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Displaying Categorical Variables Frequency Table

Displaying Categorical Variables Frequency Table. Variable. Categories of the Variable. Count of elements from sample in each category. Total = 498. Displaying Categorical Variables Relative Frequency Circle Graph. Relative Frequency or Proportion. 74/498 = 15%.

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Displaying Categorical Variables Frequency Table

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  1. Displaying Categorical VariablesFrequency Table Variable Categories of the Variable Count of elements from sample in each category. Total = 498 Section 2.1, Page 24

  2. Displaying Categorical VariablesRelative Frequency Circle Graph Relative Frequency or Proportion.74/498 = 15% Section 2.1, Page 25

  3. Displaying Categorical VariablesVertical Bar Graph Section 2.1, Page 25

  4. Displaying Categorical VariablesPareto Chart for Hate Crimes USA 1993 Cumulative count/relative frequency Frequency, Relative Frequency, and Cumulative Relative Frequency Table Section 2.1, Page 25

  5. Displaying Quantitative DataDot Plots Section 2.2, Page 26

  6. Displaying Quantitative DataStem-and-Leaf Displays To make stem-and-leaf display, first find the minimum and maximum number, 52 and 96. We then graph the tens digits in the left column, 5 – 9. We then plot each number opposite its tens digit. The plot point is the ones digit. Section 2.2, Page 27

  7. Stem-and Leaf-DisplaysProblems Section 2.2, Page 50

  8. Displaying Quantitative DataUngrouped Frequency Distribution Values of the Variable in the data set Frequency or number of times each value occurs in the data set Section 2.1, Page 29

  9. Displaying Quantitative DataGrouped Frequency Distribution Classes or bins, usually 5 to 12 of equal width 95 or more to less than 105 Section 2.2, Page 30

  10. Displaying Quantitative DataHistograms Histogram: A bar graph that represents a frequency distribution of a quantitative variable. 15 Count 10 5 5 to 12 equal sized classes or bins Section 2.2, Page 32

  11. Histograms Shapes of Distributions Section 2.2, Page 33

  12. Constructing HistogramTI-83 Calculator (50 States) Enter Data:STAT-1:Edit-ENTER Type all the data in L1 Set up Plot: 2nd Stat Plot Enter --Turn plot ON, select Histogram Icon, enter XList: as L1 and Freq: as 1 Set the Viewing WindowZoom 9: ZoomStat –Hit Trace key then arrows to view axes values. Change category size to 7Window –Make Xscl= 7. Then hit Graph Key Display class width and frequency. Trace Section 2.2 WS #21

  13. TI-83 Histogram Display # of States % College Students Enrolled in Public Institutions The leftmost class or bin shows the number of states between 44 and <51. There are 2 states in this bin. To see the next bin, hit the right arrow button. Section 2.2 WS #21

  14. Histogram Problem 2.4 Heights of NBA players selected in the June 2004 Draft. • Construct a histogram. Be sure to show the scale and the label for the x and y axes. • Describe the shape of the distribution. Section 2.2, Page 50

  15. Cumulative Frequency DistributionFinal Exam Scores for 50 Students Cumulative Relative Frequency 2/50 = .04 4/50 = .08 11/50 =.22 24/50 = .48 35/50 = .70 46/50 = .92 50/50 = 1.0 Cumulative Relative Frequency For classes <65, 11/50 = .22 Section 2.2, Page 34

  16. Measures of Central TendencyMean Find the sample mean for the set {6, 3, 8, 6, 4} Section 2.3, Page 35

  17. Measures of Central TendencyMedian The median is the value of the middle number when the date are ranked according to size. Find the median for the data the following set with an odd n: {3, 3, 5, 6, 8}, n=5. The data values are in ascending order. Depth of median = (n+1)/2. For this set: (5+1)/2 = 3The median is the 3rd number, 5. Find the median for the following data values that are in ascending order with even n: {6, 7, 8, 9, 9, 10}, n=6. Dept of median = (n+1)/2 = 3.5 The median is then the average of the 3rd and 4th number. The median is (8+9)/2 = 8.5 Section 2.3, Page 36

  18. Measures of Central TendencyMode and Midrange (L+H)/2 = (3+8)/2 = 5.5 Section 2.3, Page 37

  19. Measures of Central TendencySummary The most useful measure is the mean. However, when a set of numbers has outliers, the mean gets distorted and may not be representative of the central tendency. When this happens, the median is a better measure of central tendency because it is not affected by outliers. Section 2.3, Page 37

  20. Measures of DispersionRange Secton 2.4, Page 39

  21. Measures of DispersionVariance and Standard Deviation {6, 3, 8, 5, 3 } Section 2.4, Page 41

  22. Measures of PositionPercentiles Percentiles: Values of the variable that divide a set of ranked data into 100 equal subsets: each set of data has 99 percentiles. A specific number fromwithin the range of values In the set Section 2.5, Page 42

  23. Finding PercentilesExample Sample data set of 20 numbers in ascending rank order: {6, 12, 14, 17, 23, 27, 29, 33, 42, 51, 59, 65, 69, 74, 79, 82, 84, 88, 92, 97} Find the 21st Percentile. Sample size n=20. Calculate the depth: percentile*n/100 = 21*20/100 = 4.2. (If the depth is an integer, Pk is the average of the number and the next number. If the depth contains a decimal, Pk is the next number.) Since the depth contains a decimal, Pk is the next number, the 5th number, Pk = 23. Find the 75th Percentile:Depth = 75*20/100 = 15. Since the depth is an integer, the 75th percentile is the average of the 15th and 16th numbers, (79+82)/2=80.5. Section 2.5, Page 43

  24. Using the TI-83 to Find Percentiles Find the 21st and the 75th percentile of the following data set. {6, 12, 14, 17, 23, 27, 29, 33, 42, 51, 59, 65, 69, 74, 79, 82, 84, 88, 92, 97} STAT-EDIT: Enter the data in L1 PRGM: down arrow to PRCNTILE ENTER: (Displays Program Input Page) 2nd L1: (Enters the List name) ENTER: (Asks for Percentile) 21.0: (Enters the desired percentile) ENTER: (Displays the 21st percentile) ENTER-2ND L1-75: (Displays the 75th percentile) CLEAR: (Clears the home screen) Section 2.5, Page 43

  25. 5-Number SummaryBox and Whisker Display Q1 Q3 H L Med Interquartile Range = Q3-Q1 Range of middle 50% of values Measure of dispersion resistant to outliers. Section 2.5, Page 44

  26. TI – 83 Problem (1) • Find the mean and the standard deviation for the above sample data. • STAT – EDIT: Enter the data is L1 • STAT – CALC-1: 1-VAR Stats - ENTER • 2ND L1 - ENTER • DISPLAY: Sample Mean Sample Standard Deviation Sx • Find the variance of the sample • (2.7999)2 =7.8394 Problems, Page 50

  27. TI-83 Problem (2) • Find the 5 number summary • PRESS THE DOWN ARROW FIVE TIMES • DISPLAY: • Find the interquartile range • 32 - 28 = 4 • e. Find the range. • 34 – 25 = 9 Problems, Page 50

  28. TI-83 Problem (3) • Make a box and whisker display of the data. • 2ND STAT PLOT-ENTER • ENTER: Sets plot to ON • DOWN ARROW • RIGHT ARROW 5 TIMES: Select box Plot • DOWN ARROW – 2nd L1: Select List • Display: ZOOM – 9TRACE: Display: RIGHT-LEFT ARROW: Display 5-number summary Problems, Page 50

  29. Summary: Measures of Center and Spread The mean and median are measures of the center of a distribution. Outliers will distort the mean, so when outliers are present the mean is not a good measure of the center. The median is not distorted by outliers. The standard deviation, variance, range, and Interquartile range (IQR) are measures of the spread or variability of a distribution. Outliers will distort the standard deviation, variance, and range, so when outliers are present, these are not good measures of the spread or variability. The Interquartile range is not distorted by outliers. When outliers are present, then use the median and IQR as measures of the center and spread. When no significant outliers are present, use the mean and standard deviation as measures of center and spread. These measures allow use of the maximum number statistical tools using the distribution. Section 2.4

  30. Problem • Find the mean, variance, and standard deviation. • Find the 5-number summary. • Make a box and whisker display and label the numbers. • Calculate the Interquartile range and the range • Describe the shape of the distribution • Find the 33 percentile. Problems, Page 50

  31. Problem • Find the mean, variance, and standard deviation. • Find the 5-number summary. • Make a box and whisker display and label the numbers. • Calculate the Interquartile range and the range • Describe the shape of the distribution. • Find the 90th percentile. Problems, Page 50

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