2. Data Summary Using Descriptive Measures Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

3. Types of Descriptive Measures Central Tendency Variation Position Shape Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

4. Measures of Central Tendency Mean Median Midrange Mode Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

5. The Mean The Mean is simply the average of the data. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

6. Sample Mean Each value in the sample is represented by x thus to get the mean simply add all the values in the sample and divide by the number of values in the sample Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

7. Accident Data Set Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

8. The Median The Median (Md) of a set of data is the value in the center of the data values when they are arranged from lowest to highest. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

9. Accident Data Ordered array: 5, 6, 7, 9, 23 The value that has an equal number of items to the right and left is the median. Thus Md = 7 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

10. The Median In general if n is odd, Md is the center data value of the ordered data set. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

11. Accident Data Ordered array: 5, 6, 7, 9, 23 Md  5  1 2       st ordered value = 3rd value Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

12. The Median If n is even, Md is the average of the two center values of the ordered data set. For the ordered data set: 3, 8, 12, 14 Md  8  12 2       = 10.0 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

13. The Midrange The Midrange (Mr) provides an easy-to- grasp measure of central tendency. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

14. Accident Data Mr  5  23 2 = 14.0 x  Md = 7 Note: that the Midrange is severely affected by outliers Compare: Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

15. The Mode The Mode (Mo) of a data set is the value that occurs more than once and the most often. The Mode is not always a measure of central tendency; this value need not occur in the center of the data. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

16. Level of Measurement and Measure of Central Tendency Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

17. Measures of Variation Homogeneity refers to the degree of similarity within a set of data. The more Homogeneous a set of data is, the better the mean will represent a typical value. Variation is the tendency of data values to scatter about the mean,. x Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

18. Common Measures of Variation Range Variance Standard Deviation Coefficient of Variation Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

19. The Range For the Accident data: Range = H - L = 23 - 5 = 18 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

20. The Variance and Standard Deviation Both measures describe the variation of the values about the mean. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

21. Accident Data Data Value(x - ) (x - ) 2 5 -5 25 6 -4 16 7 -3 9 9 -1 1 23 13 169 = 220 x x ( x – x ) 2  Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

22. Definition: Sample Variance Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

23. Definition: Sample Standard Deviation s  55.0  7.416 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

24. Definition: Population Variance Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

25. Definition: Population Standard Deviation  ( x –  ) 2  N Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

26. The Coefficient of Variation The Coefficient of Variation (CV) is used to compare the variation of two or more data sets where the values of the data differ greatly. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

27. Example Data Set 1: 5, 6, 7, 9, 23 Data Set 2: 5000, 6000, 7000, 9000, 23,000 CV  7.416 100 Data Set 1 10. = 74.16 CV  7,416 100 10,000. = 74.16 Data Set 2 Thus both data sets exhibit the same relative variation Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

28. Measures of Position Percentile (Quartile) Z Score Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

29. Percentile The 35th Percentile (P 35 ) is that value such that at most 35% of the data values are less than P 35 and at most 65% of the data values are greater than P 35. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

30. Percentile Texon Industries Data 17.5 represents the position of the 35th percentile Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

31. Percentile: Location Rules If n  P/100 is not a counting number, round it up, and the Pth percentile will be the value in this position of the ordered data. If n  P/100 is a counting number, the Pth percentile is the average of the number in this location (of the ordered data) and the number in the next largest location. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

32. Quartiles Quartiles are merely particular percentiles that divide the data into quarters, namely: Q 1 = 1st quartile = 25th percentile (P 25 ) Q 2 = 2nd quartile = 50th percentile (P 50 ) Q 3 = 3rd quartile = 75th percentile (P 75 ) Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

33. Z Scores Z score determines the relative position of any particular data value x and is based on the mean and standard deviation of the data set. The Z score is expresses the number of standard deviations the value x is from the mean. A negative Z score implies that x is to the left of the mean and a positive Z score implies that x is to the right of the mean. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

34. Z Score Equation Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

35. Measures of Shape Skewness Kurtosis Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

36. Skewness Skewness measures the tendency of a distribution to stretch out in a particular direction Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

37. Skewness In a symmetrical distribution the mean, median, and mode would all be the same value. Sk = 0 (fig 3.7) A positive Sk number implies a shape which is skewed right (fig3.8). The mode < median < mean In a data set with a negative Sk value (fig3.9) the mean < Median < Mode Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

38. Figure 3.7 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

39. Figure 3.8 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

40. Figure 3.9 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

41. Skewness Calculation Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

42. Kurtosis Kurtosis measures the peakedness of the distribution. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

43. Chebyshev’s Inequality At least 75% of the data values are between x - 2s and x + 2s or At least 75% of the data values have a Z score value between -2 and +2 At least 89% of the data values are between x - 3s and x + 3s In general, at least (1-1/k 2 ) x 100% of the data values lie between x - ks and x + ks for any k>1 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

44. Empirical Rule Under the assumption of a bell shaped population Approximately 68% of the data values lie between Approximately 95% of the data values lie between Approximately 99.7% of the data values lie between Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

45. Chebyshev’s versus Empirical Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

46. Grouped Data Approximations Where: f is the frequency of the class and m is the m is the midpoint of the class Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing