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Slides by JOHN LOUCKS St. Edward’s University

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

  2. Chapter 3, Part A Descriptive Statistics: Numerical Measures • Measures of Location • Measures of Variability

  3. Measures of Location • Mean If the measures are computed for data from a sample, they are called sample statistics. • Median • Mode • Percentiles If the measures are computed for data from a population, they are called population parameters. • Quartiles A sample statistic is referred to as the point estimator of the corresponding population parameter.

  4. The sample mean is the point estimator of the population mean m. Mean • The mean of a data set is the average of all the data values.

  5. Sample Mean Sum of the values of the n observations Number of observations in the sample

  6. Population Mean m Sum of the values of the N observations Number of observations in the population

  7. Sample Mean • Example: Apartment Rents Seventy efficiency apartments were randomly sampled in a small college town. The monthly rent prices for these apartments are listed below.

  8. Sample Mean • Example: Apartment Rents

  9. Median • The median of a data set is the value in the middle • when the data items are arranged in ascending order. • Whenever a data set has extreme values, the median • is the preferred measure of central location. • 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.

  10. Median • For an odd number of observations: 26 18 27 12 14 27 19 7 observations 27 12 14 18 19 26 27 in ascending order the median is the middle value. Median = 19

  11. Median • For an even number of observations: 26 18 27 12 14 27 30 19 8 observations 27 30 12 14 18 19 26 27 in ascending order the median is the average of the middle two values. Median = (19 + 26)/2 = 22.5

  12. Median • Example: Apartment Rents Averaging the 35th and 36th data values: Median = (475 + 475)/2 = 475 Note: Data is in ascending order.

  13. 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.

  14. Mode • Example: Apartment Rents 450 occurred most frequently (7 times) Mode = 450 Note: Data is in ascending order.

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

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

  17. 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. • 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.

  18. Percentiles 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 positionsiand i+1.

  19. 80th Percentile • Example: Apartment Rents i = (p/100)n = (80/100)70 = 56 Averaging the 56th and 57th data values: 80th Percentile = (535 + 549)/2 = 542 Note: Data is in ascending order.

  20. 80th Percentile • Example: Apartment Rents “At least 80% of the items take on a value of 542 or less.” “At least 20% of the items take on a value of 542 or more.” 56/70 = .8 or 80% 14/70 = .2 or 20%

  21. Lp = (p/100)n + (1 – p/100) Using Excel’s Rank and Percentile Tool to Compute Percentiles and Quartiles • Using Excel’s Percentile Function The formula Excel uses to compute the location (Lp) of the pth percentile is Excel would compute the location of the 80th percentile for the apartment rent data as follows: L80 = (80/100)70 + (1 – 80/100) = 56 + .2 = 56.2 The 80th percentile would be 535 + .2(549 - 535) = 535 + 2.8 = 537.8

  22. Using Excel’s Rank and Percentile Tool to Compute Percentiles and Quartiles 80th percentile • Excel Formula Worksheet It is not necessary to put the data in ascending order. Note: Rows 7-71 are not shown.

  23. Using Excel’s Rank and Percentile Tool to Compute Percentiles and Quartiles • Excel Value Worksheet Note: Rows 7-71 are not shown.

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

  25. Third Quartile • Example: Apartment Rents Third quartile = 75th percentile i = (p/100)n = (75/100)70 = 52.5 = 53 Third quartile = 525 Note: Data is in ascending order.

  26. Lp = (p/100)n + (1 – p/100) Third Quartile • Using Excel’s Quartile Function Excel computes the locations of the 1st, 2nd, and 3rd quartiles by first converting the quartiles to percentiles and then using the following formula to compute the location (Lp) of the pth percentile: Excel would compute the location of the 3rd quartile (75th percentile) for the rent data as follows: L75 = (75/100)70 + (1 – 75/100) = 52.5 + .25 = 52.75 The 3rd quartile would be 515 + .75(525 - 515) = 515 + 7.5 = 522.5

  27. Third Quartile • Excel Formula Worksheet 3rd quartile It is not necessary to put the data in ascending order. Note: Rows 7-71 are not shown.

  28. Third Quartile • Excel Value Worksheet Note: Rows 7-71 are not shown.

  29. Excel’s Rank and Percentile Tool Step 1 Click the Data tab on the Ribbon Step 2 In the Analysis group, click Data Analysis Step 3 Choose Rank and Percentile from the list of Analysis Tools Step 4 When the Rank and Percentile dialog box appears (see details on next slide)

  30. Excel’s Rank and Percentile Tool Step 4 Complete the Rank and Percentile dialog box as follows:

  31. Excel’s Rank and Percentile Tool • Excel Value Worksheet Note: Rows 11-71 are not shown.

  32. Geometric Mean (GM) • The Geometric Mean is useful in finding the averages of increases in: • Percents • Ratios • Indexes • Growth Rates • The Geometric Mean will always be less than or equal to (never more than) the arithmetic mean • The GM gives a more conservative figure that is not drawn up by large values in the set

  33. Geometric Mean • The GM of a set of n positive numbers is defined as the nth root of the product of n values. The formula is:

  34. Geometric Mean Example 1:Percentage Increase

  35. Verify Geometric Mean Example

  36. Another Use Of GM:Ave. % Increase Over Time • Another use of the geometric mean is to determine the percent increase in sales, production or other business or economic series from one time period to another • Where n = number of periods

  37. Example for GM: Ave. % Increase Over Time • The total number of females enrolled in American colleges increased from 755,000 in 1992 to 835,000 in 2000. That is, the geometric mean rate of increase is 1.27%. • The annual rate of increase is 1.27% • For the years 1992 through 2000, the rate of female enrollment growth at American colleges was 1.27% per year

  38. 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.

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

  40. 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.

  41. Range • Example: Apartment Rents Range = largest value - smallest value Range = 615 - 425 = 190 Note: Data is in ascending order.

  42. 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.

  43. Interquartile Range • Example: Apartment Rents 3rd Quartile (Q3) = 525 1st Quartile (Q1) = 445 Interquartile Range = Q3 - Q1 = 525 - 445 = 80 Note: Data is in ascending order.

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

  45. Variance The variance is the average of the squared differences between each data value and the mean. The variance is computed as follows: for a sample for a population

  46. 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 interpreted than the variance.

  47. Standard Deviation The standard deviation is computed as follows: for a sample for a population

  48. Coefficient of Variation The coefficient of variation indicates how large the standard deviation is in relation to the mean. The coefficient of variation is computed as follows: for a sample for a population

  49. Sample Variance, Standard Deviation, And Coefficient of Variation • Example: Apartment Rents • Variance • Standard Deviation the standard deviation is about 11% of the mean • Coefficient of Variation

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