1. Statistics Variability

2. Descriptive Statistics
Averages tell where the data tends to pile up

3. Descriptive Statistics
Another good way to describe data is how spread out it is

5. Descriptive Statistics
Suppose you are using the mean “5” to describe each of the observations in your sample

6. For which sample would “5” be closer to the actual data values?
VARIABILITY
IN-CLASS PROBLEM
For which sample would “5” be closer to the actual data values?

7. VARIABILITY
IN-CLASS PROBLEM
In other words, for which of the two sets of data would the mean be a better descriptor?

8. VARIABILITY
IN-CLASS PROBLEM
For which of the two sets of data would the mean be a better descriptor?

9. Variability
Numbers telling how spread out our data values are are called “Measures of Variability”

10. Variability
The variability tells how close to the “average” the sample data tend to be

11. Variability
Just like measures of central tendency, there are several measures of variability

12. Variability
Range = R = max – min

13. Variability
Variance (symbolized s2) s2 =
sum of (obs – x )2
n - 1

14. Variability
An observation “x” minus the mean x is called a “deviation” The variance is sort of an average (arithmetic mean) of the squared deviations

15. Variability
In algebra, the absolute value of “deviations” are a measure of distance

16. Variability
We square them because it gets rid of the “+” “-” problem and has mathematical advantages over taking absolute values

17. Variability
Sums of squared deviations are used in the formula for a circle: r2 = (x-h)2 + (y-k)2 where r is the radius of the circle and (h,k) is its center

18. Variability
OK… so if its sort of an arithmetic mean, howcum is it divided by “n-1” not “n”?

19. Variability
Every time we estimate something in the population using our sample we have used up a bit of the “luck” that we had in getting a (hopefully) representative sample

20. Variability
To make up for that, we give a little edge to the opposing side of the story

21. Variability
Since a small variability means our sample arithmetic mean is a better estimate of the population mean than a large variability is, we bump up our estimate of variability a tad to make up for it

22. Variability
Dividing by “n” would give us a smaller variance than dividing by “n-1”, so we use that

23. Variability
Why not “n-2”?

24. Variability
Why not “n-2”? Because we only have used 1 estimate to calculate the variance: x

25. Variability
So, the variance is sort of an average (arithmetic mean) of the squared deviations bumped up a tad to make up for using an estimate ( x ) of the population mean (μ)

26. Variability
Trust me…

27. Variability
Standard deviation (symbolized “s” or “std”) s = variance

28. Variability
The standard deviation is an average square root of a sum of squared deviations We’ve used this in algebra class for distance calculations: d = (x1−x2)2 + (y1−y2)2

29. Variability
The range and standard deviation are in the same units as the original data (a good thing) The variance is in squared units (which can be confusing…)

30. Variability
Naturally, the measure of variability used most often is the hard-to-calculate one…

31. Variability
Naturally, the measure of variability used most often is the hard-to-calculate one… … the standard deviation

32. Variability
Statisticians like it because it is an average distance of all of the data from the center – the arithmetic mean

33. Variability
Range = max – min Variance = sum of (obs – x )2 n − 1 s = variance

34. Questions?

35. Variability
Range = max – min Variance = sum of (obs – x )2 n − 1 s = variance

36. VARIABILITY
IN-CLASS PROBLEM
Range = max – min Variance = sum of (obs – x )2 n − 1 s = variance Data: What is the range?

37. VARIABILITY
IN-CLASS PROBLEM
Range = max – min Variance = sum of (obs – x )2 n − 1 s = variance Data: Range = 3 – 1 = 2
Min Max

38. VARIABILITY
IN-CLASS PROBLEM
Range = max – min Variance = sum of (obs – x )2 n − 1 s = variance Data: What is the Variance?

39. VARIABILITY
IN-CLASS PROBLEM
Range = max – min Variance = sum of (obs – x )2 n − 1 s = variance Data: First find x !

40. VARIABILITY
IN-CLASS PROBLEM
Range = max – min Variance = sum of (obs – x )2 n − 1 s = variance Data: x = = 2

41. VARIABILITY
IN-CLASS PROBLEM
Range = max – min Variance = sum of (obs – x )2 n − 1 s = variance Data: Now calculate the deviations!

42. VARIABILITY
IN-CLASS PROBLEM
Range = max – min Variance = sum of (obs – x )2 n − 1 s = variance Data: Dev: 1-2=-1 1-2=-1 2-2=0 2-2=0 3-2=1 3-2=1

43. Variability
What do you get if you add up all of the deviations? Data: Dev: 1-2=-1 1-2=-1 2-2=0 2-2=0 3-2=1 3-2=1

44. Variability
Zero!

45. Variability
Zero! That’s true for ALL deviations everywhere in all times!

46. Variability
Zero! That’s true for ALL deviations everywhere in all times! That’s why they are squared in the sum of squares!

47. VARIABILITY
IN-CLASS PROBLEM
Range = max – min Variance = sum of (obs – x )2 n − 1 s = variance Data: Dev: -12=1 -12=1 02=0 02=0 12=1 12=1

48. VARIABILITY
IN-CLASS PROBLEM
Range = max – min Variance = sum of (obs – x )2 n − 1 s = variance Data: sum(obs– x )2: = 4

49. VARIABILITY
IN-CLASS PROBLEM
Range = max – min Variance = sum of (obs – x )2 n − 1 s = variance Data: Variance: 4/(6-1) = 4/5 = 0.8

50. YAY!

51. VARIABILITY
IN-CLASS PROBLEM
Range = max – min Variance = sum of (obs – x )2 n − 1 s = variance Data: What is s?

52. VARIABILITY
IN-CLASS PROBLEM
Range = max – min Variance = sum of (obs – x )2 n − 1 s = variance Data: s = 0.8 ≈ 0.89

53. VARIABILITY
IN-CLASS PROBLEM
So, for: Data: Range = max – min = 2 Variance = sum of (obs – x )2 n − 1 = 0.8 s = variance ≈ 0.89

54. Variability
Aren’t you glad Excel does all this for you???

55. Questions?

56. Variability
Just like for n and N and x and μ there are population variability symbols, too!

57. Variability
Naturally, these are going to have funny Greek-y symbols just like the averages …

58. Variability
The population variance is “σ2” called “sigma-squared” The population standard deviation is “σ” called “sigma”

59. Variability
Again, the sample statistics s2 and s values estimate population parameters σ2 and σ (which are unknown)

60. Variability
Some calculators can find x s and σ for you (Not recommended for large data sets – use EXCEL)

61. Variability
s sq “s2” vs sigma sq “σ2”

62. Variability
s2 is divided by “n-1” σ2 is divided by “N”

63. Questions?

64. Standard Deviation
What does standard deviation mean?

65. STANDARD DEVIATION
IN-CLASS PROBLEM
Suppose we have two pizza delivery drivers We’re going to give one a raise But who?

66. STANDARD DEVIATION
IN-CLASS PROBLEM
Both have the same mean delivery time of 15 minutes but Amanda’s standard deviation of delivery times = 2.6 minutes while Bethany’s standard deviation of delivery times = 8.4 minutes.

67. Who should get the raise?
STANDARD DEVIATION
IN-CLASS PROBLEM
Who should get the raise?

68. STANDARD DEVIATION
IN-CLASS PROBLEM
What are the advantages of having a data set that has a small standard deviation?

69. Questions?

70. Variability
Outliers! They can really affect your statistics!

71. Suppose we originally had data: 1 1 2 3 5 Suppose we now have data:
OUTLIERS
IN-CLASS PROBLEM
Suppose we originally had data:
Suppose we now have data:
Is the mode affected?

72. Suppose we originally had data: 1 1 2 3 5 Suppose we now have data:
OUTLIERS
IN-CLASS PROBLEM
Suppose we originally had data:
Suppose we now have data:
Original mode: 1
New mode: 1

73. Suppose we originally had data: 1 1 2 3 5 Suppose we now have data:
OUTLIERS
IN-CLASS PROBLEM
Suppose we originally had data:
Suppose we now have data:
Is the median affected?

74. Suppose we originally had data: 1 1 2 3 5 Suppose we now have data:
OUTLIERS
IN-CLASS PROBLEM
Suppose we originally had data:
Suppose we now have data:
Original median: 2
New median: 2

75. Suppose we originally had data: 1 1 2 3 5 Suppose we now have data:
OUTLIERS
IN-CLASS PROBLEM
Suppose we originally had data:
Suppose we now have data:
Is the mean affected?

76. OUTLIERS
IN-CLASS PROBLEM
Suppose we originally had data: Suppose we now have data: Original mean: 2.4 New mean: 149.6

77. Outliers!
How about measures of variability?

78. Suppose we originally had data: 1 1 2 3 5 Suppose we now have data:
OUTLIERS
IN-CLASS PROBLEM
Suppose we originally had data:
Suppose we now have data:
Is the range affected?

79. OUTLIERS
IN-CLASS PROBLEM
Suppose we originally had data: Suppose we now have data: Original range: 4 New range: 740

80. Suppose we originally had data: 1 1 2 3 5 Suppose we now have data:
OUTLIERS
IN-CLASS PROBLEM
Suppose we originally had data:
Suppose we now have data:
Is the standard deviation affected?

81. OUTLIERS
IN-CLASS PROBLEM
Suppose we originally had data: Suppose we now have data: Original s: ≈1.7 New s: ≈330.6

82. What advantages does the standard deviation have over the range?
IN-CLASS PROBLEM
What advantages does the standard deviation have over the range?

83. Questions?