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+ Chapter 1 Exploring Data Introduction: Data Analysis: Making Sense of Data 1.1 Analyzing Categorical Data 1.2 Displaying Quantitative Data with Graphs 1.3 Describing Quantitative Data with Numbers + Introduction Data Analysis: Making Sense of Data Learning Objectives After this section, you should be able to… DEFINE “Individuals” and “Variables” DISTINGUISH between “Categorical” and “Quantitative” variables DEFINE “Distribution” DESCRIBE the idea behind “Inference” Data Analysis is the process of organizing, displaying, summarizing, and asking questions about data. Definitions: Individuals – objects (people, animals, things) described by a set of data Variable - any characteristic of an individual Categorical Variable – places an individual into one of several groups or categories. Quantitative Variable – takes numerical values for which it makes sense to find an average. + is the science of data. Data Analysis Statistics + Read Example : Census at school ( P-3) Now do Exercise 3 on P- 7 Answers: a) Individuals: AP Stat students who completed a questionnaire on the first day of class. b) Categorical: gender, handedness, favorite type of music Quantitative: height( inches), amount of time the students expects to spend on HW (mins) , the total value of coin(cents) c) Female, right-handed, 58 inches tall, spends 60 mins on HW, prefers alternative music, had 76 cents in her pocket. + Do Activity : Hiring Discrimination – It just won’t fly …… Follow the directions on Page 5 Perform 5 repetitions of your simulation. Turn in your results to your teacher. (Make Groups of 4) generally takes on many different values. In data analysis, we are interested in how often a variable takes on each value. Definition: Distribution – tells us what values a variable takes and how often it takes those values Example 2009 Fuel Economy Guide MODEL 2009 Fuel Economy Guide 2009 Fuel Economy Guide MPG MPG MODEL <new>MODEL MPG 1 Acura RL 922 Dodge Avenger 1630 Mercedes-Benz E350 24 2 Audi A6 Quattro 1023 Hyundai Elantra 1733 Mercury Milan 29 3 Bentley Arnage 1114 Jaguar XF 1825 Mitsubishi Galant 27 4 BMW 5281 1228 Kia Optima 1932 Nissan Maxima 26 5 Buick Lacrosse 1328 Lexus GS 350 2026 Rolls Royce Phantom 18 6 Cadillac CTS 1425 Lincolon MKZ 2128 Saturn Aura 33 7 Chevrolet Malibu 1533 Mazda 6 2229 Toyota Camry 31 8 Chrysler Sebring 1630 Mercedes-Benz E350 2324 Volkswagen Passat 29 9 Dodge Avenger 1730 Mercury Milan 2429 Volvo S80 Variable of Interest: MPG 25 <new> Dotplot of MPG Distribution + A variable 2009 Fuel Economy Guide Examine each variable by itself. Then study relationships among the variables. MODEL + 2009 Fuel Economy Guide 2009 Fuel Economy Guide MPG MPG MODEL <new>MODEL MPG 1 Acura RL 9 22 Dodge Avenger 1630 Mercedes-Benz E350 24 2 Audi A6 Quattro 1023 Hyundai Elantra 1733 Mercury Milan 29 3 Bentley Arnage 1114 Jaguar XF 1825 Mitsubishi Galant 27 4 BMW 5281 1228 Kia Optima 1932 Nissan Maxima 26 5 Buick Lacrosse 1328 Lexus GS 350 2026 Rolls Royce Phantom 18 6 Cadillac CTS 1425 Lincolon MKZ 2128 Saturn Aura 33 7 Chevrolet Malibu 1533 Mazda 6 2229 Toyota Camry 31 8 Chrysler Sebring 1630 Mercedes-Benz E350 2324 Volkswagen Passat 29 9 Dodge Avenger 1730 Mercury Milan 2429 Volvo S80 25 Start with a graph or graphs Add numerical summaries Data Analysis How to Explore Data <new> Population Sample + Data Analysis From Data Analysis to Inference Collect data from a representative Sample... Make an Inference about the Population. Perform Data Analysis, keeping probability in mind… + Inference: Inference is the process of making a conclusion about a population based on a sample set of data. + Introduction Data Analysis: Making Sense of Data Summary In this section, we learned that… A dataset contains information on individuals. For each individual, data give values for one or more variables. Variables can be categorical or quantitative. The distribution of a variable describes what values it takes and how often it takes them. Inference is the process of making a conclusion about a population based on a sample set of data. + Do Notes worksheet : First page + Looking Ahead… In the next Section… We’ll learn how to analyze categorical data. Bar Graphs Pie Charts Two-Way Tables Conditional Distributions We’ll also learn how to organize a statistical problem. + Section 1.1 Analyzing Categorical Data Learning Objectives After this section, you should be able to… CONSTRUCT and INTERPRET bar graphs and pie charts RECOGNIZE “good” and “bad” graphs CONSTRUCT and INTERPRET two-way tables DESCRIBE relationships between two categorical variables ORGANIZE statistical problems The values of a categorical variable are labels for the different categories The distribution of a categorical variable lists the count or percent of individuals who fall into each category. Example, page 8 Frequency Table Format Variable Values Relative Frequency Table Count of Stations Format Percent of Stations Adult Contemporary 1556 Adult Contemporary Adult Standards 1196 Adult Standards 8.6 Contemporary Hit 4.1 Contemporary Hit 569 11.2 Country 2066 Country 14.9 News/Talk 2179 News/Talk 15.7 Oldies 1060 Oldies Religious 2014 Religious Rock 869 Spanish Language 750 Other Formats Total 1579 13838 7.7 14.6 Rock 6.3 Count Spanish Language 5.4 Other Formats 11.4 Total 99.9 Percent Analyzing Categorical Data Variables place individuals into one of several groups or categories + Categorical + categorical data Frequency tables can be difficult to read. Sometimes is is easier to analyze a distribution by displaying it with a bar graph or pie chart. Frequency Table Format Relative Frequency Table Count of Stations Format Percent of Stations Adult Contemporary 1556 Adult Contemporary Adult Standards 1196 Adult Standards 8.6 Contemporary Hit 4.1 Contemporary Hit 569 11.2 Country 2066 Country 14.9 News/Talk 2179 News/Talk 15.7 Oldies 1060 Oldies Religious 2014 Religious 7.7 14.6 Rock 869 Rock 6.3 Spanish Language 750 Spanish Language 5.4 Other Formats Total 1579 13838 Other Formats 11.4 Total 99.9 Analyzing Categorical Data Displaying Good and Bad Bar graphs compare several quantities by comparing the heights of bars that represent those quantities. Our eyes react to the area of the bars as well as height. Be sure to make your bars equally wide. Avoid the temptation to replace the bars with pictures for greater appeal…this can be misleading! Alternate Example This ad for DIRECTV has multiple problems. How many can you point out? + Graphs: + Example: Bar graph What personal media do you own? Here are the percents of 15-18 year olds who own the following personal media devices. Device % who won Cell Phone 85 MP3 player 83 Handheld Video game player 41 Laptop 38 Portable CD/tape player 20 a) Make a well-labeled bar graph. What do you see? b) Would it be appropriate to make a pie chart for these data? Why or why not? When a dataset involves two categorical variables, we begin by examining the counts or percents in various categories for one of the variables. Definition: Two-way Table – describes two categorical variables, organizing counts according to a row variable and a column variable. Example, p. 12 Young adults by gender and chance of getting rich Female Male Total Almost no chance 96 98 194 Some chance, but probably not 426 286 712 A 50-50 chance 696 720 1416 A good chance 663 758 1421 Almost certain 486 597 1083 Total 2367 2459 4826 What are the variables described by this twoway table? How many young adults were surveyed? + Tables and Marginal Distributions Analyzing Categorical Data Two-Way Definition: The Marginal Distribution of one of the categorical variables in a two-way table of counts is the distribution of values of that variable among all individuals described by the table. Note: Percents are often more informative than counts, especially when comparing groups of different sizes. To examine a marginal distribution, 1)Use the data in the table to calculate the marginal distribution (in percents) of the row or column totals. 2)Make a graph to display the marginal distribution. + Tables and Marginal Distributions Analyzing Categorical Data Two-Way + Two-Way Tables and Marginal Distributions Example, p. 13 Young adults by gender and chance of getting rich Female Male Total Almost no chance 96 98 194 Some chance, but probably not 426 286 712 A 50-50 chance 696 720 1416 A good chance 663 758 1421 Almost certain 486 597 1083 Total 2367 2459 4826 Almost no chance Some chance Chance of being wealthy by age 30 Percent 194/4826 = 4.0% 712/4826 = 14.8% A 50-50 chance 1416/4826 = 29.3% A good chance 1421/4826 = 29.4% Almost certain 1083/4826 = 22.4% Percent Response Examine the marginal distribution of chance of getting rich. 35 30 25 20 15 10 5 0 Almost none Some 50-50 Good chance chance chance Survey Response Almost certain Between Categorical Variables Marginal distributions tell us nothing about the relationship between two variables. Definition: A Conditional Distribution of a variable describes the values of that variable among individuals who have a specific value of another variable. To examine or compare conditional distributions, 1)Select the row(s) or column(s) of interest. 2)Use the data in the table to calculate the conditional distribution (in percents) of the row(s) or column(s). 3)Make a graph to display the conditional distribution. • Use a side-by-side bar graph or segmented bar graph to compare distributions. + Relationships Tables and Conditional Distributions Chance of being wealthy by age 30 + Two-Way Example, p. 15 100% Female Male Total Almost no chance 96 98 194 Some chance, but probably not 426 286 712 A 50-50 chance 696 720 1416 A good chance 663 758 1421 Almost certain 486 597 1083 Total 2367 2459 4826 Female Almost no chance 98/2459 = 4.0% 96/2367 = 4.1% 286/2459 = 11.6% 426/2367 = 18.0% A 50-50 chance 720/2459 = 29.3% 696/2367 = 29.4% A good chance 758/2459 = 30.8% 663/2367 = 28.0% 597/2459 = 24.3% 486/2367 = 20.5% Some chance Almost certain Almost certain Good chance 50-50 chance Some chance Almost no chance Chance of being wealthy by age 30 Chance of being wealthy by age 30 35 30 25 20 15 10 5 0 Percent Male 10% 0% Males Females Opinion Percent Response Calculate the conditional 90% 80% distribution of opinion 70% among males. 60% Examine the relationship 50% 40% between gender and 30% opinion. 20% Percent Young adults by gender and chance of getting rich 35 30 25 20 15 10 5 0 Males Males Females no Some Almost noAlmost Some 50-50 50-50 Good Good AlmostAlmost chance chance chance chance chance chance chancechance certaincertain Opinion Opinion As you learn more about statistics, you will be asked to solve more complex problems. Here is a four-step process you can follow. How to Organize a Statistical Problem: A Four-Step Process State: What’s the question that you’re trying to answer? Plan: How will you go about answering the question? What statistical techniques does this problem call for? Do: Make graphs and carry out needed calculations. Conclude: Give your practical conclusion in the setting of the real-world problem. + a Statistical Problem Analyzing Categorical Data Organizing + Do : P- 25 # 26. State: Do the data support the idea that people who get angry easily tend to have more heart disease? Plan: We suspect that people with different anger levels will have different rates of CHD, so we will compare the conditional distributions of CHD for each anger level. Do: We will display the conditional distributions in a table to compare the rate of CHD occurrence for each of the 3 anger levels Conclude: The conditional distributions show that while CHD occurrence is quite small overall, the percent of the population with CHD does increase as eth anger level increases. + Section 1.1 Analyzing Categorical Data Summary In this section, we learned that… The distribution of a categorical variable lists the categories and gives the count or percent of individuals that fall into each category. Pie charts and bar graphs display the distribution of a categorical variable. A two-way table of counts organizes data about two categorical variables. The row-totals and column-totals in a two-way table give the marginal distributions of the two individual variables. There are two sets of conditional distributions for a two-way table. + Section 1.1 Analyzing Categorical Data Summary, continued In this section, we learned that… We can use a side-by-side bar graph or a segmented bar graph to display conditional distributions. To describe the association between the row and column variables, compare an appropriate set of conditional distributions. Even a strong association between two categorical variables can be influenced by other variables lurking in the background. You can organize many problems using the four steps state, plan, do, and conclude. + Looking Ahead… In the next Section… We’ll learn how to display quantitative data. Dotplots Stemplots Histograms We’ll also learn how to describe and compare distributions of quantitative data. + Section 1.2 Displaying Quantitative Data with Graphs Learning Objectives After this section, you should be able to… CONSTRUCT and INTERPRET dotplots, stemplots, and histograms DESCRIBE the shape of a distribution COMPARE distributions USE histograms wisely One of the simplest graphs to construct and interpret is a dotplot. Each data value is shown as a dot above its location on a number line. How to Make a Dotplot 1)Draw a horizontal axis (a number line) and label it with the variable name. 2)Scale the axis from the minimum to the maximum value. 3)Mark a dot above the location on the horizontal axis corresponding to each data value. Number of Goals Scored Per Game by the 2004 US Women’s Soccer Team 3 0 2 7 8 2 4 3 5 1 1 4 5 3 1 1 3 3 3 2 1 2 2 2 4 3 5 6 1 5 5 1 1 5 Displaying Quantitative Data + Dotplots The purpose of a graph is to help us understand the data. After you make a graph, always ask, “What do I see?” How to Examine the Distribution of a Quantitative Variable In any graph, look for the overall pattern and for striking departures from that pattern. Describe the overall pattern of a distribution by its: •Shape Don’t forget your •Center SOCS! •Spread •Outliers Note individual values that fall outside the overall pattern. These departures are called outliers. + Examining the Distribution of a Quantitative Variable Displaying Quantitative Data + this data Example, page 28 The table and dotplot below displays the Environmental Protection Agency’s estimates of highway gas mileage in miles per gallon (MPG) for a sample of 24 model year 2009 midsize cars. 2009 Fuel Economy Guide MODEL 2009 Fuel Economy Guide 2009 Fuel Economy Guide MPG MPG MODEL <new>MODEL MPG 1 Acura RL 922 Dodge Avenger 1630 Mercedes-Benz E350 24 2 Audi A6 Quattro 1023 Hyundai Elantra 1733 Mercury Milan 29 3 Bentley Arnage 1114 Jaguar XF 1825 Mitsubishi Galant 27 4 BMW 5281 1228 Kia Optima 1932 Nissan Maxima 26 5 Buick Lacrosse 1328 Lexus GS 350 2026 Rolls Royce Phantom 18 6 Cadillac CTS 1425 Lincolon MKZ 2128 Saturn Aura 33 7 Chevrolet Malibu 1533 Mazda 6 2229 Toyota Camry 31 8 Chrysler Sebring 1630 Mercedes-Benz E350 2324 Volksw agen Passat 29 9 Dodge Avenger 1730 Mercury Milan 2429 Volvo S80 25 <new> Describe the shape, center, and spread of the distribution. Are there any outliers? Displaying Quantitative Data Examine When you describe a distribution’s shape, concentrate on the main features. Look for rough symmetry or clear skewness. Definitions: A distribution is roughly symmetric if the right and left sides of the graph are approximately mirror images of each other. A distribution is skewed to the right (right-skewed) if the right side of the graph (containing the half of the observations with larger values) is much longer than the left side. It is skewed to the left (left-skewed) if the left side of the graph is much longer than the right side. Symmetric Skewed-left Skewed-right Displaying Quantitative Data Shape + Describing + You Do: Smart Phone Battery Life Here is the estimated battery life for each of 9 different smart phones (in minutes) according to http://cellphones.toptenreviews.com/smartphones/. Smart Phone Battery Life (minutes) Apple iPhone 300 Motorola Droid 385 Palm Pre 300 Blackberry Bold Blackberry Storm Motorola Cliq Samsung Moment Blackberry Tour HTC Droid 360 330 360 330 300 460 Make a dot plot. * Then describe the shape, center, and spread. of the distribution. Are there any outliers? + Solution: Collection 1 300 300 300 300 Dot Plot 340 380 420 460 BatteryLife (minutes) 330 330 360 360 385 460 Shape: There is a peak at 300 and the distribution has a long tail to the right (skewed to the right). Center: The middle value is 330 minutes. Spread: The range is 460 – 300 = 160 minutes. Outliers: There is one phone with an unusually long battery life, the HTC Droid at 460 minutes. Distributions Some of the most interesting statistics questions involve comparing two or more groups. Always discuss shape, center, spread, and possible outliers whenever you compare distributions of a quantitative variable. U.K Place South Africa Example, page 32 Compare the distributions of household size for these two countries. Don’t forget your SOCS! ( Compare each characteristic) + Comparing Another simple graphical display for small data sets is a stemplot. Stemplots give us a quick picture of the distribution while including the actual numerical values. How to Make a Stemplot 1)Separate each observation into a stem (all but the final digit) and a leaf (the final digit). 2)Write all possible stems from the smallest to the largest in a vertical column and draw a vertical line to the right of the column. 3)Write each leaf in the row to the right of its stem. 4)Arrange the leaves in increasing order out from the stem. 5)Provide a key that explains in context what the stems and leaves represent. Displaying Quantitative Data (Stem-and-Leaf Plots) + Stemplots These data represent the responses of 20 female AP Statistics students to the question, “How many pairs of shoes do you have?” Construct a stemplot. 50 26 26 31 57 19 24 22 23 38 13 50 13 34 23 30 49 13 15 51 1 1 93335 1 33359 2 2 664233 2 233466 3 3 1840 3 0148 4 4 9 4 9 5 5 0701 5 0017 Stems Add leaves Order leaves Key: 4|9 represents a female student who reported having 49 pairs of shoes. Add a key Displaying Quantitative Data (Stem-and-Leaf Plots) + Stemplots Stems and Back-to-Back Stemplots When data values are “bunched up”, we can get a better picture of the distribution by splitting stems. Two distributions of the same quantitative variable can be compared using a back-to-back stemplot with common stems. Females + Splitting Males 50 26 26 31 57 19 24 22 23 38 14 7 6 5 12 38 8 7 10 10 13 50 13 34 23 30 49 13 15 51 10 11 4 5 22 7 5 10 35 7 0 0 1 1 2 2 3 3 4 4 5 5 Females “split stems” 333 95 4332 66 410 8 9 100 7 Males 0 0 1 1 2 2 3 3 4 4 5 5 4 555677778 0000124 2 58 Key: 4|9 represents a student who reported having 49 pairs of shoes. + From both sides, compare and write about the center , spread , shape and possible outliers. Do CYU : P- 34 . Check Your Understanding, page 34: 1. In general, it appears that females have more pairs of shoes than males. The median report for the males was 9 pairs while the female median was 26 pairs. The females also have a larger range of 5713=44 in comparison to the range of 38-4=34 for the males . Finally, both males and females have distributions that are skewed to the right, though the distribution for the males is more heavily skewed as evidenced by the three likely outliers at 22, 35 and 38. The females do not have any likely outliers. 2. b 3. b 4. b Quantitative variables often take many values. A graph of the distribution may be clearer if nearby values are grouped together. The most common graph of the distribution of one quantitative variable is a histogram. How to Make a Histogram 1)Divide the range of data into classes of equal width. 2)Find the count (frequency) or percent (relative frequency) of individuals in each class. 3)Label and scale your axes and draw the histogram. The height of the bar equals its frequency. Adjacent bars should touch, unless a class contains no individuals. + Histograms a Histogram The table on page 35 presents data on the percent of residents from each state who were born outside of the U.S. Class Count 0 to <5 20 5 to <10 13 10 to <15 9 15 to <20 5 20 to <25 2 25 to <30 1 Total 50 Number of States Frequency Table Percent of foreign-born residents Displaying Quantitative Data Making + Example, page 35 Histograms Wisely Here are several cautions based on common mistakes students make when using histograms. Cautions 1)Don’t confuse histograms and bar graphs. 2)Use percents instead of counts on the vertical axis when comparing distributions with different numbers of observations. 3)Just because a graph looks nice, it’s not necessarily a meaningful display of data. ( P- 40: Read the example) + Using + Try Check your understanding: P – 39 Many people……….. (Also put these #s in the calculator.) Solution: 2. The distribution is roughly symmetric and bell-shaped. The median IQ appears to be between 110 and 120 and the IQ’s vary from 80 to 150. There do not appear to be any outliers. + Section 1.2 Displaying Quantitative Data with Graphs Summary In this section, we learned that… You can use a dotplot, stemplot, or histogram to show the distribution of a quantitative variable. When examining any graph, look for an overall pattern and for notable departures from that pattern. Describe the shape, center, spread, and any outliers. Don’t forget your SOCS! Some distributions have simple shapes, such as symmetric or skewed. The number of modes (major peaks) is another aspect of overall shape. When comparing distributions, be sure to discuss shape, center, spread, and possible outliers. Histograms are for quantitative data, bar graphs are for categorical data. Use relative frequency histograms when comparing data sets of different sizes. + Do # 56 P- 46 + Looking Ahead… In the next Section… We’ll learn how to describe quantitative data with numbers. Mean and Standard Deviation Median and Interquartile Range Five-number Summary and Boxplots Identifying Outliers We’ll also learn how to calculate numerical summaries with technology and how to choose appropriate measures of center and spread. + Section 1.3 Describing Quantitative Data with Numbers Learning Objectives After this section, you should be able to… MEASURE center with the mean and median MEASURE spread with standard deviation and interquartile range IDENTIFY outliers CONSTRUCT a boxplot using the five-number summary CALCULATE numerical summaries with technology The most common measure of center is the ordinary arithmetic average, or mean. Definition: To find the mean x (pronounced “x-bar”) of a set of observations, add their values and divide by the number of observations. If the n observations are x1, x2, x3, …, xn, their mean is: sum of observations x1 x 2 ... x n x n n In mathematics, the capital Greek letter Σis short for “add them all up.” Therefore, the formula for the mean can be written in more compact notation: x x n i Describing Quantitative Data Center: The Mean + Measuring Another common measure of center is the median. In section 1.2, we learned that the median describes the midpoint of a distribution. Definition: The median M is the midpoint of a distribution, the number such that half of the observations are smaller and the other half are larger. To find the median of a distribution: 1)Arrange all observations from smallest to largest. 2)If the number of observations n is odd, the median M is the center observation in the ordered list. 3)If the number of observations n is even, the median M is the average of the two center observations in the ordered list. Describing Quantitative Data Center: The Median + Measuring Use the data below to calculate the mean and median of the commuting times (in minutes) of 20 randomly selected New York workers. Example, page 53 10 30 5 25 40 20 10 15 30 20 15 20 85 15 65 15 60 60 40 45 10 30 5 25 ... 40 45 x 31.25 minutes 20 0 1 2 3 4 5 6 7 8 5 005555 0005 Key: 4|5 00 represents a 005 005 5 New York worker who reported a 45minute travel time to work. 20 25 M 22.5 minutes 2 Describing Quantitative Data Center + Measuring The mean and median measure center in different ways, and both are useful. Don’t confuse the “average” value of a variable (the mean) with its “typical” value, which we might describe by the median. Comparing the Mean and the Median The mean and median of a roughly symmetric distribution are close together. If the distribution is exactly symmetric, the mean and median are exactly the same. In a skewed distribution, the mean is usually farther out in the long tail than is the median. + Comparing the Mean and the Median Describing Quantitative Data A measure of center alone can be misleading. A useful numerical description of a distribution requires both a measure of center and a measure of spread. How to Calculate the Quartiles and the Interquartile Range To calculate the quartiles: 1)Arrange the observations in increasing order and locate the median M. 2)The first quartile Q1 is the median of the observations located to the left of the median in the ordered list. 3)The third quartile Q3 is the median of the observations located to the right of the median in the ordered list. The interquartile range (IQR) is defined as: IQR = Q3 – Q1 Describing Quantitative Data Spread: The Interquartile Range (IQR) + Measuring + Do CYU P- 55 and Interpret the IQR + Find Travel times to work for 20 randomly selected New Yorkers 10 30 5 25 40 20 10 15 30 20 15 20 85 15 65 15 60 60 40 45 5 10 10 15 15 15 15 20 20 20 25 30 30 40 40 45 60 60 65 85 Q1 = 15 M = 22.5 Q3= 42.5 IQR = Q3 – Q1 = 42.5 – 15 = 27.5 minutes Interpretation: The range of the middle half of travel times for the New Yorkers in the sample is 27.5 minutes. Describing Quantitative Data Example, page 57 In addition to serving as a measure of spread, the interquartile range (IQR) is used as part of a rule of thumb for identifying outliers. Definition: The 1.5 x IQR Rule for Outliers Call an observation an outlier if it falls more than 1.5 x IQR above the third quartile or below the first quartile. Example, page 57 In the New York travel time data, we found Q1=15 minutes, Q3=42.5 minutes, and IQR=27.5 minutes. 0 1 2 For these data, 1.5 x IQR = 1.5(27.5) = 41.25 3 Q1 - 1.5 x IQR = 15 – 41.25 = -26.25 4 Q3+ 1.5 x IQR = 42.5 + 41.25 = 83.75 5 Any travel time shorter than -26.25 minutes or longer than 6 7 83.75 minutes is considered an outlier. 8 5 005555 0005 00 005 005 5 Describing Quantitative Data Outliers + Identifying + Five-Number Summary The minimum and maximum values alone tell us little about the distribution as a whole. Likewise, the median and quartiles tell us little about the tails of a distribution. To get a quick summary of both center and spread, combine all five numbers. Definition: The five-number summary of a distribution consists of the smallest observation, the first quartile, the median, the third quartile, and the largest observation, written in order from smallest to largest. Minimum Q1 M Q3 Maximum Describing Quantitative Data The + Calculator: Command to get 5 number summary: Put all #s in the list. Stat -> Edit-> List L1-> Put the numbers Now, for the command for 5 number summary, Stat -> Calc -> choose 1:1-VarStats L1 Enter The five-number summary divides the distribution roughly into quarters. This leads to a new way to display quantitative data, the boxplot. How to Make a Boxplot •Draw and label a number line that includes the range of the distribution. •Draw a central box from Q1 to Q3. •Note the median M inside the box. •Extend lines (whiskers) from the box out to the minimum and maximum values that are not outliers. + Boxplots (Box-and-Whisker Plots) Describing Quantitative Data a Boxplot + Construct Consider our NY travel times data. Construct a boxplot. 10 30 5 25 40 20 10 15 30 20 15 20 85 15 65 15 60 60 40 45 5 10 10 15 15 15 15 20 20 20 25 30 30 40 40 45 60 60 65 85 Min=5 Q1 = 15 M = 22.5 Q3= 42.5 Max=85 Recall, this is an outlier by the 1.5 x IQR rule Describing Quantitative Data Example The most common measure of spread looks at how far each observation is from the mean. This measure is called the standard deviation. Let’s explore it! Consider the following data on the number of pets owned by a group of 9 children. 1) Calculate the mean. 2) Calculate each deviation. deviation = observation – mean deviation: 1 - 5 = -4 deviation: 8 - 5 = 3 x =5 Describing Quantitative Data Spread: The Standard Deviation + Measuring Spread: The Standard Deviation (xi-mean) 1 1 - 5 = -4 (-4)2 = 16 3 3 - 5 = -2 (-2)2 = 4 3) Square each deviation. 4 4 - 5 = -1 (-1)2 = 1 4) Find the “average” squared deviation. Calculate the sum of the squared deviations divided by (n-1)…this is called the variance. 4 4 - 5 = -1 (-1)2 = 1 4 4 - 5 = -1 (-1)2 = 1 5 5-5=0 (0)2 = 0 7 7-5=2 (2)2 = 4 8 8-5=3 (3)2 = 9 9 9-5=4 (4)2 = 16 5) Calculate the square root of the variance…this is the standard deviation. Sum=? “average” squared deviation = 52/(9-1) = 6.5 Standard deviation = square root of variance = Describing Quantitative Data (xi-mean)2 xi + Measuring Sum=? This is the variance. 6.5 2.55 Spread: The Standard Deviation The standard deviation sx measures the average distance of the observations from their mean. It is calculated by finding an average of the squared distances and then taking the square root. This average squared distance is called the variance. (x1 x ) 2 (x 2 x ) 2 ... (x n x ) 2 1 variance = s (x i x ) 2 n 1 n 1 2 x 1 2 standard deviation = sx (x x ) i n 1 Describing Quantitative Data Definition: + Measuring We now have a choice between two descriptions for center and spread Mean and Standard Deviation Median and Interquartile Range Choosing Measures of Center and Spread •The median and IQR are usually better than the mean and standard deviation for describing a skewed distribution or a distribution with outliers. •Use mean and standard deviation only for reasonably symmetric distributions that don’t have outliers. •NOTE: Numerical summaries do not fully describe the shape of a distribution. ALWAYS PLOT YOUR DATA! + Choosing Measures of Center and Spread Describing Quantitative Data + Section 1.3 Describing Quantitative Data with Numbers Summary In this section, we learned that… A numerical summary of a distribution should report at least its center and spread. The mean and median describe the center of a distribution in different ways. The mean is the average and the median is the midpoint of the values. When you use the median to indicate the center of a distribution, describe its spread using the quartiles. The interquartile range (IQR) is the range of the middle 50% of the observations: IQR = Q3 – Q1. + Section 1.3 Describing Quantitative Data with Numbers Summary In this section, we learned that… An extreme observation is an outlier if it is smaller than Q1–(1.5xIQR) or larger than Q3+(1.5xIQR) . The five-number summary (min, Q1, M, Q3, max) provides a quick overall description of distribution and can be pictured using a boxplot. The variance and its square root, the standard deviation are common measures of spread about the mean as center. The mean and standard deviation are good descriptions for symmetric distributions without outliers. The median and IQR are a better description for skewed distributions. + Looking Ahead… In the next Chapter… We’ll learn how to model distributions of data… • Describing Location in a Distribution • Normal Distributions + 92 (a) The median is the average of the ranked scores in the middle two positions (the 15th and 16th ranked scores). The median is 87.75. Half of the students scored less than 87.75 and half scored more than 87.75. Q1 is the score one-quarter up the list of ordered scores, 82. Q3 is IQR = 93-82 =11 The middle 50% of the scores have a range of 11 points. Any observation above Q3 +1.5 IQR = 93 + 1.5(11) = 109.5 or below Q1-1.5 IQR = 82 -1.5 (11) = the score three-quarters up the ordered list of scores, 93. 65.5 is considered an outlier. Thus, the scores 43 and 45 are outliers. +