1. Welcome to .
Week 08 Tues .
MAT135 Statistics

2. Non-normal Distributions
Last class we studied a lot about the normal distribution Some distributions are not normal …

3. Non-normal Distributions
Skewness – the data are “bunched” to one side vs a normal curve

4. Non-normal Distributions
Scores that are "bunched" at the right or high end of the scale are said to have a “negative skew”

5. Non-normal Distributions
In a “positive skew”, scores are bunched near the left or low end of a scale

6. Non-normal Distributions
Note: this is exactly the opposite of how most people use the terms!

7. NON-NORMAL DISTRIBUTIONS
PROJECT QUESTIONS 1,2
Which is positively skewed? Which is negatively skewed?

8. Non-normal Distributions
Kurtosis - how tall or flat your curve is compared to a normal curve

9. Non-normal Distributions
Curves taller than a normal curve are called “Leptokurtic” Curves that are flatter than a normal curve are called ”Platykurtic”

10. Non-normal Distributions
Platykurtic Leptokurtic
W.S. Gosset 1908

11. NON-NORMAL DISTRIBUTIONS
PROJECT QUESTIONS 3,4
Which is platykurtic? Which is leptokurtic?

12. Questions?

13. Normal Distributions
We use normal distributions a lot in statistics because lots of things have graphs this shape!
heights
weights
IQ test scores
bull’s eyes

14. Normal Distributions
Also, even data which are not normally distributed have averages which DO have normal distributions

15. Normal Distributions
If you take a gazillion samples and find the means for each of the gazillion samples You would have a new population: the gazillion means

16. Normal Distributions

17. Normal Distributions
If you plotted the frequency of the gazillion mean values, it is called a SAMPLING DISTRIBUTION

18. Sampling Distributions
The shape of the plot of the gazillion sample means would have a normal-ish distribution NO MATTER WHAT THE ORIGINAL DATA LOOKED LIKE

19. Sampling Distributions
But … the shape of the distribution of your gazillion means changes with the size “n” of the samples you took

20. Sampling Distributions
Graphs of a gazillion means for different n values

21. Sampling Distributions
As “n” increases, the distributions of the means become closer and closer to normal

22. Sampling Distributions
This also works for discrete data

23. Sampling Distributions
as “n” increases, variability (spread) also decreases

24. Sampling Distributions
We usually say the sample mean will be normally distributed if n is ≥ 20 (or 30…) (the “good-enuff” value)

25. Sampling Distributions
I call it: 20or30

26. Sampling Distributions
The statistical principle that allows us to conclude that sample means have a normal distribution if the sample size is 20or30 or more is called the Central Limit Theorem

27. Sampling Distributions
If you can assume the distribution of the sample means is normal, you can use the normal distribution probabilities for making probability statements about µ

28. Sampling Distributions
Sample means from platykurtic, leptokurtic, and bimodal distributions become “normal enough” when your sample size n is 20or30 or more

29. Sampling Distributions
Means from samples of skewed populations do not become “normal enough” very easily You sometimes need a mega-huge sample size to “normalize” a badly skewed distribution

30. Sampling Distributions
A wild outlier might indicate a badly skewed distribution

31. SAMPLING DISTRIBUTIONS
PROJECT QUESTION 5
From which of these would you expect the distribution of the sample means to be normal? Original population normal Samples taken of size 10 Sample taken of size 50 Highly skewed population

32. Questions?

33. Graphs of 𝒙
Graph of 𝒙 values

34. Graphs of 𝒙
Averages (measures of central tendency) show where the data tend to pile up
Graph of 𝒙 values

35. The place where 𝒙 tends to pile up is at μ
Graphs of 𝒙
The place where 𝒙 tends to pile up is at μ
Graph of 𝒙 values

36. Graphs of 𝒙
So, the most likely value for 𝒙 is μ
Graph of 𝒙 values

37. Graphs of 𝒙
As you move away from μ on the graph, 𝒙 is less likely to have these values
Graph of 𝒙 values

38. Graphs of 𝒙

39. GRAPHS OF 𝒙
PROJECT QUESTION 6
Population Sample mean mean μ 𝒙 What is the best estimate we have for the unknown population mean µ ?

40. 𝒙 is the best estimate we have for the unknown population mean µ
GRAPHS OF 𝒙
PROJECT QUESTION 6 𝒙
is the best estimate
we have for the unknown
population mean µ

41. The mean of all of the gazillion 𝒙 values will be µ
Graphs of 𝒙
The mean of all of the gazillion 𝒙 values will be µ

42. Graph of likely values for µ:
GRAPHS OF 𝒙
PROJECT QUESTION 7
Graph of likely values for µ: ?

43. Graph of likely values for µ:
GRAPHS OF 𝒙
PROJECT QUESTION 7
Graph of likely values for µ: 𝒙

44. Questions?

45. Estimation
We will use the sample mean 𝒙 to estimate the unknown population mean µ

46. Estimation
Using the sample mean 𝒙 to estimate the unknown population mean µ is called “making inferences”

47. Estimation
The sample standard deviation “s” is the best estimate we have for the unknown population standard deviation “σ”

48. Using s to estimate σ is also an inference
Estimation
Using s to estimate σ is also an inference

49. Estimation 𝒙 -3s 𝒙 -2s 𝒙 -s 𝒙 𝒙 +s 𝒙 +2s 𝒙 +3s
You would think, since we use 𝒙 to estimate µ and s to estimate σ that the graph of 𝒙 would be:
𝒙 -3s 𝒙 -2s 𝒙 -s 𝒙 𝒙 +s 𝒙 +2s 𝒙 +3s

50. Estimation
It’s not…

51. Remember, as “n” increases, the variability decreases:
Estimation
Remember, as “n” increases, the variability decreases:

52. Estimation
While s is a good estimate for the original population standard deviation σ, s IS NOT the measure of variability in the new population of 𝒙 s

53. It needs to be decreased to take sample size into account!
Estimation
It needs to be decreased to take sample size into account!

54. Estimation
We use: s/ n for the measure of variability in the new population of 𝒙 s

55. Estimation
The standard deviation of the 𝒙 s: s/ n is called the “standard error” abbreviated “se”

56. Estimation
BTW: you now know ALL of the items on the Descriptive Statistics list in Excel !

57. Estimation
So our curve is:
𝒙 -3se 𝒙 -2se 𝒙 -se 𝒙 𝒙 +se 𝒙 +2se 𝒙 +3se

58. ESTIMATION
PROJECT QUESTION 8
Suppose we have a population of 𝒙 s from samples of size 49 The mean of the 𝒙 s is 150 The standard deviation is 56 Can we assume the population of 𝒙 s forms a normal distribution?

59. ESTIMATION
PROJECT QUESTION 8
Suppose we have a population of 𝒙 s from samples of size 49 The mean of the 𝒙 s is 150 The standard deviation is 56 Because the sample size 49 is above the usual “good-enuff” value of 20or30, unless the original distribution is very skewed, it will be normal

60. ESTIMATION
PROJECT QUESTION 9
Suppose we have a population of 𝒙 s from samples of size 49 The mean of the 𝒙 s is 150 The standard deviation is 56 What is our best estimate of the original population mean?

61. ESTIMATION
PROJECT QUESTION 9
Suppose we have a population of 𝒙 s from samples of size 49 The mean of the 𝒙 s is 150 The standard deviation is 56 What is our best estimate of the original population mean? 150

62. ESTIMATION
PROJECT QUESTION 10
Suppose we have a population of 𝒙 s from samples of size 49 The mean of the 𝒙 s is 150 The standard deviation is 56 What is our best estimate of the original population standard deviation?

63. ESTIMATION
PROJECT QUESTION 10
Suppose we have a population of 𝒙 s from samples of size 49 The mean of the 𝒙 s is 150 The standard deviation is 56 What is our best estimate of the original population standard deviation? 56

64. ESTIMATION
PROJECT QUESTION 11
Suppose we have a population of 𝒙 s from samples of size 49 The mean of the 𝒙 s is 150 The standard deviation is 56 What is our estimate of the standard error?

65. ESTIMATION
PROJECT QUESTION 11
Suppose we have a population of 𝒙 s from samples of size 49 The mean of the 𝒙 s is 150 The standard deviation is 56 What is our estimate of the standard error? 56/ 49 = 56/7 = 8

66. ESTIMATION
PROJECT QUESTION 12
Suppose we have a population of 𝒙 s from samples of size 49 The mean of the 𝒙 s is 150 The standard deviation is 56 What will be the normal curve for the 𝒙 s ?

67. ESTIMATION
PROJECT QUESTION 12
Our curve is:

68. Our curve is: 126 134 142 150 158 166 174 ESTIMATION
PROJECT QUESTION 12
Our curve is:

69. ESTIMATION
PROJECT QUESTION 13
Suppose we have a population of 𝒙 s from samples of size 49 The mean of the 𝒙 s is 150 The standard deviation is 56 What is the probability that the true population mean is greater between 142 and 158?

70. ESTIMATION
PROJECT QUESTION 13
Suppose we have a population of 𝒙 s from samples of size 49 The mean of the 𝒙 s is 150 The standard deviation is 56 What is the probability that the true population mean is between 142 and 158? 68%

71. ESTIMATION
PROJECT QUESTION 14
Suppose we have a population of 𝒙 s from samples of size 49 The mean of the 𝒙 s is 150 The standard deviation is 56 What is the probability that the true population mean is 150?

72. ESTIMATION
PROJECT QUESTION 14
Suppose we have a population of 𝒙 s from samples of size 49 The mean of the 𝒙 s is 150 The standard deviation is 56 What is the probability that the true population mean is 150? 0%

73. ESTIMATION
PROJECT QUESTION 15
Suppose we have a population of 𝒙 s from samples of size 49 The mean of the 𝒙 s is 150 The standard deviation is 56 What is the range of values that would ensure with 95% probability that we include the mean?

74. ESTIMATION
PROJECT QUESTION 15
Suppose we have a population of 𝒙 s from samples of size 49 The mean of the 𝒙 s is 150 The standard deviation is 56 What is the range of values that would ensure with 95% probability that we include the mean?

75. ESTIMATION
PROJECT QUESTION 16
Suppose we have a population of 𝒙 s from samples of size 49 The mean of the 𝒙 s is 150 The standard deviation is 56 What is the probability the true p lies between 130 and 145?

76. ESTIMATION
PROJECT QUESTION 16
Suppose we have a population of 𝒙 s from samples of size 49 The mean of the 𝒙 s is 150 The standard deviation is 56 What is the probability the true p lies between 130 and 145? 26%

77. Questions?

78. You survived!
Turn in your homework! Don’t forget your homework due next class! See you Thursday!