1. 1 Measures of Center

2. 2 Measure of Center  Measure of Center the value at the center or middle of a data set 1.Mean 2.Median 3.Mode 4.Midrange (rarely used)

3. 3 Mean  Arithmetic Mean (Mean) the measure of center obtained by adding the values and dividing the total by the number of values What most of us call an average.

4. 4 Notation ∑ denotes the sum of a set of values. x is the variable used to represent the individual data values. n represents the number of data values in a sample. N represents the number of data values in a population.

5. 5 µ is pronounced ‘mu’ and denotes the mean of all values in a population x = n ∑ x∑ x is pronounced ‘x-bar’ and denotes the mean of a set of sample values x N µ = ∑ x∑ x This is the sample mean This is the population mean

6. 6  Advantages Is relatively reliable. Takes every data value into account Mean  Disadvantage Is sensitive to every data value, one extreme value can affect it dramatically; is not a resistant measure of center

7. 7 Mean  Example  Major in Geography at University of North Carolina

8. 8 Median  Median the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude  often denoted by x (pronounced ‘x-tilde’) ~  is not affected by an extreme value - is a resistant measure of the center

9. 9 Finding the Median 1.If the number of data values is odd, the median is the value located in the exact middle of the list. 2.If the number of data values is even, the median is found by computing the mean of the two middle numbers. First sort the values (arrange them in order), then follow one of these rules:

10. 10 5.40 1.10 0.420.73 0.48 1.10 0.66 Example 1

11. 11 5.40 1.10 0.420.73 0.48 1.10 0.66 Example 1 0.42 0.48 0.660.73 1.10 1.10 5.40 Order from smallest to largest:

12. 12 5.40 1.10 0.420.73 0.48 1.10 0.66 Example 1 0.42 0.48 0.660.73 1.10 1.10 5.40 Order from smallest to largest: exact middle MEDIAN is 0.73

13. 13 5.40 1.10 0.420.73 0.48 1.10 Example 2

14. 14 5.40 1.10 0.420.73 0.48 1.10 Example 2 0.42 0.48 0.731.10 1.10 5.40 Order from smallest to largest:

15. 15 5.40 1.10 0.420.73 0.48 1.10 Example 2 0.42 0.48 0.731.10 1.10 5.40 Order from smallest to largest: Middle values

16. 16 5.40 1.10 0.420.73 0.48 1.10 Example 2 0.42 0.48 0.731.10 1.10 5.40 Order from smallest to largest: Middle values 0.73 + 1.10 2 = 0.915

17. 17 5.40 1.10 0.420.73 0.48 1.10 Example 2 0.42 0.48 0.731.10 1.10 5.40 Order from smallest to largest: Middle values MEDIAN is 0.915 0.73 + 1.10 2 = 0.915

18. 18 Mode  Mode the value that occurs with the greatest frequency  Data set can have one, more than one, or no mode Bimodal two data values occur with the same greatest frequency Multimodalmore than two data values occur with the same greatest frequency No Modeno data value is repeated

19. 19 a. 5.40 1.10 0.42 0.73 0.48 1.10 b. 27 27 27 55 55 55 88 88 99 c. 1 2 3 6 7 8 9 10 Mode - Examples

20. 20 a. 5.40 1.10 0.42 0.73 0.48 1.10 b. 27 27 27 55 55 55 88 88 99 c. 1 2 3 6 7 8 9 10 Mode - Examples  Mode is 1.10  Bimodal - 27 & 55  No Mode

21. 21  Midrange  the value midway between the maximum and minimum values in the original data set Definition Midrange = maximum value + minimum value 2

22. 22  Sensitive to extremes because it uses only the maximum and minimum values.  Midrange is rarely used in practice Midrange

23. 23 Carry one more decimal place than is present in the original set of values. Round-off Rule for Measures of Center

24. 24 Common Distributions

25. 25  Symmetric distribution of data is symmetric if the left half of its histogram is roughly a mirror image of its right half  Skewed distribution of data is skewed if it is not symmetric and extends more to one side than the other Skewed and Symmetric

26. 26 Symmetry and skewness

27. 27 Measures of Variation

28. 28 Measures of Variation spread, variability of data width of a distribution 1.Standard deviation 2.Variance 3.Range (rarely used)

29. 29 Standard deviation The standard deviation of a set of sample values, denoted by s, is a measure of variation of values about the mean.

30. 30 Sample Standard Deviation Formula Σ (x – x) 2 n – 1 s =s =

31. 31 Sample Standard Deviation (Shortcut Formula) n (n – 1) s =s = nΣ ( x 2 ) – (Σx) 2

32. 32 Σ (x – µ) Population Standard Deviation 2 N σ = σ is pronounced ‘sigma’ This formula only has a theoretical significance, it cannot be used in practice.

33. 33 Example Values: 1, 3, 14 Find the sample standard deviation: Find the population standard deviation:

34. 34 Example Values: 1, 3, 14 Find the sample standard deviation: s = 7.0 Find the population standard deviation: σ = 5.7

35. 35  Population variance: σ 2 - Square of the population standard deviation σ Variance  The variance is a measure of variation equal to the square of the standard deviation.  Sample variance: s 2 - Square of the sample standard deviation s

36. 36 Variance - Notation s = sample standard deviation s 2 = sample variance σ = population standard deviation σ 2 = population variance

37. 37 Example Values: 1, 3, 14 s = 7.0 s 2 = 49.0 σ = 5.7 σ 2 = 32.7

38. 38 Range (Rarely used) The difference between the maximum data value and the minimum data value. Range = (maximum value) – (minimum value) It is very sensitive to extreme values; therefore range is not as useful as the other measures of variation.

39. 39 Using Excel

40. 40 Using Excel Enter values into first column

41. 41 Using Excel In C1, type “=average(a1:a6)”

42. 42 Using Excel Then, Enter

43. 43 Using Excel Same thing with “=stdev(a1:a6)”

44. 44 Using Excel Same with “=median(a1:a6)” - and add some labels

45. 45 Using Excel Same with min, max, and mode

46. 46 Usual and Unusual Events

47. 47 Usual values in a data set are those that are typical and not too extreme. Minimum usual value = (mean) – 2 * (standard deviation) Maximum usual value = (mean) + 2 * (standard deviation)

48. 48 Usual values in a data set are those that are typical and not too extreme.

49. 49 Rule of Thumb Based on the principle that for many data sets, the vast majority (such as 95%) of sample values lie within two standard deviations of the mean. A value is unusual if it differs from the mean by more than two standard deviations.

50. 50 Empirical (or 68-95-99.7) Rule For data sets having a distribution that is approximately bell shaped, the following properties apply:  About 68% of all values fall within 1 standard deviation of the mean.  About 95% of all values fall within 2 standard deviations of the mean.  About 99.7% of all values fall within 3 standard deviations of the mean.

51. 51 The Empirical Rule

52. 52 The Empirical Rule

53. 53 The Empirical Rule

54. 54 Measures of Relative Standing

55. 55  Z-score (or standardized value) T he number of standard deviations that a given value x is above or below the mean Z-score

56. 56 Sample Population x – µ z = σ Round z scores to 2 decimal places Measure of Position: Z-score z = x – x s

57. 57 Interpreting Z-scores Whenever a value is less than the mean, its corresponding z score is negative Ordinary values: –2 ≤ Z-score ≤ 2 Unusual values: Z-score 2

58. 58 Percentiles Measures of location. There are 99 percentiles denoted P 1, P 2,... P 99, which divide a set of data into 100 groups with about 1% of the values in each group.

59. 59 Finding the Percentile of a Data Value Percentile of value x = 100 number of values less than x total number of values Round it off to the nearest whole number

60. 60 Example 2, pg 116 35 sorted values: Find the percentile of 29 4.556.5720 29 303540 415052 606568 70 72747580100113116 120125132150160200225

61. 61 Example 2, pg 116 35 sorted values: Find the percentile of 29 Percentile of 29 = 17 (rounded) 4.556.5720 29 303540 415052 606568 70 72747580100113116 120125132150160200225

62. 62 n total number of values in the data set k percentile being used Llocator that gives the position of a value P k kth percentile L = n k 100 Notation Converting from the kth Percentile to the Corresponding Data Value

63. 63 Example 3, pg 116 35 sorted values: Find P 60 4.556.5720 29 303540 415052 606568 70 72747580100113116 120125132150160200225

64. 64 Example 3, pg 116 35 sorted values: Find P 60 P 60 = 71 4.556.5720 29 303540 415052 606568 70 72747580100113116 120125132150160200225

65. 65 Converting from the kth Percentile to the Corresponding Data Value

66. 66 Quartiles  Q 1 (First Quartile) separates the bottom 25% of sorted values from the top 75%.  Q 2 (Second Quartile) same as the median; separates the bottom 50% of sorted values from the top 50%.  Q 3 (Third Quartile) separates the bottom 75% of sorted values from the top 25%. Measures of location, denoted Q 1, Q 2, and Q 3, which divide a set of data into four groups with about 25% of the values in each group.

67. 67 Quartiles To calculate the quartile for homework and other CourseCompass work, using Excel: 1.Sort the data 2.Enter =quartile(,1) 3.Find the result in the sorted data 4.If the result is not in the sorted data, go to the next higher value

68. 68 Example - Quartile =quartile(A1:G5,1) give 37.5 37.5 is between 35 and 40 The 1 st quartile value is 40 4.556.5720 29 303540 415052 606568 70 72747580100113116 120125132150160200225

69. 69 Q 1, Q 2, Q 3 divide ranked scores into four equal parts Quartiles 25% Q3Q3 Q2Q2 Q1Q1 (minimum)(maximum) (median)

70. 70  Interquartile Range (or IQR): Q 3 – Q 1  10 - 90 Percentile Range: P 90 – P 10  Semi-interquartile Range: 2 Q 3 – Q 1  Midquartile: 2 Q 3 + Q 1 Some Other Statistics

71. 71 For a set of data, the 5-number summary consists of the ● minimum value ●first quartile Q 1 ●median (or second quartile Q 2 ) ●third quartile, Q 3 ●maximum value. 5-Number Summary

72. 72 Example 35 sorted values: Find the 5-number summary 4.556.5720 29 303540 415052 606568 70 72747580100113116 120125132150160200225

73. 73 Example Min = 4.5 Q1 = 40 Median = 50 Q3 = 1130 Max = 225