2. Descriptive Statistics (Part 1) Numerical Description Numerical Description Central Tendency Central Tendency Dispersion Chapter 44

3. Statistics are descriptive measures derived from a sample (n items).Statistics are descriptive measures derived from a sample (n items). Parameters are descriptive measures derived from a population (N items).Parameters are descriptive measures derived from a population (N items). Numerical Description

4. Three key characteristics of numerical data:Three key characteristics of numerical data: CharacteristicInterpretation Central Tendency Where are the data values concentrated? What seem to be typical or middle data values? Numerical Description Dispersion How much variation is there in the data? How spread out are the data values? Are there unusual values? Shape Are the data values distributed symmetrically? Skewed? Sharply peaked? Flat? Bimodal?

5. Numerical statistics can be used to summarize this random sample of brands.Numerical statistics can be used to summarize this random sample of brands. Defect rate = total no. defectsDefect rate = total no. defects no. inspected x 100 Must allow for sampling error since the analysis is based on sampling.Must allow for sampling error since the analysis is based on sampling. Numerical Description  Example: Vehicle Quality Consider the data set of vehicle defect rates from J. D. Power and Associates.Consider the data set of vehicle defect rates from J. D. Power and Associates.

6. Numerical Description Number of defects per 100 vehicles, 1004 models.Number of defects per 100 vehicles, 1004 models.

7. To begin, sort the data in Excel.

8. Sorted data provides insight into central tendency and dispersion.Sorted data provides insight into central tendency and dispersion. Numerical Description

9. The dot plot offers a visual impression of the data.The dot plot offers a visual impression of the data.  Visual Displays Numerical Description

10. Histograms with 5 bins (suggested by Sturge’s Rule) and 10 bins are shown below.Histograms with 5 bins (suggested by Sturge’s Rule) and 10 bins are shown below. Both are symmetric with no extreme values and show a modal class toward the low end.Both are symmetric with no extreme values and show a modal class toward the low end.  Visual Displays Numerical Description

11. Descriptive Statistics in Excel Go to Tools | Data Analysis and select Descriptive Statistics

12. Highlight the data range, specify a cell for the upper-left corner of the output range, check Summary Statistics and click OK.

13. Here is the resulting analysis.

14. Descriptive Statistics in MegaStat

15. Here is the resulting MegaStat analysis:

16. The central tendency is the middle or typical values of a distribution.The central tendency is the middle or typical values of a distribution. Central tendency can be assessed using a dot plot, histogram or more precisely with numerical statistics.Central tendency can be assessed using a dot plot, histogram or more precisely with numerical statistics. Central Tendency

17. StatisticFormula Excel Formula ProCon Mean=AVERAGE(Data) Familiar and uses all the sample information. Influenced by extreme values. Central Tendency  Six Measures of Central Tendency Median Middle value in sorted array =MEDIAN(Data) Robust when extreme data values exist. Ignores extremes and can be affected by gaps in data values.

18. StatisticFormula Excel Formula ProCon Mode Most frequently occurring data value =MODE(Data) Useful for attribute data or discrete data with a small range. May not be unique, and is not helpful for continuous data. Central Tendency  Six Measures of Central Tendency Midrange=0.5*(MIN(Data)+MAX(Data)) Easy to understand and calculate. Influenced by extreme values and ignores most data values.

19. StatisticFormula Excel Formula ProCon Geometric mean (G) =GEOMEAN(Data) Useful for growth rates and mitigates high extremes. Less familiar and requires positive data. Trimmed mean Same as the mean except omit highest and lowest k% of data values (e.g., 5%) =TRMEAN(Data, %) Mitigates effects of extreme values. Excludes some data values that could be relevant. Central Tendency  Six Measures of Central Tendency

20. A familiar measure of central tendency.A familiar measure of central tendency. In Excel, use function =AVERAGE(Data) where Data is an array of data values.In Excel, use function =AVERAGE(Data) where Data is an array of data values. Population FormulaSample Formula Central Tendency  Mean

21. For the sample of n = 37 car brands:For the sample of n = 37 car brands: Central Tendency  Mean

22. Arithmetic mean is the most familiar average.Arithmetic mean is the most familiar average. Affected by every sample item.Affected by every sample item. The balancing point or fulcrum for the data.The balancing point or fulcrum for the data. Central Tendency  Characteristics of the Mean

23. Regardless of the shape of the distribution, absolute distances from the mean to the data points always sum to zero.Regardless of the shape of the distribution, absolute distances from the mean to the data points always sum to zero. Central Tendency  Characteristics of the Mean Consider the following asymmetric distribution of quiz scores whose mean = 65.Consider the following asymmetric distribution of quiz scores whose mean = 65. = (42 – 65) + (60 – 65) + (70 – 65) + (75 – 65) + (78 – 65) = (-23) + (-5) + (5) + (10) + (13) = -28 + 28 = 0

24. The median (M) is the 50 th percentile or midpoint of the sorted sample data.The median (M) is the 50 th percentile or midpoint of the sorted sample data. M separates the upper and lower half of the sorted observations.M separates the upper and lower half of the sorted observations. If n is odd, the median is the middle observation in the data array.If n is odd, the median is the middle observation in the data array. If n is even, the median is the average of the middle two observations in the data array.If n is even, the median is the average of the middle two observations in the data array. Central Tendency  Median

25. Central Tendency  Median For n = 8, the median is between the fourth and fifth observations in the data array.For n = 8, the median is between the fourth and fifth observations in the data array.

26. Central Tendency  Median For n = 9, the median is the fifth observation in the data array.For n = 9, the median is the fifth observation in the data array.

27. Consider the following n = 6 data values: 11 12 15 17 21 32Consider the following n = 6 data values: 11 12 15 17 21 32 What is the median?What is the median? M = (x 3 +x 4 )/2 = (15+17)/2 = 16 11 12 15 16 17 21 32 For even n, Median = n/2 = 6/2 = 3 and n/2+1 = 6/2 + 1 = 4 Central Tendency  Median

28. Consider the following n = 7 data values: 12 23 23 25 27 34 41Consider the following n = 7 data values: 12 23 23 25 27 34 41 What is the median?What is the median? M = x 4 = 25 12 23 23 25 27 34 41 For odd n, Median = (n+1)/2 = (7+1)/2 = 8/2 = 4 Central Tendency  Median

29. Use Excel’s function =MEDIAN(Data) where Data is an array of data values.Use Excel’s function =MEDIAN(Data) where Data is an array of data values. For the 37 vehicle quality ratings (odd n) the position of the median is (n+1)/2 = (37+1)/2 = 19.For the 37 vehicle quality ratings (odd n) the position of the median is (n+1)/2 = (37+1)/2 = 19. So, the median is x 19 = 121.So, the median is x 19 = 121. When there are several duplicate data values, the median does not provide a clean “50-50” split in the data.When there are several duplicate data values, the median does not provide a clean “50-50” split in the data. Central Tendency  Median

30. The median is insensitive to extreme data values.The median is insensitive to extreme data values. For example, consider the following quiz scores for 3 students:For example, consider the following quiz scores for 3 students: Tom’s scores: 20, 40, 70, 75, 80 Mean =57, Median = 70, Total = 285 Jake’s scores: 60, 65, 70, 90, 95 Mean = 76, Median = 70, Total = 380 Mary’s scores: 50, 65, 70, 75, 90 Mean = 70, Median = 70, Total = 350 What does the median for each student tell you?What does the median for each student tell you? Central Tendency  Characteristics of the Median

31. The most frequently occurring data value.The most frequently occurring data value. Similar to mean and median if data values occur often near the center of sorted data.Similar to mean and median if data values occur often near the center of sorted data. May have multiple modes or no mode.May have multiple modes or no mode. Central Tendency  Mode

32. Lee’s scores: 60, 70, 70, 70, 80Mean =70, Median = 70, Mode = 70 Pat’s scores: 45, 45, 70, 90, 100Mean = 70, Median = 70, Mode = 45 Sam’s scores: 50, 60, 70, 80, 90Mean = 70, Median = 70, Mode = none Xiao’s scores: 50, 50, 70, 90, 90Mean = 70, Median = 70, Modes = 50,90 Central Tendency  Mode For example, consider the following quiz scores for 3 students:For example, consider the following quiz scores for 3 students: What does the mode for each student tell you?What does the mode for each student tell you?

33. Easy to define, not easy to calculate in large samples.Easy to define, not easy to calculate in large samples. Use Excel’s function =MODE(Array) - will return #N/A if there is no mode. - will return first mode found if multimodal.Use Excel’s function =MODE(Array) - will return #N/A if there is no mode. - will return first mode found if multimodal. May be far from the middle of the distribution and not at all typical.May be far from the middle of the distribution and not at all typical. Central Tendency  Mode

34. Generally isn’t useful for continuous data since data values rarely repeat.Generally isn’t useful for continuous data since data values rarely repeat. Best for attribute data or a discrete variable with a small range (e.g., Likert scale).Best for attribute data or a discrete variable with a small range (e.g., Likert scale). Central Tendency  Mode

35. Consider the following P/E ratios for a random sample of 68 Standard & Poor’s 500 stocks.Consider the following P/E ratios for a random sample of 68 Standard & Poor’s 500 stocks. What is the mode?What is the mode? Central Tendency  Example: Price/Earnings Ratios and Mode 78810 1213 14 15 16 1718 19 20 21 22 23 242526 2729 303134363740414548556891

36. Excel’s descriptive statistics results are:Excel’s descriptive statistics results are: The mode 13 occurs 7 times, but what does the dot plot show?The mode 13 occurs 7 times, but what does the dot plot show? Mean22.7206 Median19 Mode13 Range84 Minimum7 Maximum91 Sum1545 Count68 Central Tendency  Example: Price/Earnings Ratios and Mode

37. The dot plot shows local modes (a peak with valleys on either side) at 10, 13, 15, 19, 23, 26, 29.The dot plot shows local modes (a peak with valleys on either side) at 10, 13, 15, 19, 23, 26, 29. These multiple modes suggest that the mode is not a stable measure of central tendency.These multiple modes suggest that the mode is not a stable measure of central tendency. Central Tendency  Example: Price/Earnings Ratios and Mode

38. Points scored by the winning NCAA football team tends to have modes in multiples of 7 because each touchdown yields 7 points.Points scored by the winning NCAA football team tends to have modes in multiples of 7 because each touchdown yields 7 points. Central Tendency  Example: Rose Bowl Winners’ Points Consider the dot plot of the points scored by the winning team in the first 87 Rose Bowl games.Consider the dot plot of the points scored by the winning team in the first 87 Rose Bowl games. What is the mode?What is the mode?

39. A bimodal distribution refers to the shape of the histogram rather than the mode of the raw data.A bimodal distribution refers to the shape of the histogram rather than the mode of the raw data. Occurs when dissimilar populations are combined in one sample. For example,Occurs when dissimilar populations are combined in one sample. For example, Central Tendency  Mode

40. Compare mean and median or look at histogram to determine degree of skewness.Compare mean and median or look at histogram to determine degree of skewness. Central Tendency  Skewness

41. Distribution’s Shape Histogram Appearance Statistics Skewed left (negative skewness) Long tail of histogram points left (a few low values but most data on right) Mean < Median Central Tendency  Symptoms of Skewness Symmetric Tails of histogram are balanced (low/high values offset) (low/high values offset) Mean  Median Skewed right (positive skewness) Long tail of histogram points right (most data on left but a few high values) Mean > Median

42. For the sample of J.D. Power quality ratings, the mean (125.38) exceeds the median (121). What does this suggest?For the sample of J.D. Power quality ratings, the mean (125.38) exceeds the median (121). What does this suggest? Central Tendency  Skewness

43. The geometric mean (G) is a multiplicative average.The geometric mean (G) is a multiplicative average. For the J. D. Power quality data (n=37):For the J. D. Power quality data (n=37): In Excel use =GEOMEAN(Array)In Excel use =GEOMEAN(Array) The geometric mean tends to mitigate the effects of high outliers.The geometric mean tends to mitigate the effects of high outliers. Central Tendency  Geometric Mean

44. A variation on the geometric mean used to find the average growth rate for a time series.A variation on the geometric mean used to find the average growth rate for a time series. For example, from 1998 to 2002, Spirit Airlines revenues are:For example, from 1998 to 2002, Spirit Airlines revenues are: YearRevenue (mil) 1998131 1999227 2000311 2001354 2002403 Central Tendency  Growth Rates

45. The average growth rate is given by taking the geometric mean of the ratios of each year’s revenue to the preceding year.The average growth rate is given by taking the geometric mean of the ratios of each year’s revenue to the preceding year. Due to cancellations, only the first and last years are relevant:Due to cancellations, only the first and last years are relevant: = 1.242  1 =.242 or 24.2% per year = 1.242  1 =.242 or 24.2% per year In Excel use =(403/131)^(1/5)-1In Excel use =(403/131)^(1/5)-1 Central Tendency  Growth Rates

46. The midrange is the point halfway between the lowest and highest values of X.The midrange is the point halfway between the lowest and highest values of X. Easy to use but sensitive to extreme data values.Easy to use but sensitive to extreme data values. Midrange = For the J. D. Power quality data (n=37):For the J. D. Power quality data (n=37): Midrange = = Here, the midrange (130) is higher than the mean (125.38) or median (121).Here, the midrange (130) is higher than the mean (125.38) or median (121). Central Tendency  Midrange

47. To calculate the trimmed mean, first remove the highest and lowest k percent of the observations.To calculate the trimmed mean, first remove the highest and lowest k percent of the observations. For example, for the n = 68 P/E ratios, we want a 5 percent trimmed mean (i.e., k =.05).For example, for the n = 68 P/E ratios, we want a 5 percent trimmed mean (i.e., k =.05). To determine how many observations to trim, multiply k x n = 0.05 x 68 = 3.4 or 3 observations.To determine how many observations to trim, multiply k x n = 0.05 x 68 = 3.4 or 3 observations. So, we would remove the three smallest and three largest observations before averaging the remaining values.So, we would remove the three smallest and three largest observations before averaging the remaining values. Central Tendency  Trimmed Mean

48. Here is a summary of all the measures of central tendency for the n = 68 P/E values.Here is a summary of all the measures of central tendency for the n = 68 P/E values. The trimmed mean mitigates the effects of very high values, but still exceeds the median.The trimmed mean mitigates the effects of very high values, but still exceeds the median. Mean:22.72 =AVERAGE(PERatio) =AVERAGE(PERatio) Median:19.00 =MEDIAN(PERatio) =MEDIAN(PERatio) Mode:13.00 =MODE(PERatio) =MODE(PERatio) Geometric Mean: 19.85 =GEOMEAN(PERatio) =GEOMEAN(PERatio) Midrange:49.00 =(MIN(PERatio)+MAX(PERatio))/2 =(MIN(PERatio)+MAX(PERatio))/2 5% Trim Mean: 21.10 =TRIMMEAN(PERatio,0.1) =TRIMMEAN(PERatio,0.1) Central Tendency  Trimmed Mean

49. Central Tendency  Trimmed Mean The Federal Reserve uses a 16% trimmed mean to mitigate the effects of extremes in its analysis of the Consumer Price Index.The Federal Reserve uses a 16% trimmed mean to mitigate the effects of extremes in its analysis of the Consumer Price Index.

50. Variation is the “spread” of data points about the center of the distribution in a sample. Consider the following measures of dispersion:Variation is the “spread” of data points about the center of the distribution in a sample. Consider the following measures of dispersion: StatisticFormulaExcelProCon Range x max – x min =MAX(Data)- MIN(Data) Easy to calculate Sensitive to extreme data values. DispersionDispersion Variance (s 2 ) =VAR(Data) Plays a key role in mathematical statistics. Non-intuitive meaning.  Measures of Variation

51. StatisticFormulaExcelProCon Standard deviation (s) =STDEV(Data) Most common measure. Uses same units as the raw data ($, £, ¥, etc.). Non-intuitive meaning. DispersionDispersion  Measures of Variation Coef- ficient. of variation (CV) None Measures relative variation in percent so can compare data sets. Requires non- negative data.

52. StatisticFormulaExcelProCon Mean absolute deviation (MAD) =AVEDEV(Data) Easy to understand. Lacks “nice” theoretical properties. DispersionDispersion  Measures of Variation

53. The difference between the largest and smallest observation.The difference between the largest and smallest observation. Range = x max – x min For example, for the n = 68 P/E ratios,For example, for the n = 68 P/E ratios, Range = 91 – 7 = 84 DispersionDispersion  Range

54. The population variance (  2 ) is defined as the sum of squared deviations around the mean  divided by the population size.The population variance (  2 ) is defined as the sum of squared deviations around the mean  divided by the population size. For the sample variance (s 2 ), we divide by n – 1 instead of n, otherwise s 2 would tend to underestimate the unknown population variance  2.For the sample variance (s 2 ), we divide by n – 1 instead of n, otherwise s 2 would tend to underestimate the unknown population variance  2. DispersionDispersion  Variance

55. The square root of the variance.The square root of the variance. Units of measure are the same as X.Units of measure are the same as X. Population standard deviation Sample standard deviation Explains how individual values in a data set vary from the mean.Explains how individual values in a data set vary from the mean. DispersionDispersion  Standard Deviation

56. Excel’s built in functions areExcel’s built in functions are Statistic Excel population formula Excel sample formula Variance=VARP(Array)=VAR(Array) Standard deviation =STDEVP(Array)=STDEV(Array) DispersionDispersion  Standard Deviation

57. Consider the following five quiz scores for Stephanie.Consider the following five quiz scores for Stephanie. DispersionDispersion  Calculating a Standard Deviation

58. Now, calculate the sample standard deviation:Now, calculate the sample standard deviation: Somewhat easier, the two-sum formula can also be used:Somewhat easier, the two-sum formula can also be used: DispersionDispersion  Calculating a Standard Deviation

59. The standard deviation is nonnegative because deviations around the mean are squared.The standard deviation is nonnegative because deviations around the mean are squared. When every observation is exactly equal to the mean, the standard deviation is zero.When every observation is exactly equal to the mean, the standard deviation is zero. Standard deviations can be large or small, depending on the units of measure.Standard deviations can be large or small, depending on the units of measure. Compare standard deviations only for data sets measured in the same units and only if the means do not differ substantially.Compare standard deviations only for data sets measured in the same units and only if the means do not differ substantially. DispersionDispersion  Calculating a Standard Deviation

60. Useful for comparing variables measured in different units or with different means.Useful for comparing variables measured in different units or with different means. A unit-free measure of dispersionA unit-free measure of dispersion Expressed as a percent of the mean.Expressed as a percent of the mean. Only appropriate for nonnegative data. It is undefined if the mean is zero or negative.Only appropriate for nonnegative data. It is undefined if the mean is zero or negative. DispersionDispersion  Coefficient of Variation

61. For example:For example: Defect rates (n = 37) s = 22.89 = 125.38 gives CV = 100 × (22.89)/(125.38) = 18% ATM deposits (n = 100) s = 280.80 = 233.89 gives CV = 100 × (280.80)/(233.89) = 120% P/E ratios (n = 68) s = 14.28 = 22.72 gives CV = 100 × (14.08)/(22.72) = 62% DispersionDispersion  Coefficient of Variation

62. The Mean Absolute Deviation (MAD) reveals the average distance from an individual data point to the mean (center of the distribution).The Mean Absolute Deviation (MAD) reveals the average distance from an individual data point to the mean (center of the distribution). Uses absolute values of the deviations around the mean.Uses absolute values of the deviations around the mean. Excel’s function is =AVEDEV(Array)Excel’s function is =AVEDEV(Array) DispersionDispersion  Mean Absolute Deviation

63. Consider the histograms of hole diameters drilled in a steel plate during manufacturing.Consider the histograms of hole diameters drilled in a steel plate during manufacturing. The desired distribution is outlined in red.The desired distribution is outlined in red. DispersionDispersion Machine A Machine B  Central Tendency vs. Dispersion: Manufacturing

64. Desired mean (5mm) but too much variation. Acceptable variation but mean is less than 5 mm. Take frequent samples to monitor quality.Take frequent samples to monitor quality. Machine A Machine B DispersionDispersion  Central Tendency vs. Dispersion: Manufacturing

65. Consider student ratings of four professors on eight teaching attributes (10-point scale).Consider student ratings of four professors on eight teaching attributes (10-point scale). DispersionDispersion  Central Tendency vs. Dispersion: Job Performance

66. Jones and Wu have identical means but different standard deviations.Jones and Wu have identical means but different standard deviations. DispersionDispersion  Central Tendency vs. Dispersion: Job Performance

67. Smith and Gopal have different means but identical standard deviations.Smith and Gopal have different means but identical standard deviations. DispersionDispersion  Central Tendency vs. Dispersion: Job Performance

68. A high mean (better rating) and low standard deviation (more consistency) is preferred. Which professor do you think is best?A high mean (better rating) and low standard deviation (more consistency) is preferred. Which professor do you think is best? DispersionDispersion  Central Tendency vs. Dispersion: Job Performance

69. Applied Statistics in Business and Economics End of Part 1 of Chapter 4