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MM207-Statistics Unit 2 Seminar-Descriptive Statistics Dr Bridgette Stevens AIM:BStevensKaplan (add me to your Buddy list) 1 Chapter Outline • 2.1 Frequency Distributions and Their Graphs • 2.2 More Graphs and Displays • 2.3 Measures of Central Tendency • 2.4 Measures of Variation • 2.5 Measures of Position Larson/Farber 4th ed. 2 Section 2.1 Frequency Distributions and Their Graphs Larson/Farber 4th ed. 3 Frequency Distribution Frequency Distribution • A table that shows classes or Class width 6–1=5 intervals of data with a count of the number of entries in each class. • The frequency, f, of a class is the number of data entries in the class. Class Frequency, f 1–5 5 6 – 10 8 11 – 15 6 16 – 20 8 21 – 25 5 26 – 30 4 Lower class limits Larson/Farber 4th ed. Upper class limits 4 Constructing a Frequency Distribution 1. Decide on the number of classes. • Usually between 5 and 20; otherwise, it may be difficult to detect any patterns. 2. Find the class width. • • • Determine the range of the data. Divide the range by the number of classes. Round up to the next convenient number. Larson/Farber 4th ed. 5 Constructing a Frequency Distribution 3. Find the class limits. • • • • You can use the minimum data entry as the lower limit of the first class. Find the remaining lower limits (add the class width to the lower limit of the preceding class). Find the upper limit of the first class. Remember that classes cannot overlap. Find the remaining upper class limits. Larson/Farber 4th ed. 6 Example: Constructing a Frequency Distribution The following sample data set lists the number of minutes 50 Internet subscribers spent on the Internet during their most recent session. Construct a frequency distribution that has seven classes. 50 40 41 17 11 7 22 44 28 21 19 23 37 51 54 42 86 41 78 56 72 56 17 7 69 30 80 56 29 33 46 31 39 20 18 29 34 59 73 77 36 39 30 62 54 67 39 31 53 44 Larson/Farber 4th ed. 7 Solution: Constructing a Frequency Distribution 50 40 41 17 11 7 22 44 28 21 19 23 37 51 54 42 86 41 78 56 72 56 17 7 69 30 80 56 29 33 46 31 39 20 18 29 34 59 73 77 36 39 30 62 54 67 39 31 53 44 1. Number of classes = 7 (given) 2. Find the class width max min 86 7 11.29 #classes 7 Round up to 12 Larson/Farber 4th ed. 8 Solution: Constructing a Frequency Distribution 3. Use 7 (minimum value) as first lower limit. Add the class width of 12 to get the lower limit of the next class. 7 + 12 = 19 Find the remaining lower limits. Larson/Farber 4th ed. Lower Upper limit limit Class width = 12 7 19 31 43 55 67 79 9 Solution: Constructing a Frequency Distribution The upper limit of the first class is 18 (one less than the lower limit of the second class). Add the class width of 12 to get the upper limit of the next class. 18 + 12 = 30 Find the remaining upper limits. Larson/Farber 4th ed. Lower limit Upper limit 7 19 31 43 55 18 30 42 54 66 67 79 78 90 Class width = 12 10 Solution: Constructing a Frequency Distribution 4. Make a tally mark for each data entry in the row of the appropriate class. 5. Count the tally marks to find the total frequency f for each class. Class 7 – 18 Tally Frequency, f IIII I 6 19 – 30 IIII IIII 10 31 – 42 IIII IIII III 13 43 – 54 IIII III 8 55 – 66 IIII 5 67 – 78 IIII I 6 79 – 90 II 2 Σf = 50 Larson/Farber 4th ed. 11 Determining the Midpoint Midpoint of a class (Lower class limit) (Upper class limit) 2 Class 7 – 18 Midpoint 7 18 12.5 2 19 – 30 19 30 24.5 2 31 – 42 31 42 36.5 2 Larson/Farber 4th ed. Frequency, f 6 Class width = 12 10 13 12 Determining the Relative Frequency Relative Frequency of a class • Portion or percentage of the data that falls in a particular class. • class frequency f relative frequency Sample size n Larson/Farber 4th ed. Class Frequency, f 7 – 18 6 19 – 30 10 31 – 42 13 Relative Frequency 6 0.12 50 10 0.20 50 13 0.26 50 13 Graphs of Frequency Distributions frequency Frequency Histogram • A bar graph that represents the frequency distribution. • The horizontal scale is quantitative and measures the data values. • The vertical scale measures the frequencies of the classes. • Consecutive bars must touch. data values Larson/Farber 4th ed. 14 Example: Frequency Histogram Construct a frequency histogram for the Internet usage frequency distribution. Class Class boundaries Midpoint 7 – 18 6.5 – 18.5 12.5 6 19 – 30 18.5 – 30.5 24.5 10 31 – 42 30.5 – 42.5 36.5 13 43 – 54 42.5 – 54.5 48.5 8 55 – 66 54.5 – 66.5 60.5 5 67 – 78 66.5 – 78.5 72.5 6 79 – 90 78.5 – 90.5 84.5 2 Larson/Farber 4th ed. Frequency, f 15 Solution: Frequency Histogram (using Midpoints) Larson/Farber 4th ed. 16 Graphs of Frequency Distributions frequency Frequency Polygon • A line graph that emphasizes the continuous change in frequencies. data values Larson/Farber 4th ed. 17 Solution: Frequency Polygon Internet Usage Frequency The graph should begin and end on the horizontal axis, so extend the left side to one class width before the first class midpoint and extend the right side to one class width after the last class midpoint. 14 12 10 8 6 4 2 0 0.5 12.5 24.5 36.5 48.5 60.5 72.5 84.5 96.5 Time online (in minutes) You can see that the frequency of subscribers increases up to 36.5 minutes and then decreases. Larson/Farber 4th ed. 18 Graphs of Frequency Distributions relative frequency Relative Frequency Histogram • Has the same shape and the same horizontal scale as the corresponding frequency histogram. • The vertical scale measures the relative frequencies, not frequencies. data values Larson/Farber 4th ed. 19 Example: Relative Frequency Histogram Construct a relative frequency histogram for the Internet usage frequency distribution. Class Class boundaries Frequency, f Relative frequency 7 – 18 6.5 – 18.5 6 0.12 19 – 30 18.5 – 30.5 10 0.20 31 – 42 30.5 – 42.5 13 0.26 43 – 54 42.5 – 54.5 8 0.16 55 – 66 54.5 – 66.5 5 0.10 67 – 78 66.5 – 78.5 6 0.12 79 – 90 78.5 – 90.5 2 0.04 Larson/Farber 4th ed. 20 Solution: Relative Frequency Histogram 6.5 18.5 30.5 42.5 54.5 66.5 78.5 From this graph you can see that 20% of Internet subscribers spent between 18.5 minutes and 30.5 minutes online. 21 90.5 Graphs of Frequency Distributions cumulative frequency Cumulative Frequency Graph or Ogive • A line graph that displays the cumulative frequency of each class at its upper class boundary. • The upper boundaries are marked on the horizontal axis. • The cumulative frequencies are marked on the vertical axis. data values Larson/Farber 4th ed. 22 Solution: Ogive Internet Usage Cumulative frequency 60 50 40 30 20 10 0 6.5 18.5 Time online (in minutes) 30.5 42.5 54.5 66.5 78.5 90.5 From the ogive, you can see that about 40 subscribers spent 60 minutes or less online during their last session. The greatest increase in usage occurs between 30.5 minutes and 42.5 minutes. 23 Section 2.2 More Graphs and Displays Larson/Farber 4th ed. 24 Section 2.2 Objectives • Graph quantitative data using stem-and-leaf plots and dot plots • Graph qualitative data using pie charts and Pareto charts • Graph paired data sets using scatter plots and time series charts Larson/Farber 4th ed. 25 Graphing Quantitative Data Sets Stem-and-leaf plot • Each number is separated into a stem and a leaf. • Similar to a histogram. • Still contains original data values. Data: 21, 25, 25, 26, 27, 28, 30, 36, 36, 45 Larson/Farber 4th ed. 26 2 1 5 5 6 7 8 3 0 6 6 4 5 26 Example: Constructing a Stem-andLeaf Plot The following are the numbers of text messages sent last month by the cellular phone users on one floor of a college dormitory. Display the data in a stem-and-leaf plot. 155 118 139 129 159 118 139 112 144 129 105 145 126 116 130 114 122 112 112 142 126 108 122 121 109 140 126 119 113 117 118 109 109 119 122 78 133 126 123 145 121 134 124 119 132 133 124 126 148 147 Larson/Farber 4th ed. 27 Solution: Constructing a Stem-and-Leaf Plot Include a key to identify the values of the data. From the display, you can conclude that more than 50% of the cellular phone users sent between 110 and 130 text messages. 28 Graphing Quantitative Data Sets Dot plot • Each data entry is plotted, using a point, above a horizontal axis Data: 21, 25, 25, 26, 27, 28, 30, 36, 36, 45 26 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 29 Solution: Constructing a Dot Plot 155 118 139 129 159 118 139 112 144 129 105 145 126 116 130 114 122 112 112 142 126 108 122 121 109 140 126 119 113 117 118 109 109 119 122 78 133 126 123 145 121 134 124 119 132 133 124 126 148 147 From the dot plot, you can see that most values cluster between 105 and 148 and the value that occurs the most is 126. You can also see that 78 is an unusual data value. 30 Graphing Qualitative Data Sets Pie Chart • A circle is divided into sectors that represent categories. • The area of each sector is proportional to the frequency of each category. Larson/Farber 4th ed. 31 Example: Constructing a Pie Chart The numbers of motor vehicle occupants killed in crashes in 2005 are shown in the table. Use a pie chart to organize the data. (Source: U.S. Department of Transportation, National Highway Traffic Safety Administration) Larson/Farber 4th ed. Vehicle type Killed Cars 18,440 Trucks 13,778 Motorcycles 4,553 Other 823 32 Solution: Constructing a Pie Chart • Find the relative frequency (percent) of each category. Vehicle type Frequency, f Relative frequency Cars 18,440 Trucks 13,778 Motorcycles Other 4,553 823 18440 0.49 37594 13778 0.37 37594 4553 0.12 37594 823 0.02 37594 37,594 33 Solution: Constructing a Pie Chart Relative Vehicle type frequency Central angle Cars 0.49 176º Trucks 0.37 133º Motorcycles 0.12 43º Other 0.02 7º From the pie chart, you can see that most fatalities in motor vehicle crashes were those involving the occupants of cars. 34 Graphing Qualitative Data Sets Frequency Pareto Chart • A vertical bar graph in which the height of each bar represents frequency or relative frequency. • The bars are positioned in order of decreasing height, with the tallest bar positioned at the left. Categories Larson/Farber 4th ed. 35 Constructing a Pareto Chart Cause Admin. error $ (million) 7.8 Employee theft 15.6 Shoplifting 14.7 Vendor fraud 2.9 From the graph, it is easy to see that the causes of inventory shrinkage that should be addressed first are employee theft and shoplifting. 36 Graphing Paired Data Sets Paired Data Sets • Each entry in one data set corresponds to one entry in a second data set. • Graph using a scatter plot. • The ordered pairs are graphed as points in a coordinate plane. • Used to show the relationship between two quantitative variables. y x 37 Example: Interpreting a Scatter Plot As the petal length increases, what tends to happen to the petal width? Each point in the scatter plot represents the petal length and petal width of one flower. 38 Graphing Paired Data Sets Time Series • Data set is composed of quantitative entries taken at regular intervals over a period of time. • Use a time series chart to graph. Quantitative data • e.g., The amount of precipitation measured each day for one month. time 39 Example: Constructing a Time Series Chart The table lists the number of cellular telephone subscribers (in millions) for the years 1995 through 2005. Construct a time series chart for the number of cellular subscribers. (Source: Cellular Telecommunication & Internet Association) Larson/Farber 4th ed. 40 Solution: Constructing a Time Series Chart The graph shows that the number of subscribers has been increasing since 1995, with greater increases recently. 41 Section 2.3 Measures of Central Tendency Larson/Farber 4th ed. 42 Measures of Central Tendency Measure of central tendency • A value that represents a typical, or central, entry of a data set. • Most common measures of central tendency: • Mean • Median • Mode Larson/Farber 4th ed. 43 Measure of Central Tendency: Mode Mode • The data entry that occurs with the greatest frequency. • If no entry is repeated the data set has no mode. • If two entries occur with the same greatest frequency, each entry is a mode (bimodal). Larson/Farber 4th ed. 44 Comparing the Mean, Median, and Mode • All three measures describe a typical entry of a data set. • Advantage of using the mean: • The mean is a reliable measure because it takes into account every entry of a data set. • Disadvantage of using the mean: • Greatly affected by outliers (a data entry that is far removed from the other entries in the data set). Larson/Farber 4th ed. 45 Example: Comparing the Mean, Median, and Mode Find the mean, median, and mode of the sample ages of a class shown. Which measure of central tendency best describes a typical entry of this data set? Are there any outliers? Ages in a class Larson/Farber 4th ed. 20 20 20 20 20 20 21 21 21 21 22 22 22 23 23 23 23 24 24 65 46 Solution: Comparing the Mean, Median, and Mode Ages in a class Mean: Median: Mode: 20 20 20 20 20 20 21 21 21 21 22 22 22 23 23 23 23 24 24 65 x 20 20 ... 24 65 x 23.8 years n 20 21 22 21.5 years 2 20 years (the entry occurring with the greatest frequency) Larson/Farber 4th ed. 47 Solution: Comparing the Mean, Median, and Mode Mean ≈ 23.8 years Median = 21.5 years Mode = 20 years • The mean takes every entry into account, but is influenced by the outlier of 65. • The median also takes every entry into account, and it is not affected by the outlier. • In this case the mode exists, but it doesn't appear to represent a typical entry. Larson/Farber 4th ed. 48 Solution: Comparing the Mean, Median, and Mode Sometimes a graphical comparison can help you decide which measure of central tendency best represents a data set. In this case, it appears that the median best describes the data set. Larson/Farber 4th ed. 49 Weighted Mean Weighted Mean • The mean of a data set whose entries have varying weights. • where w is the weight of each entry x ( x w) x w Larson/Farber 4th ed. 50 Example: Finding a Weighted Mean You are taking a class in which your grade is determined from five sources: 50% from your test mean, 15% from your midterm, 20% from your final exam, 10% from your computer lab work, and 5% from your homework. Your scores are 86 (test mean), 96 (midterm), 82 (final exam), 98 (computer lab), and 100 (homework). What is the weighted mean of your scores? If the minimum average for an A is 90, did you get an A? Larson/Farber 4th ed. 51 Solution: Finding a Weighted Mean Source x∙w Score, x Weight, w Test Mean 86 0.50 86(0.50)= 43.0 Midterm 96 0.15 96(0.15) = 14.4 Final Exam 82 0.20 82(0.20) = 16.4 Computer Lab 98 0.10 98(0.10) = 9.8 Homework 100 0.05 100(0.05) = 5.0 Σw = 1 Σ(x∙w) = 88.6 ( x w) 88.6 x 88.6 w 1 Your weighted mean for the course is 88.6. You did not get an A. Larson/Farber 4th ed. 52 Mean of Grouped Data Mean of a Frequency Distribution • Approximated by ( x f ) x xand f are the midpoints n fand frequencies of a where n class, respectively Larson/Farber 4th ed. 53 Finding the Mean of a Frequency Distribution In Words 1. Find the midpoint of each class. In Symbols (lower limit)+(upper limit) x 2 2. Find the sum of the products of the midpoints and the frequencies. ( x f ) 3. Find the sum of the frequencies. n f 4. Find the mean of the frequency distribution. Larson/Farber 4th ed. ( x f ) x n 54 Example: Find the Mean of a Frequency Distribution Use the frequency distribution to approximate the mean number of minutes that a sample of Internet subscribers spent online during their most recent session. Larson/Farber 4th ed. Class Midpoint Frequency, f 7 – 18 12.5 6 19 – 30 24.5 10 31 – 42 36.5 13 43 – 54 48.5 8 55 – 66 60.5 5 67 – 78 72.5 6 79 – 90 84.5 2 55 Solution: Find the Mean of a Frequency Distribution Class Midpoint, x Frequency, f (x∙f) 7 – 18 12.5 6 12.5∙6 = 75.0 19 – 30 24.5 10 24.5∙10 = 245.0 31 – 42 36.5 13 36.5∙13 = 474.5 43 – 54 48.5 8 48.5∙8 = 388.0 55 – 66 60.5 5 60.5∙5 = 302.5 67 – 78 72.5 6 72.5∙6 = 435.0 79 – 90 84.5 2 84.5∙2 = 169.0 n = 50 Σ(x∙f) = 2089.0 ( x f ) 2089 x 41.8 minutes n 50 Larson/Farber 4th ed. 56 The Shape of Distributions Symmetric Distribution • A vertical line can be drawn through the middle of a graph of the distribution and the resulting halves are approximately mirror images. Larson/Farber 4th ed. 57 The Shape of Distributions Uniform Distribution (rectangular) • All entries or classes in the distribution have equal or approximately equal frequencies. • Symmetric. Larson/Farber 4th ed. 58 The Shape of Distributions Skewed Left Distribution (negatively skewed) • The “tail” of the graph elongates more to the left. • The mean is to the left of the median. Larson/Farber 4th ed. 59 The Shape of Distributions Skewed Right Distribution (positively skewed) • The “tail” of the graph elongates more to the right. • The mean is to the right of the median. Larson/Farber 4th ed. 60 Section 2.3 Summary • Determined the mean, median, and mode of a population and of a sample • Determined the weighted mean of a data set and the mean of a frequency distribution • Described the shape of a distribution as symmetric, uniform, or skewed and compared the mean and median for each Larson/Farber 4th ed. 61 Section 2.4 Measures of Variation Larson/Farber 4th ed. 62 Deviation, Variance, and Standard Deviation Population Variance • ( x ) N 2 2 Sum of squares, SSx Population Standard Deviation • 2 ( x ) 2 N Larson/Farber 4th ed. 63 Example: Using Technology to Find the Standard Deviation Sample office rental rates (in dollars per square foot per year) for Miami’s central business district are shown in the table. Use a calculator or a computer to find the mean rental rate and the sample standard deviation. (Adapted from: Cushman & Wakefield Inc.) Larson/Farber 4th ed. Office Rental Rates 35.00 33.50 37.00 23.75 26.50 31.25 36.50 40.00 32.00 39.25 37.50 34.75 37.75 37.25 36.75 27.00 35.75 26.00 37.00 29.00 40.50 24.50 33.00 38.00 64 Solution: Using Technology to Find the Standard Deviation Sample Mean Sample Standard Deviation Larson/Farber 4th ed. 65 Interpreting Standard Deviation • Standard deviation is a measure of the typical amount an entry deviates from the mean. • The more the entries are spread out, the greater the standard deviation. 66 Interpreting Standard Deviation: Empirical Rule (68 – 95 – 99.7 Rule) For data with a (symmetric) bell-shaped distribution, the standard deviation has the following characteristics: • About 68% of the data lie within one standard deviation of the mean. • About 95% of the data lie within two standard deviations of the mean. • About 99.7% of the data lie within three standard deviations of the mean. 67 Interpreting Standard Deviation: Empirical Rule (68 – 95 – 99.7 Rule) 99.7% within 3 SD 95% within 2 SD 68% within 1 SD 34% 2.35% x 3s x 2s Larson/Farber 4th ed. 34% 13.5% x s 2.35% 13.5% x xs x 2s 68 x 3s Example: Using the Empirical Rule In a survey conducted by the National Center for Health Statistics, the sample mean height of women in the United States (ages 20-29) was 64 inches, with a sample standard deviation of 2.71 inches. Estimate the percent of the women whose heights are between 64 inches and 69.42 inches. Larson/Farber 4th ed. 69 Solution: Using the Empirical Rule • Because the distribution is bell-shaped, you can use the Empirical Rule. 34% 13.5% 55.8 x 3s 7 58.5 x 2s 8 61.2 9x s 64 x 66.7 xs 1 69.4 x 2s 2 34% + 13.5% = 47.5% of women are between 64 and 69.42 inches tall. Larson/Farber 4th ed. 70 72.1 3x 3s Chebychev’s Theorem • The portion of any data set lying within k standard deviations (k > 1) of the mean is at least: 1 1 2 k • k = 2: In any data set, at least 1 3 1 2 or 75% 2 4 of the data lie within 2 standard deviations of the mean. • k = 3: In any data set, at least 1 8 1 2 or 88.9% 3 9 of the data lie within 3 standard deviations of the mean. Larson/Farber 4th ed. 71 Box-and-Whisker Plot Box-and-whisker plot • Exploratory data analysis tool. • Highlights important features of a data set. • Requires (five-number summary): • • • • • Minimum entry First quartile Q1 Median Q2 Third quartile Q3 Maximum entry Larson/Farber 4th ed. 72 Drawing a Box-and-Whisker Plot 1. 2. 3. 4. Find the five-number summary of the data set. Construct a horizontal scale that spans the range of the data. Plot the five numbers above the horizontal scale. Draw a box above the horizontal scale from Q1 to Q3 and draw a vertical line in the box at Q2. 5. Draw whiskers from the box to the minimum and maximum entries. Whiske r Minimu m entry Q1 Larson/Farber 4th ed. Box Median, Q2 Whiske r Maximu Q m entry 3 73