1. Welcome to Week 03 Tues MAT135 Statistics

2. Review

3. Types of Statistics
Descriptive statistics – describe our sample – we’ll use this to make inferences about the population Inferential statistics – make inferences about the population with a level of probability attached

4. Descriptive Statistics
graphs n max min each observation frequencies

5. Descriptive Statistics
And… averages!

6. Measures of Central Tendency
In statistics class, averages are called “measures of central tendency” (where the data “tend to center”)

7. Measures of Central Tendency
arithmetic mean (sample x , population μ) x is your best estimate of the population mean μ

8. Measures of Central Tendency
The arithmetic mean is the balance point for a data set

9. Measures of Central Tendency3 4 5 8 9
The median (50th percentile) is also a measure of central tendency (aka: average)

10. Measures of Central Tendency
The mode (the most commonly-occurring observation) is another!

11. Statistics vs Parameters
Statistic Parameter n N x μ

12. Questions?

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

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

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

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

18. VARIABILITY
IN-CLASS PROBLEM 5
In other words, for which of the two sets of data would the mean be a better descriptor?

19. VARIABILITY
IN-CLASS PROBLEM 6
For which of the two sets of data would the mean be a better descriptor?

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

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

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

23. Variability
Range = R = max – min

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

25. 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

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

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

28. 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

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

30. 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

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

32. 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

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

34. Variability
Why not “n-2”?

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

36. 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)

37. Variability
Trust me…

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

39. 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

40. 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…)

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

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

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

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

45. Questions?

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

47. VARIABILITY
IN-CLASS PROBLEM 7
Range = max – min Variance = sum of (obs – x )2 n − 1 s = variance Data: What is the range?

48. VARIABILITY
IN-CLASS PROBLEM 7
Range = max – min Variance = sum of (obs – x )2 n − 1 s = variance Data: Range = 3 – 1 = 2
Min Max

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

50. VARIABILITY
IN-CLASS PROBLEM 8
Range = max – min Variance = sum of (obs – x )2 n − 1 s = variance Data: First find x !

51. VARIABILITY
IN-CLASS PROBLEM 8
Range = max – min Variance = sum of (obs – x )2 n − 1 s = variance Data: x = = 2

52. VARIABILITY
IN-CLASS PROBLEM 9
Range = max – min Variance = sum of (obs – x )2 n − 1 s = variance Data: Now calculate the deviations!

53. VARIABILITY
IN-CLASS PROBLEM 9
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

54. 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

55. Variability
Zero!

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

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

58. VARIABILITY
IN-CLASS PROBLEM 10
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

59. VARIABILITY
IN-CLASS PROBLEM 11
Range = max – min Variance = sum of (obs – x )2 n − 1 s = variance Data: sum(obs– x )2: = 4

60. VARIABILITY
IN-CLASS PROBLEM 12a,b
Range = max – min Variance = sum of (obs – x )2 n − 1 s = variance Data: Variance: 4/(6-1) = 4/5 = 0.8

61. YAY!

62. VARIABILITY
IN-CLASS PROBLEM 13
Range = max – min Variance = sum of (obs – x )2 n − 1 s = variance Data: What is s?

63. VARIABILITY
IN-CLASS PROBLEM 13
Range = max – min Variance = sum of (obs – x )2 n − 1 s = variance Data: s = 0.8 ≈ 0.89

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

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

66. Questions?

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

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

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

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

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

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

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

74. Questions?

75. Standard Deviation
What does standard deviation mean?

76. STANDARD DEVIATION
IN-CLASS PROBLEM 14
Suppose we have two pizza delivery drivers We’re going to give one a raise But who?

77. STANDARD DEVIATION
IN-CLASS PROBLEM 14
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.

78. Who should get the raise?
STANDARD DEVIATION
IN-CLASS PROBLEM 14
Who should get the raise?

79. STANDARD DEVIATION
IN-CLASS PROBLEM 15
What are the advantages of having a data set that has a small standard deviation?

80. Questions?

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

82. Suppose we originally had data: 1 1 2 3 5 Suppose we now have data:
OUTLIERS
IN-CLASS PROBLEM 16
Suppose we originally had data:
Suppose we now have data:
Is the mode affected?

83. Suppose we originally had data: 1 1 2 3 5 Suppose we now have data:
OUTLIERS
IN-CLASS PROBLEM 16
Suppose we originally had data:
Suppose we now have data:
Original mode: 1
New mode: 1

84. Suppose we originally had data: 1 1 2 3 5 Suppose we now have data:
OUTLIERS
IN-CLASS PROBLEM 17
Suppose we originally had data:
Suppose we now have data:
Is the median affected?

85. Suppose we originally had data: 1 1 2 3 5 Suppose we now have data:
OUTLIERS
IN-CLASS PROBLEM 17
Suppose we originally had data:
Suppose we now have data:
Original median: 2
New median: 2

86. Suppose we originally had data: 1 1 2 3 5 Suppose we now have data:
OUTLIERS
IN-CLASS PROBLEM 18
Suppose we originally had data:
Suppose we now have data:
Is the mean affected?

87. OUTLIERS
IN-CLASS PROBLEM 18
Suppose we originally had data: Suppose we now have data: Original mean: 2.4 New mean: 149.6

88. Outliers!
How about measures of variability?

89. Suppose we originally had data: 1 1 2 3 5 Suppose we now have data:
OUTLIERS
IN-CLASS PROBLEM 19
Suppose we originally had data:
Suppose we now have data:
Is the range affected?

90. OUTLIERS
IN-CLASS PROBLEM 19
Suppose we originally had data: Suppose we now have data: Original range: 4 New range: 740

91. Suppose we originally had data: 1 1 2 3 5 Suppose we now have data:
OUTLIERS
IN-CLASS PROBLEM 20
Suppose we originally had data:
Suppose we now have data:
Is the standard deviation affected?

92. OUTLIERS
IN-CLASS PROBLEM 20
Suppose we originally had data: Suppose we now have data: Original s: ≈1.7 New s: ≈330.6

93. What advantages does the standard deviation have over the range?
IN-CLASS PROBLEM 21
What advantages does the standard deviation have over the range?

94. In-class Project
Turn in your classwork! Don’t forget your homework due next class! See you Thursday!