1. 1 Describing distributions with numbers William P. Wattles Psychology 302

2. 2 Measuring the Center of a distribution n Mean – The arithmetic average – Requires measurement data n Median – The middle value n Mode – The most common value

3. 3 Measuring the center with the Mean

4. 4 Our first formula

5. 5 The Mean n One number that tells us about the middle using all the data. n The group not the individual has a mean.

6. Population Sample

7. 6 Sample mean

8. 7 Mu, the population mean

9. Population Sample

10. 8 Calculate the mean with Excel n Save the file psy302 to your hard drive –right click on the file –save to desktop or temp n Open file psy302 n Move flower trivia score to new sheet

11. 9 Calculate the mean with Excel n Rename Sheet – double click sheet tab, type flower n Calculate the sum – type label: total n Calculate the mean – type label: mean n Check with average function

12. 10 Measuring the center with the Median n Rank order the values n If the number of observations is odd the median is the center observation n If the number of observations is even the median is the mean of the middle two observations. (half way between them)

13. 11 Measuring the center with the Median

14. 12 The mean versus the median n The Mean – uses all the data – has arithmetic properties n The Median – less influenced by Outliers and extreme values

15. Mean vs. Median

16. 5 The Mean n The mean uses all the data. n The group not the individual has a mean. n We calculate the mean on Quantitative Data Three things to remember

17. n The mean tells us where the middle of the data lies. n We also need to know how spread out the data are.

19. Measuring Spread n Knowing about the middle only tells us part of the story of the data. n We need to know how spread out the data are.

20. Variability n Variety is the spice of life n Without variability things are just boring

21. Why is the mean alone not enough to describe a distribution? n Outliers is NOT the answer!!!!

22. The mean tells us the middle but not how spread out the scores are.

23. 14 Example of Spread n New York n mean annual high temperature 62

24. 14 Example of Spread n San Francisco n mean annual high temperature 65

26. 16 Example of Spread n New York n meanmaxminrangesd n 6284394517.1 n San Francisco n 65735518 6.4

27. Example of Variability

28. 17 Measuring Spread n Range n Quartiles n Five-number summary – Minimum – first quartile – median – third quartile – Maximum n Standard Deviation

29. n Mean 50.63% n Mean 33.19% Std Dev 21.4% Std Dev 13.2%

30. 19 Deviation score n Each individual has a deviation score. It measures how far that individual deviates from the mean. n Deviation scores always sum to zero. n Deviation scores contain information. – How far and in which direction the individual lies from the mean

31. 18 Measuring spread with the standard deviation n Measures spread by looking at how far the observations are from their mean. n The standard deviation is the square root of the variance. n The variance is also a measure of spread

33. Individual deviation scores

34. Standard deviation n One number that tells us about the spread using all the data. n The group not the individual has a standard deviation. Note !!

35. 23 Standard Deviation

36. 22 Variance

37. 24 Properties of the standard deviation n s measures the spread about the mean n s=0 only when there is no spread. This happens when all the observations have the same value. n s is strongly influenced by extreme values

38. n New Column headed deviation n Deviation score = X – the mean

39. 25 Calculate Standard Deviation with Excel n In new column type heading: dev2 n Enter formula to square deviation n Total squared deviations – type label: sum of squares n Divide sum of squares by n-1 – type label: variance

40. Moore page 50

42. n To Calculate Standard Deviation: n Total raw scores n divide by n to get mean n calculate deviation score for each subject (X minus the mean) n Square each deviation score n Sum the deviation scores to obtain sum of squares n Divide by n-1 to obtain variance n Take square root of variance to get standard deviation.

43. Population Sample

44. 26 Sample variance

45. 27 Population variance

46. Population Variance Sample Variance

47. 28 Little sigma, the Population standard deviation

48. 29 Sample standard deviation

49. Population Standard Deviation Sample Standard Deviation

51. To analyze data n 1. Make a frequency distribution and plot the data n Look for overall pattern and outliers or skewness n Create a numerical summary: mean and standard deviation.

52. 41 Start with a list of scores

53. 42 Make a frequency distribution

54. 43 Frequency distribution

55. 44 Represent with a chart (histogram)

56. 45 Represent with line chart

57. Density Curve n Replaces the histogram when we have many observations.

58. Transform a score n Hotel Atlantico n 200 pesos n Peso a unit of measure

59. Transform a score n 1 dollar = 28.38 pesos n 200/28.38=$7.05 n Dollar a unit of measure

60. 31 n standardized observations or values. n To standardize is to transform a score into standard deviation units. n Frequently referred to as z-scores n A z-score tells how many standard deviations the score or observation falls from the mean and in which direction

61. 32 Standard Scores (Z-scores) n individual scores expressed in terms of the mean and standard deviation of the sample or population. n Z = X minus the mean/standard deviation

62. 33 Z-score

63. 34 new symbols

64. 35 Calculate Z-scores for trivia data n Label column E as Z-score n Type formula deviation score/std dev n Make std dev reference absolute (use F4 to insert dollar signs) n Copy formula down. n Check: should sum to zero

65. File extensions n Word.doc n Excel.xls n Text files.txt

66. To view File extensions n Open Windows Explorer n Choose Tools/Folder Options/View n uncheck “hide extensions for known file types.

68. 37 Z Scores n Height of young women – Mean = 64 – Standard deviation = 2.7 n How tall in deviations is a woman 70 inches? n A woman 5 feet tall (60 inches) is how tall in standard deviations?

69. 38 Z scores n Height of young women – Mean = 64 – Standard deviation = 2.7 n How tall in deviations is a woman 70 inches? z = 2.22 n A woman 5 feet tall (60 inches) is how tall in standard deviations? z = -1.48

70. 39 Calculating Z scores

71. Calculating X from Z scores

72. 72 Types of data n Categorical or Qualitative data –Nominal: Assign individuals to mutually exclusive categories. F exhaustive: everyone is in one category –Ordinal: Involves putting individuals in rank order. Categories are still mutually exclusive and exhaustive, but the order cannot be changed.

73. 73 Types of data n Measurement or Quantitative Data –Interval data: There is a consistent interval or difference between the numbers. Zero point is arbitrary –Ratio data: Interval scale plus a meaningful zero. Zero means none. Weight, money and Celsius scales exemplify ratio data –Measurement data allows for arithmetic operations.

74. Review n Video2 Video2

75. 60 The End

76. Mean vs. Median