is possible to show the values of the five-point summary by moving the cursor over the boxplot. Using the Casio ClassPad In the following consider the set of marks: 28 21 21 3 22 31 35 26 27 33 36 35 23 24 43 31 30 34 48 SA M enter the data into list1. Tap SetGraph, Setting . . . and the tab for Graph 2, enter the In and tap SET. (Note that on settings shown including the tick box the Classpad you can store settings for a number of different graphs and return to them quickly.) Tap SetGraph, StatGraph2 and tap the box to tick and select the graph (de-select any other graphs). to produce the graph. The boxplot is Tap produced as shown. With the graph window selected (bold border), tap 6 to adjust the viewing window for the graph. Tap Analysis, Trace and use the navigator key to move between the outlier(s), Minimum, Q1, Median, Q3 and Maximum scores. Starting from the left of the plot, we see that the: Minimum value is 3: min X = 3. It is also an outlier Lower adjacent value is 21: X = 21 First quartile is 23: Q1 = 23 Median is 30: Med = 30 Second quartile is 35: Q3 = 35 Maximum value is 48: max X = 48. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> Chapter 22 — Describing the distribution of a single variable 539 Exercise 22G Example 17 1 The heights (in centimetres) of a class of girls are 160 154 165 159 123 149 143 167 154 176 180 163 133 154 123 167 157 168 157 132 135 145 140 143 140 157 150 156 Example 18 PL E a Determine the five-figure summary for this data set. b Draw a boxplot of the data. c Describe the pattern of heights in the class in terms of shape, centre and spread. 2 A researcher is interested in the number of books people borrow from a library. She decided to select a sample of 38 cards and record the number of books each person has borrowed in the previous year. Here are her results. 7 28 0 2 38 18 0 0 4 0 0 2 1 1 14 1 8 27 0 52 4 0 12 28 10 1 0 2 0 1 11 5 11 0 13 0 a Determine the five-figure summary for this data set. b Determine if there are any outliers. c Draw a boxplot of the data, showing any outliers. d Describe the number of books borrowed in terms of shape, centre and spread. 13 15 3 The winnings of the top 25 male tennis players in 2004 are given in the following table. Winnings 6 357 547 2 766 051 2 604 590 2 273 283 1 697 155 1 639 171 1 508 177 1 448 209 1 177 254 1 045 985 927 344 861 357 854 533 SA M Player Roger Federer Lleyton Hewitt Andy Roddick Marat Safin Guillermo Coria Gaston Gaudio Tim Henman Carlos Moya Andre Agassi David Nalbandian Jonas Bjorkman Tommy Robredo Nicolas Massu Player Joachim Johansson Jiri Novak Dominik Hrbaty Guillermo Canas Fernando Gonzalez Sebastian Grosjean Feliciano Lopez Max Mirnyi Juan Ignacio Chela Mikhail Youzhny Radek Stepanek Vincent Spadea Winnings 828 744 813 792 808 944 780 701 766 416 755 795 748 662 742 196 727 736 725 948 706 387 704 105 a Draw a boxplot of the data, indicating any outliers. b Describe the data in terms of shape, centre, spread and outliers. 4 The hourly rate of pay for a group of students engaged in part-time work was found to be: $4.75 $8.50 $17.23 $9.00 $12.00 $11.69 $6.25 $7.50 $8.89 $6.75 $7.90 $12.46 $10.80 $8.40 $12.34 $10.90 $11.65 $10.00 $10.00 $13.00 Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> 540 Essential Advanced General Mathematics a Draw a boxplot of the data, indicating any outliers. b Describe the hourly pay rate for the students in terms of shape, centre, spread and outliers. 5 The daily circulation of several newspapers in Australia is: 570 000 217 284 98 158 327 654 214 000 77 500 299 797 212 770 56 000 273 248 171 568 43 330 258 700 170 000 17 398 230 487 125 778 a Draw a boxplot of the data, indicating any outliers. b Describe the daily newspaper circulation in terms of shape, centre, spread and outliers. 22.8 PL E P1: FXS/ABE Using boxplots to compare distributions Boxplots are extremely useful for comparing two or more sets of data collected on the same variable, such as marks on the same assignment for two different groups of students. By drawing boxplots on the same axis, both the centre and spread for the distributions are readily identified and can be compared visually. Example 19 The number of hours spent by individual students on the project referred to in Example 17 at another school were: 152 106 226 80 82 14 17 54 30 18 9 16 16 173 156 106 SA M 53 57 136 24 136 102 19 6 21 86 107 38 11 227 24 3 1 48 42 55 12 21 128 45 176 Use boxplots to compare the time spent on the project by students at this school with those in Example 17. Solution The five-figure summary for this data set is: median, m = 48; first quartile, Q 1 = 17.5; third quartile, Q 3 = 106.5; minimum = 1; maximum = 227 In order to compare the time spent on the project by the students at each school, boxplots for both data sets are drawn on the same axis. School 1 * School 2 0 100 200 300 Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> Chapter 22 — Describing the distribution of a single variable 541 From the boxplots the distributions of time for the two schools can be compared in terms of shape; centre, spread and outliers. Clearly the two distributions for both schools are positively skewed, indicating a larger range of values in the upper half of the distributions. The centre for School 1 is higher than the centre for School 2 (71 hours compared to 48 hours). As can be seen by comparing the box widths, which indicate the IQR, the spread of the data is comparable for both distributions. There is one outlier, a student who attended School 1 and spent 264 hours on the project. PL E The boxplot is useful for summarising large data sets and for comparing several sets of data. It focuses attention on important features of the data and gives a picture of the data which is easy to interpret. When a single data set is being investigated a stem-and-leaf plot is sometimes better, as a boxplot may hide the local detail of the data set. Exercise 22H 19 1 To test the effect of a physical fitness course the number of sit-ups that a person could do in 1 minute, both before and after the course, were recorded. Twenty randomly selected participants scored as follows. Before 29 23 After 28 25 22 22 25 26 29 26 26 30 24 12 31 17 46 21 34 20 28 30 26 24 25 30 35 34 33 30 36 15 32 29 54 21 50 19 43 34 SA M Example a Construct boxplots of these two sets of data on the same axis. b Describe the effect of the physical fitness course on the number of sit-ups achieved in terms of shape, centre, spread and outliers. 2 The number of hours spent on homework per week by a group of students in Year 8 and a group of students in Year 12 are shown in the tables. Year 8 1 1 2 3 4 4 2 3 4 3 4 1 5 7 3 2 7 1 7 3 2 1 4 4 3 1 3 0 Year 12 1 2 2 3 3 1 5 1 6 4 7 7 7 8 6 9 7 6 8 7 7 8 5 7 4 2 1 3 Draw boxplots of these two sets of data on the same axis and use them to answer the following questions. a Which group does the most homework? b Which group varies more in the number of hours homework they do? Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> 542 Essential Advanced General Mathematics 3 The ages of mothers at the birth of their first child were noted, for the first forty such births, at a particular hospital in 1970 and again in 1990. 1970 21 37 24 16 29 22 21 21 25 26 22 25 32 31 36 26 37 26 22 34 30 27 25 27 24 19 31 18 36 21 20 39 23 33 18 24 19 17 20 21 1990 24 19 26 25 22 33 18 35 35 44 28 31 32 24 32 23 17 18 43 19 28 27 28 46 38 24 26 29 20 33 28 23 30 29 41 34 39 23 28 29 PL E P1: FXS/ABE a Construct boxplots of these two sets of data on the same axis. b Compare the ages of the mothers in 1970 and 1990 in terms of shape, centre, spread and outliers. Using a CAS calculator with statistics I How to construct a histogram Use a TI-89 graphics calculator to display the following set of marks in the form of a histogram. SA M 16 11 4 25 15 7 14 13 14 12 15 13 16 14 15 12 18 22 17 18 23 15 13 17 18 22 23 Enter the data into your calculator by pressing APPS , and moving to the Stats/List Editor. Type the data into list1. Your screen should look like the one shown. Next, set up the calculator to plot a statistical graph. a Press F2 to access the Plots menu. b Select 1:Plot Setup. This will take you to the Plot Setup menu. c With Plot 1: highlighted, press F1 . This will take you to the Define Plot1 dialogue box. d Complete the dialogue box as follows: r For Plot Type: select 4:Histogram r Leave Mark: as Box. r For x type in list1. r For Hist.Bucket Width, type in 3.5. ‘Bucket width’ means ‘interval width’. Choose a minimum of five intervals and divide the range of values by the number to get the ‘bucket width’. If we choose six 25 − 4 = 4.2, or 4 as a more convenient number. intervals, the bucket width is 5 Note: Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> Chapter 22 — Describing the distribution of a single variable 543 Pressing ENTER confirms your selection and returns you to the Plot Setup menu. Set the viewing window ( WINDOW ) with the following entries. Remember the interval width is 4. r xmin= 4 r xmax= 29 (4 greater than the highest data value) r xscl= 0 (no tick marks will appear on the scale) r ymin= –5 (to allow space below the histogram) r ymax= 13 (a first guess at the maximum height PL E P1: FXS/ABE of the histogram; half the number of data values) r yscl= 0 (leave as is) r xres= 2 (leave as is) Pressing GRAPH plots the histogram. SA M Pressing F3 places a marker at the top of the first column of the histogram and tells us that the first class interval contains all values ranging from 4 to less than 8. For this interval, the count is two (n:2). To find out the counts in the other intervals, use the to move from interval to interval. horizontal arrow key How to construct a boxplot with outliers Use a TI-89 graphics calculator to display the following set of marks in the form of a boxplot with outliers. 28 21 21 3 22 31 35 26 27 33 36 35 23 24 43 31 30 34 48 Enter the data into your calculator by pressing APPS , moving to Stats/List Editor and pressing ENTER to select. Type the data into list1. Your screen should look like the one shown. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> 544 Essential Advanced General Mathematics Set up the calculator to plot a statistical graph. Press F2 to access the Plots menu. Press ENTER to select 1:Plot Setup. Press F1 to Define Plot1. Complete the dialogue box as follows: r For Plot Type: select 5:Mod Box Plot r Leave Mark: as Box. r For x type in list1. Pressing ENTER confirms your selection and returns you to the Plot Setup menu. PL E a b c d SA M Pressing F5 (Zoom Data) in Plot Setup automatically plots the box plot in a properly scaled window. Key values can be read from the boxplot by pressing F3 . This places a marker on the boxplot. You can then use the horizontal arrow keys ( and ) to move from point to point on the boxplot and read off the associated values. Starting at the far left of the plot, we see that the r minimum value is 3: minX=3. It is also an outlier. r lower adjacent value is 21: X=21 r first quartile is 23: Q1=23. r median is 30: Med=30. r second quartile is 35: Q1=35. r maximum value is 48: maxX=48. See the figure opposite. How to calculate the mean and standard deviation The following are all heights (in cm) of a group of women: 176 160 163 157 168 172 173 169 Calculate the mean and standard deviation. Enter the data into your calculator using the Stats/List Editor. Type the data into list1. Your screen should look like the one shown. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c22.xml CUAU033-EVANS September 12, 2008 9:40 Back to Menu >>> Chapter 22 — Describing the distribution of a single variable 545 Press F4 to access the Calculate menu. With 1:1-Var Stats highlighted, press ENTER to select. This will take you to the 1-Var Stats dialogue box. SA M PL E Complete the dialogue box: r For List:, type in list1. This is not necessary if list1 is already shown. Press ENTER to obtain the results. Write down your answers to the required degree of accuracy (two decimal places). Note: The value of the standard deviation is given by Sx. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c22-1.xml CUAU033-EVANS September 12, 2008 10:56 Back to Menu >>> Essential Advanced General Mathematics Chapter summary PL E Variables may be classified as categorical or numerical. Numerical data may be discrete or continuous. Examination of a data set should always begin with a visual display. A bar chart is the appropriate visual display for categorical data. When a data set is small, a stem-and-leaf plot is the most appropriate visual display for numerical data. When a data set is larger, a histogram, frequency polygon or boxplot is a more appropriate visual display for numerical data. Cumulative frequency distributions and cumulative relative frequency distributions are useful for answering questions about the number or proportion of data values greater than or less than a particular value. These are graphically represented in cumulative frequency polygons or cumulative relative frequency polygons. From a stem-and-leaf plot, histogram or boxplot, insight can be gained into the shape, centre and spread of the distribution, and whether or not there are any outliers. An outlier is a value which sits away from the main body of the data in a plot. It is formally defined as a value more than 1.5IQR below Q1 , or more than 1.5IQR above Q3 . For numerical data it is also very useful to calculate some summary statistics. n 1 The mean is defined as x̄ = xi . n i=1 n + 1 th If n, the number of observations, is odd, then the median is the observation 2 from the end of the ordered list. If n is even, then the median is found by averaging the two n th n th middle observations in the list, i.e., the and the + 1 observations are added 2 2 together and divided by 2. The mode is the most common observation in a group of data. The most useful measures of centre are the median and the mean. To find the interquartile range of a distribution: r Arrange all observations in order according to size. r Divide the observations into two equal sized groups. If n, the number of observations, is odd, then the median is omitted from both groups. r Locate Q , the first quartile, which is the median of the lower half of the 1 observations, and Q3 , the third quartile, which is the median of the upper half of the observations. r The interquartile range IQR is defined as the difference between the quartiles. That is IQR = Q 3 − Q 1 n 1 The standard deviation is defined as s = (xi − x̄)2 . n − 1 i=1 The most useful measures of spread are the interquartile range and the standard deviation. SA M Review 546 Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P2: FXS 9780521740494c22-1.xml CUAU033-EVANS September 12, 2008 10:56 Back to Menu >>> Chapter 22 — Describing the distribution of a single variable 547 min median Q1 max Q3 When the data set is symmetric any of the summary statistics are appropriate. When the data set is not symmetric or when there are outliers the median and the interquartile range are the preferred summary statistics. In general, 95% of the values of the data set will fall within two standard deviations of the mean. When comparing the distribution of two or more data sets the comparison should be made in terms of the shape, centre, spread and outliers for each distribution. Multiple-choice questions SA M 1 In a survey a number of subjects were asked to indicate how much they exercise by selecting one of the following options. 1 Never 2 Seldom 3 Occasionally 4 Regularly The resulting variable was named Level of Exercise, and the level of measurement of this variable is A variable B numerical C constant D categorical E metric Questions 2, 3 and 4 relate to the following information. The numbers of hours worked per week by employees in a large company are shown in this percentage frequency histogram. 40 Percentage Frequency 30 20 10 0 20 40 60 Hours worked weekly 80 Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard Review The five-figure summary of a set of data consists of the minimum, Q 1 , median, Q 3 , and the maximum. A boxplot is a diagrammatic representation of this, e.g. PL E P1: FXS/ABE P1: FXS/ABE P2: FXS 9780521740494c22-1.xml CUAU033-EVANS September 12, 2008 10:56 Back to Menu >>> Essential Advanced General Mathematics 2 The percentage of employees who work from 20 to less than 30 hours per week is closest to A 1% B 2% C 6% D 10% E 33% 3 The median number of hours worked is in the interval A 10 to less than 20 B 20 to less than 30 C 30 to less than 40 D 40 to less than 50 E 50 to less than 60 0 1 2 3 4 1 0 2 1 2 3 2 4 4 PL E Questions 4 and 5 relate to the following information. A group of 19 employees of a company was asked to record the number of meetings that they attended in the last month. Their responses are summarised in the following stem-and-leaf plot. 3 4 4 6 5 5 4 The median number of meetings is A 6 B 6.5 C 7 D 7.5 6 6 7 9 E 9 6 The cumulative frequency polygon shown gives the examination scores in Mathematics for a group of 200 students. The number of students who scored less than 70 on the examination is closest to A 30 B 100 C 150 D 175 E 200 Number of students 5 The interquartile range (IQR) of number of meetings is A 0 B 4 C 9.5 D 10 E 14 SA M Review 548 200 100 0 40 50 60 70 80 Exam Score 90 Questions 7 and 8 relate to the following information. The number of years that a sample of people has lived that their current address is summarised in this boxplot. Years lived this address 0 10 20 30 40 50 7 The shape of the distribution of years lived at this address is: A positively skewed B negatively skewed C bimodal D symmetric E symmetric with outliers 8 The interquartile range years lived at this address is approximately equal to: A 5 B 8 C 17 D 12 E 50 Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P2: FXS 9780521740494c22-1.xml CUAU033-EVANS September 12, 2008 10:56 Back to Menu >>> Chapter 22 — Describing the distribution of a single variable 549 The amount paid per week to the employees of each of five large companies are shown in the boxplots: Company 2 Company 3 0 PL E Company 1 20000 40000 60 000 80 000 Yearly income 100 000 120 000 9 The company with the lowest typical wage is A Company 1 B Company 2 C Company 3 D Company 1 and Company 2 E Company 2 and Company 3 SA M 10 The company with the largest variation in wage is A Company 1 B Company 2 C Company 3 D Company 1 and Company 2 E Company 2 and Company 3 Short-answer questions (technology-free) 1 Classify the data which arise from the following situations as categorical or numerical. a The number of phones calls a hotel receptionist receives each day. b Interest in politics on a scale from 1 to 5 where 1 = very interested, 2 = quite interested, 3 = somewhat interested, 4 = not very interested, and 5 = uninterested. 2 This bar chart shows the percentage of people working who are employed in private companies, work for the Government or are self-employed in a certain town. a What kind of measurement is the ‘Type of company worked for’? b Approximately what percentage of the people are self-employed? 50 40 30 20 10 0 Private Government Self-employed Type of company worked for Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard Review Questions 9 and 10 relate to the following data. Percent P1: FXS/ABE P1: FXS/ABE P2: FXS 9780521740494c22-1.xml CUAU033-EVANS September 12, 2008 10:56 Back to Menu >>> Essential Advanced General Mathematics 3 A researcher asked a group of people to record how many cigarettes they had smoked on a particular day. Here are her results: 0 5 0 0 9 17 10 14 23 3 25 6 0 0 0 33 34 23 32 0 0 32 0 13 30 21 0 22 4 6 Using an appropriate class interval, construct a histogram of these data. 56 54 57 52 47 69 PL E 4 A teacher recorded the time taken (in minutes) by each of a class of students to complete a test. 68 72 52 65 51 45 43 44 22 55 59 56 51 49 39 50 a Make a stem-and-leaf plot of these times, using one row per stem. b Use this stem-and-leaf-plot to find the median and quartiles for the time taken. 5 The weekly rentals, in dollars, for apartments in a particular suburb are given in the following table. 285 265 185 300 210 210 215 270 320 190 680 245 280 315 Find the mean and the median of the weekly rental. 6 Geoff decided to record the time it takes him to complete his mail delivery round each working day for four weeks. His data are recorded in the following table. SA M Review 550 170 164 182 189 176 167 201 161 188 183 187 211 168 180 174 182 201 193 161 147 185 166 188 183 167 186 173 176 The mean of the time taken, x̄, is 179 and the standard deviation, s, is 14. a Determine the percentage of observations falling within two standard deviations of the mean. b Is this what you would expect to find? 7 A group of students were asked to record the number of SMS messages that they sent in one 24-hour period, and the following five-figure summary was obtained from the data set. Use it to construct a simple boxplot of these data. Min = 0, Q 1 = 3, Median = 5, Q 3 = 12, Max = 24 8 The following table gives the number of students absent each day from a large secondary college on each of 36 randomly chosen school days. 7 7 15 22 3 16 12 21 13 15 30 21 21 13 10 16 2 16 23 7 11 23 12 4 17 18 3 23 14 0 8 14 31 16 0 44 Construct a boxplot of these data, with outliers. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P2: FXS 9780521740494c22-1.xml CUAU033-EVANS September 12, 2008 10:56 Back to Menu >>> Chapter 22 — Describing the distribution of a single variable 551 1 The divorce rates (in percentages) of 19 countries are 27 26 18 8 14 14 25 5 28 15 6 32 32 6 44 19 53 9 0 What is the level of measurement of the variable, ‘divorce rate’? Construct an ordered stem-and-leaf plot of divorce rates, with one row per stem. What shape is the divorce rates? What percentage of countries have divorce rates greater than 30? Calculate the mean and median of the divorce rates for the 19 countries. Construct a histogram of the data with class intervals of width 10. i What is the shape of the histogram? ii How many countries had divorce rates from 10% to less than 20%? g Construct a cumulative percentage frequency polygon of divorce rates. i What percentage of countries has divorce rates less than 20%? ii Use the cumulative frequency distribution to estimate the median percentage divorce rate. a b c d e f SA M 2 Hillside Trains have decided to improve their service on the Lilydale line. Trains were timed on the run from Lilydale to Flinders Street, and their times recorded over a period of six weeks at the same time each day. The time taken for each journey is shown below. 60 90 63 58 61 59 67 64 70 86 74 69 72 70 78 59 68 77 65 62 80 64 68 63 76 57 82 89 65 65 89 74 69 60 75 60 79 68 62 82 60 64 a Construct a histogram of the times taken for the journey from Lilydale to Flinders Street, using class intervals 55–59, 60–64, 65–69 etc. i On how many days did the trip take from 65–69 minutes? ii What shape is the histogram? iii What percentage of trains took less than 65 minutes to reach Flinders Street? b Calculate the following summary statistics for the time taken (correct to two decimal places). x s Min Q1 M Q3 Max c Use the summary statistics to complete the following report. i The mean time taken from Lilydale to Flinders Street (in minutes) was . . . ii 50% of the trains took more than . . . minutes to travel from Lilydale to Flinders Street. iii The range of travelling times was . . . minutes while the interquartile range was . . ... minutes. (cont’d) Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard Review Extended-response questions PL E P1: FXS/ABE P1: FXS/ABE P2: FXS 9780521740494c22-1.xml CUAU033-EVANS September 12, 2008 10:56 Back to Menu >>> Essential Advanced General Mathematics iv 25% of trains took more than . . . minutes to travel to Flinders Street. v The standard deviation of travelling times was . . . vi Approximately 95% of trains took between . . . and . . . minutes to travel to Flinders St. d Summary statistics for the year before Hillside Trains took over the Lilydale line from the Met are indicated below: Q 1 = 65 Median = 70 Q 3 = 89 Max = 99 PL E Min = 55 Draw simple boxplots for the last year the Met ran the line and the data from Hillside trains on the same axis. e Use the information from the boxplots to compare travelling times for the two transport corporations in terms of shape, centre and spread. 3 In a small company, upper management wants to know if there is a difference in the three methods used to train its machine operators. One method uses a hands-on approach. A second method uses a combination of classroom instruction and on-the-job training. The third method is based completely on classroom training. Fifteen trainees are assigned to each training technique. The following data are the results of a test undertaken by the machine operators after completion of one of the different training methods. Method 1 98 100 89 90 81 85 97 95 87 70 69 75 91 92 93 Method 2 79 62 61 89 69 99 87 62 65 88 98 79 73 96 83 Method 3 70 74 60 72 65 49 71 75 55 65 70 59 77 67 80 SA M Review 552 a Draw boxplots of the data sets, on the same axis. b Write a paragraph comparing the three training methods in terms of shape, centre, spread and outliers. c Which training method would you recommend? Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P2: FXS 9780521740494c22-1.xml CUAU033-EVANS September 12, 2008 10:56 Back to Menu >>> 553 Chapter 22 — Describing the distribution of a single variable Family First-born Second-born Third-born 1 38 9 12 2 45 40 12 3 30 24 12 4 29 16 25 5 34 16 9 6 19 21 11 7 35 34 20 8 40 29 12 9 25 22 10 10 50 29 20 11 44 20 16 12 36 19 13 13 26 18 10 SA M a Draw boxplots of the data sets on the same axis. b Write a paragraph comparing the independence scores of first-, secondand third-born children. Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard Review 4 It has been argued that there is a relationship between a child’s level of independence and the order in which they were born in the family. Suppose that the children in thirteen three-children families are rated on a 50-point scale of independence. This is done when all children are adults, thus eliminating age effects. The results are as follows. PL E P1: FXS/ABE