Normal or skewed distributions (inferential) - Copyright updated

1. Are the distributions all normal or is at least
one skewed?
Normal? Skewed?

2. The Normal Distribution is a distribution that has most
of the data in the center with decreasing amounts
evenly distributed to the left and the right.
Skewed Distribution is distribution with data clumped
up on one side or the other with decreasing amounts
trailing off to the left or the right.
Central Tendency, Spread, or Symmetry?

3. The Normal Distribution is a distribution that has most
of the data in the center with decreasing amounts
evenly distributed to the left and the right.
The Skewed Distribution is distribution with data
clumped up on one side or the other with decreasing
amounts trailing off to the left or the right.
Central Tendency, Spread, or Symmetry?
Right skewed Left skewed

4. Why is this important to know?

5. Why is this important to know?
Because the means and standard deviations of
samples with skewed distributions do not
generalize to the population well.
This is because the skewed distributions means
are pulled toward the tail of the distribution.

6. Why is this important to know?
Because the means and standard deviations of
samples with skewed distributions do not
generalize to the population well.
This is because the skewed distributions means
are pulled toward the tail of the distribution.
The Population
Skewed
Sample

7. Why is this important to know?
Because the means and standard deviations of
samples with skewed distributions do not
generalize to the population well.
This is because the skewed distributions means
are pulled toward the tail of the distribution.
The Population
Skewed
Sample

8. Why is this important to know?
Because the means and standard deviations of
samples with skewed distributions do not
generalize to the population well.
This is because the skewed distributions means
are pulled toward the tail of the distribution.
The Population
Skewed
Sample

9. Why is this important to know?
Because the means and standard deviations of
samples with skewed distributions do not
generalize to the population well.
This is because the skewed distributions means
are pulled toward the tail of the distribution.
The Population
Skewed
Sample
mean mean

10. How can you tell if a distribution is
• Normal?

11. How can you tell if a distribution is
• Normal?
or
• Skewed?

12. We will show you the answer to that question
within the context of a problem.

13. Problem
Is there a significant difference between drivers of
old cars and drivers of new cars in terms of average
freeway driving speed?
First let’s determine if the old and new car
distributions are normal or skewed.

14. Problem
Is there a significant difference between drivers of
old cars and drivers of new cars in terms of average
freeway driving speed?
First let’s determine if the old and new car
distributions are normal or skewed.

15. Data Set
Access the data set below:
Link to data set

16. SPSS
Copy and paste the data into SPSS by following
the instructions at this link.
Note – go to the 2nd page of the instructions

17. How to check for skew in SPSS

18. Select Analyze
Step 1

19. Descriptive – Compare Means - Means
Step 2

20. Select the Dependent Variable – click the arrow
Step 3

21. Select the Dependent Variable – click the arrow
Step 3

22. Select the Independent Variable – click the arrow
Step 3

23. Click Options
Step 4

24. Bring Skew and Standard Error of the Skew over
Step 5

25. Click OK
Step 5

26. Output

27. Let’s Interpret!

28. Let’s Interpret!
First, we have to determine if this is a descriptive
or inferential question.

29. Let’s Interpret!
If descriptive, we look just at this value

30. Let’s Interpret!
For the new car distribution the skewness is -.953.

31. Let’s Interpret!
For the new car distribution the skewness is -.953.
If the skewness is below -2.0 or above +2.0 then
the distribution is considered skewed.

32. Let’s Interpret!
For the new car distribution the skewness is -.953.
Because -.953 is between -2.0 and +2.0 then the
distribution is considered Normal

33. Let’s Interpret!
Therefore the new car distribution is considered
Normal if we are dealing with descriptive statistics

34. But …

35. We are NOT dealing with descriptive statistics.
We are dealing with inferential statistics.
Therefore, we must take a different approach.
Everyone in
the population

36. We are NOT dealing with descriptive statistics.
We are dealing with inferential statistics.
Therefore, we must take a different approach.
Everyone in
the population
Everyone in
the population
Sample

37. We are NOT dealing with descriptive statistics.
We are dealing with inferential statistics.
Therefore, we must take a different approach.
Everyone in
the population
Everyone in
the population
Sample

38. We must . . .
. . divide the skewness by the standard error of
skewness.

39. We must . . .
. . divide the skewness by the standard error of
skewness.

40. We must . . .
. . divide the skewness by the standard error of
skewness.

41. We must . . .
. . divide the skewness by the standard error of
skewness.

42. We must . . .
. . divide the skewness by the standard error of
skewness.

43. We must . . .
. . divide the skewness by the standard error of
skewness.

44. We must . . .
. . divide the skewness by the standard error of
skewness.
-1.69

45. We must . . .
The reason we do this will be explained later on in
the course.
-1.69

46. Let’s Interpret!
-1.69

47. Let’s Interpret!
For the new car distribution the skewness is -1.69.
-1.69

48. Let’s Interpret!
For the new car distribution the skewness is -1.69.
-1.69
If the skewness is below -2.0 or above +2.0 then
the distribution is considered skewed.

49. Let’s Interpret!
Because -1.69 is between -2.0 and +2.0 then the
distribution is considered Normal
For the new car distribution the skewness is -1.69.
-1.69

50. Let’s Interpret!
-1.69
Therefore the new car distribution is considered
Normal if we are dealing with inferential statistics

51. Now the focus turns to the skewness
of the old car distribution.

52. Calculating Skew
. . divide the skewness by the standard error of
skewness.

53. Calculating Skew
. . divide the skewness by the standard error of
skewness.
-4.26

54. -4.26
Let’s Interpret!

55. Let’s Interpret!
-4.26
For the old car distribution the skewness is -4.26.

56. Let’s Interpret!
If the skewness is below -2.0 or above +2.0 then
the distribution is considered skewed.
-4.26
For the old car distribution the skewness is -4.26.

57. Let’s Interpret!
Because -4.26 is less than -2.0 then the
distribution is considered negatively skewed
-4.26
For the old car distribution the skewness is -4.26.

58. Let’s Interpret!
If the skewness had been positive +4.26 it would
have been positively or right skewed:
For the old car distribution the skewness is -4.26.

59. Let’s Interpret!
But in this case, the old car distribution is considered
negative or left skewed if we are dealing with
inferential statistics
-4.26

60. Let’s practice

61. Old car / new car skew problem with
different data set.

62. Is the old car data set skewed or
normal (inferential study)?

63. Is the old car data set skewed or
normal?
Report
speed
new_old_car Mean N Std. Deviation Skewness
Std. Error of
Skewness
New car 72.81 16 6.595 -.953 .564
Old car 79.94 17 18.081 3.344 .550
Total 76.48 33 14.034 -1.832 .409
Normal? Skewed?

64. Report
speed
new_old_car Mean N Std. Deviation Skewness
Std. Error of
Skewness
New car 72.81 16 6.595 -.953 .564
Old car 79.94 17 18.081 3.344 .550
Total 76.48 33 14.034 -1.832 .409
Is the old car data set skewed or
normal?
+6.06
Normal? Skewed?

65. Report
speed
new_old_car Mean N Std. Deviation Skewness
Std. Error of
Skewness
New car 72.81 16 6.595 -.953 .564
Old car 79.94 17 18.081 3.344 .550
Total 76.48 33 14.034 -1.832 .409
Is the old car data set skewed or
normal?
+6.06
Normal? Skewed?

66. Report
speed
new_old_car Mean N Std. Deviation Skewness
Std. Error of
Skewness
New car 72.81 16 6.595 -.953 .564
Old car 79.94 17 18.081 3.344 .550
Total 76.48 33 14.034 -1.832 .409
Is the old car data set skewed or
normal?
+6.06
Normal? Skewed?
If the skewness is below -2.0 or above +2.0 then
the distribution is considered skewed.

67. Report
speed
new_old_car Mean N Std. Deviation Skewness
Std. Error of
Skewness
New car 72.81 16 6.595 -.953 .564
Old car 79.94 17 18.081 3.344 .550
Total 76.48 33 14.034 -1.832 .409
Is the old car data set skewed or
normal?
+6.06
Normal? Skewed?
Because +6.06 is greater than +2.0 then the
distribution is considered positively or right skewed.

68. Is the new car data set skewed or
normal?
Normal? Skewed?

69. Is the new car data set skewed or
normal?
Normal? Skewed?
Report
speed
new_old_car Mean N Std. Deviation Skewness
Std. Error of
Skewness
New car 72.81 16 6.595 1.813 .993
Old car 79.94 17 18.081 -2.344 .550
Total 76.48 33 14.034 -1.832 .409

70. Report
speed
new_old_car Mean N Std. Deviation Skewness
Std. Error of
Skewness
New car 72.81 16 6.595 1.813 .993
Old car 79.94 17 18.081 -2.344 .550
Total 76.48 33 14.034 -1.832 .409
Is the new car data set skewed or
normal?
Normal? Skewed?
1.83

71. Report
speed
new_old_car Mean N Std. Deviation Skewness
Std. Error of
Skewness
New car 72.81 16 6.595 1.813 .993
Old car 79.94 17 18.081 -2.344 .550
Total 76.48 33 14.034 -1.832 .409
Is the new car data set skewed or
normal?
Normal? Skewed?
1.83

72. Report
speed
new_old_car Mean N Std. Deviation Skewness
Std. Error of
Skewness
New car 72.81 16 6.595 1.813 .993
Old car 79.94 17 18.081 -2.344 .550
Total 76.48 33 14.034 -1.832 .409
Is the new car data set skewed or
normal?
Normal? Skewed?
1.83
If the skewness is between -2.0 and +2.0 then the
distribution is considered normal.

73. Report
speed
new_old_car Mean N Std. Deviation Skewness
Std. Error of
Skewness
New car 72.81 16 6.595 1.813 .993
Old car 79.94 17 18.081 -2.344 .550
Total 76.48 33 14.034 -1.832 .409
Is the new car data set skewed or
normal?
Normal? Skewed?
1.83
Because 1.83 is between -2.0 and +2.0 then the
distribution is considered normal.

74. After calculating the skew for the data set in your
original problem, determine if the distributions are
all normal or is there at least one that is skewed?
Normal? Skewed?