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Slides Prepared by JOHN S. LOUCKS St. Edward’s University

Slides Prepared by JOHN S. LOUCKS St. Edward’s University. Chapter 3 Descriptive Statistics: Numerical Methods, Part A. Measures of Location Measures of Variability. . . %. x. Measures of Location. Mean Median Mode Percentiles Quartiles. Example: Apartment Rents.

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Slides Prepared by JOHN S. LOUCKS St. Edward’s University

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  1. Slides Prepared by JOHN S. LOUCKS St. Edward’s University

  2. Chapter 3 Descriptive Statistics: Numerical Methods, Part A • Measures of Location • Measures of Variability   % x

  3. Measures of Location • Mean • Median • Mode • Percentiles • Quartiles

  4. Example: Apartment Rents Given below is a sample of monthly rent values ($) for one-bedroom apartments. The data is a sample of 70 apartments in a particular city. The data are presented in ascending order.

  5. Mean • The mean of a data set is the average of all the data values. • If the data are from a sample, the mean is denoted by . • If the data are from a population, the mean is denoted by m (mu).

  6. Example: Apartment Rents • Mean

  7. Median • The median is the measure of location most often reported for annual income and property value data. • A few extremely large incomes or property values can inflate the mean.

  8. Median • The median of a data set is the value in the middle when the data items are arranged in ascending order. • For an odd number of observations, the median is the middle value. • For an even number of observations, the median is the average of the two middle values.

  9. Example: Apartment Rents • Median Median = 50th percentile i = (p/100)n = (50/100)70 = 35.5 Averaging the 35th and 36th data values: Median = (475 + 475)/2 = 475

  10. Mode • The mode of a data set is the value that occurs with greatest frequency. • The greatest frequency can occur at two or more different values. • If the data have exactly two modes, the data are bimodal. • If the data have more than two modes, the data are multimodal.

  11. Example: Apartment Rents • Mode 450 occurred most frequently (7 times) Mode = 450

  12. Using Excel to Computethe Mean, Median, and Mode • Formula Worksheet Note: Rows 7-71 are not shown.

  13. Using Excel to Computethe Mean, Median, and Mode • Value Worksheet Note: Rows 7-71 are not shown.

  14. Percentiles • A percentile provides information about how the data are spread over the interval from the smallest value to the largest value. • Admission test scores for colleges and universities are frequently reported in terms of percentiles.

  15. Percentiles • The pth percentile of a data set is a value such that at least p percent of the items take on this value or less and at least (100 - p) percent of the items take on this value or more. • Arrange the data in ascending order. • Compute index i, the position of the pth percentile. i = (p/100)n • If i is not an integer, round up. The pth percentile is the value in the ith position. • If i is an integer, the pth percentile is the average of the values in positions i and i+1.

  16. Example: Apartment Rents • 90th Percentile i = (p/100)n = (90/100)70 = 63 Averaging the 63rd and 64th data values: 90th Percentile = (580 + 590)/2 = 585

  17. Quartiles • Quartiles are specific percentiles • First Quartile = 25th Percentile • Second Quartile = 50th Percentile = Median • Third Quartile = 75th Percentile

  18. Example: Apartment Rents • Third Quartile Third quartile = 75th percentile i = (p/100)n = (75/100)70 = 52.5 = 53 Third quartile = 525

  19. Using Excel to ComputePercentiles and Quartiles • Unsorted Monthly Rent ($) Note: Rows 7-71 are not shown.

  20. Using Excel to ComputePercentiles and Quartiles • Sorting Data Step 1 Select any cell containing data in column B Step 2 Select the Data pull-down menu Step 3 Choose the Sort option Step 4 When the Sort dialog box appears: In the Sort by box, make sure that Monthly Rent ($) appears and that Ascending is selected In the My list has box, make sure that Header row is selected Click OK

  21. Using Excel to ComputePercentiles and Quartiles • Sorted Monthly Rent ($) Note: Rows 7-71 are not shown.

  22. Using Excel to ComputePercentiles and Quartiles • Formula Worksheet for 90th Percentile’s Index Note: Rows 7-71 are not shown.

  23. Using Excel to ComputePercentiles and Quartiles • Value Worksheet for 90th Percentile’s Index Note: Rows 7-71 are not shown.

  24. Using Excel to ComputePercentiles and Quartiles • Value Worksheet for 3rd Quartile’s Index Note: Rows 7-71 are not shown.

  25. Measures of Variability • It is often desirable to consider measures of variability (dispersion), as well as measures of location. • For example, in choosing supplier A or supplier B we might consider not only the average delivery time for each, but also the variability in delivery time for each.

  26. Measures of Variability • Range • Interquartile Range • Variance • Standard Deviation • Coefficient of Variation

  27. Range • The range of a data set is the difference between the largest and smallest data values. • It is the simplest measure of variability. • It is very sensitive to the smallest and largest data values.

  28. Example: Apartment Rents • Range Range = largest value - smallest value Range = 615 - 425 = 190

  29. Interquartile Range • The interquartile range of a data set is the difference between the third quartile and the first quartile. • It is the range for the middle 50% of the data. • It overcomes the sensitivity to extreme data values.

  30. Example: Apartment Rents • Interquartile Range 3rd Quartile (Q3) = 525 1st Quartile (Q1) = 445 Interquartile Range = Q3 - Q1 = 525 - 445 = 80

  31. Variance • The variance is a measure of variability that utilizes all the data. • It is based on the difference between the value of each observation (xi) and the mean (x for a sample, m for a population).

  32. Variance • The variance is the average of the squared differences between each data value and the mean. • If the data set is a sample, the variance is denoted by s2. • If the data set is a population, the variance is denoted by  2.

  33. Standard Deviation • The standard deviation of a data set is the positive square root of the variance. • It is measured in the same units as the data, making it more easily comparable, than the variance, to the mean. • If the data set is a sample, the standard deviation is denoted s. • If the data set is a population, the standard deviation is denoted  (sigma).

  34. Coefficient of Variation • The coefficient of variation indicates how large the standard deviation is in relation to the mean. • If the data set is a sample, the coefficient of variation is computed as follows: • If the data set is a population, the coefficient of variation is computed as follows:

  35. Example: Apartment Rents • Variance • Standard Deviation • Coefficient of Variation

  36. Using Excel to Compute the Sample Variance, Standard Deviation, and Coefficient of Variation • Formula Worksheet Note: Rows 8-71 are not shown.

  37. Using Excel to Compute the Sample Variance, Standard Deviation, and Coefficient of Variation • Value Worksheet Note: Rows 8-71 are not shown.

  38. Using Excel’sDescriptive Statistics Tool Step 1 Select the Tools pull-down menu Step 2 Choose the Data Analysis option Step 3 Choose Descriptive Statistics from the list of Analysis Tools … continued

  39. Using Excel’sDescriptive Statistics Tool Step 4 When the Descriptive Statistics dialog box appears: Enter B1:B71 in the Input Range box Select Grouped By Columns Select Labels in First Row Select Output Range Enter D1 in the Output Range box Select Summary Statistics Click OK

  40. Using Excel’sDescriptive Statistics Tool • Value Worksheet (Partial) Note: Rows 9-71 are not shown.

  41. Using Excel’sDescriptive Statistics Tool • Value Worksheet (Partial) Note: Rows 1-8 and 17-71 are not shown.

  42.  % x Descriptive Statistics: Numerical Methods, Part B • Measures of Relative Location and Detecting Outliers • Exploratory Data Analysis • Measures of Association Between Two Variables • The Weighted Mean and Working with Grouped Data

  43. Measures of Relative Locationand Detecting Outliers • z-Scores • Chebyshev’s Theorem • Empirical Rule • Detecting Outliers

  44. z-Scores • The z-score is often called the standardized value. • It denotes the number of standard deviations a data value xi is from the mean. • A data value less than the sample mean will have a z-score less than zero. • A data value greater than the sample mean will have a z-score greater than zero. • A data value equal to the sample mean will have a z-score of zero.

  45. Example: Apartment Rents • z-Score of Smallest Value (425) Standardized Values for Apartment Rents

  46. Chebyshev’s Theorem At least (1 - 1/k2) of the items in any data set will be within k standard deviations of the mean, where k is any value greater than 1. • At least 75% of the items must be within k = 2 standard deviations of the mean. • At least 89% of the items must be within k = 3 standard deviations of the mean. • At least 94% of the items must be within k = 4 standard deviations of the mean.

  47. Example: Apartment Rents • Chebyshev’s Theorem Let k = 1.5 with = 490.80 and s = 54.74 At least (1 - 1/(1.5)2) = 1 - 0.44 = 0.56 or 56% of the rent values must be between - k(s) = 490.80 - 1.5(54.74) = 409 and + k(s) = 490.80 + 1.5(54.74) = 573

  48. Example: Apartment Rents • Chebyshev’s Theorem (continued) Actually, 86% of the rent values are between 409 and 573.

  49. Empirical Rule For data having a bell-shaped distribution: • Approximately 68% of the data values will be within onestandard deviation of the mean.

  50. Empirical Rule For data having a bell-shaped distribution: • Approximately 95% of the data values will be within twostandard deviations of the mean.

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