1. The Normal Distribution
James H. Steiger

2. Types of Probability
Distributions
There are two fundamental types of
probability distributions
Discrete
Continuous

3. Discrete Probability Distributions
These are used to model a situation where
The number of events (things that can happen)
is countable (i.e., can be placed in
correspondence with integers)
Probabilities sum to 1
Each event has a probability

4. A Discrete Probability Function
Discrete Uniform (1,6) Distribution
0.167
0.000
0 1 2 3 4 5 6 7

5. Continuous Probability
Distributions
These are used to model situations where the
number of things that can happen is not countable
Probability of a particular outcome cannot be
defined
Interval probabilities can be defined: Probability
of an interval is the area under the probability
density curve between the endpoints of the interval

6. Probability Density Function
Uniform(0,6) Distribution
0.167
0.000
-2 -1 0 1 2 3 4 5 6 7 8

7. The Normal Distribution Family
Standard Normal Distribution
0.4
34.13%
0.3
σ
0.2
0.1
0.0
µ µ+σ

8. The Standard Normal
Distribution
Standard Normal Distribution
0.4
34.13%
0.3
σ
0.2
0.1
0.0
-3 -2 -1 0 1 2 3

9. The Normal Curve Table
Z F(Z) Z F(Z)
0.25 59.87 1.75 95.99
0.50 69.15 1.96 97.5
0.75 77.34 2.00 97.72
1.00 84.13 2.25 98.78
1.25 89.44 2.326 99.00
1.50 93.32 2.50 99.38
1.645 95.00 2.576 99.50

10. Some Typical “Routine”
Problems
Distribution of IQ Scores
Normal(100,15)
0.028
0.024
0.020
0.016
0.012
0.008
0.004
0.000
55 70 85 100 115 130 145

11. Some Typical “Routine”
Problems
1.What percentage of people have IQ scores
between 100 and 115?
2.What percentage of people have IQ scores
greater than 130?
3.What percentage of people have IQ scores
between 70 and 85?
4. What is the percentile rank of an IQ score of
122.5?
5. What IQ score is at the 99th percentile?

12. General Strategy for Normal
Curve Problems
ALWAYS draw the picture
Estimate the answer from the picture,
remembering:
Symmetry
Total area is 1.0
Area to the left or right of center is .50
Convert to Z-score form
Compute interval probability by subtraction

13. Problem #1
Distribution of IQ Scores
Normal(100,15)
0.028
0.024
0.020
0.016
0.012
0.008
0.004
0.000
55 70 85 100 115 130 145

14. Problem #2
Distribution of IQ Scores
Normal(100,15)
0.028
0.024
0.020
0.016
0.012
0.008
0.004
0.000
55 70 85 100 115 130 145

15. Problem #2 (ctd)
130 − 100
Z130 = = +2.00
15

16. The Normal Curve Table
Z F(Z) Z F(Z)
0.25 59.87 1.75 95.99
0.50 69.15 1.96 97.5
0.75 77.34 2.00 97.72
1.00 84.13 2.25 98.78
1.25 89.44 2.326 99.00
1.50 93.32 2.50 99.38
1.645 95.00 2.576 99.50

17. Problem #3
Distribution of IQ Scores
Normal(100,15)
0.028
0.024
0.020
0.016
0.012
0.008
0.004
0.000
55 70 85 100 115 130 145

18. Problem #3
70 − 100
Z 70 = = −2.00
15
85 − 100
Z 85 = = −1.00
15

19. The Normal Curve Table
Z F(Z) Z F(Z)
0.25 59.87 1.75 95.99
0.50 69.15 1.96 97.5
0.75 77.34 2.00 97.72
1.00 84.13 2.25 98.78
1.25 89.44 2.326 99.00
1.50 93.32 2.50 99.38
1.645 95.00 2.576 99.50

20. Problem #4
Distribution of IQ Scores
Normal(100,15)
0.028
0.024
0.020
0.016
0.012
0.008
0.004
0.000
55 70 85 100 115 130 145

21. Problem #4
122.5 − 100
Z122.5 = = 1.50
15

22. The Normal Curve Table
Z F(Z) Z F(Z)
0.25 59.87 1.75 95.99
0.50 69.15 1.96 97.5
0.75 77.34 2.00 97.72
1.00 84.13 2.25 98.78
1.25 89.44 2.326 99.00
1.50 93.32 2.50 99.38
1.645 95.00 2.576 99.50

23. Problem #5
Distribution of IQ Scores
Normal(100,15)
0.028
0.024
0.020
0.016
0.012
0.008
0.004
0.000
55 70 85 100 115 130 145

24. Problem #5
Find Z-score corresponding to 99th
percentile.
Convert to a raw score.

25. Relative Tail Probability
Problems
These problems involve analyzing relative
probabilities of extreme events in two
populations. Such problems:
Are seldom found in textbooks
Are often relevant to real world problems
Sometimes produce results that are
counterintuitive

26. Relative Tail Probability #1
Group A has a mean of 100 and a standard
deviation of 15. Group B has a mean of 100
and a standard deviation of 17. What is the
relative likelihood of finding a person with
a score above 145 in Group B, relative to
Group A?

27. The Normal Curve Table
Z Area Above Z
2.6 .0047
2.65 .00405
2.7 .0035
3.0 .00135
Answer: About 3 to 1.

28. Relative Tail Probability #2
Group A has a mean of 100 and a standard
deviation of 15. Group B has a mean of 100
and a standard deviation of 17. What is the
relative likelihood of finding a person with
a score above 160 in Group B, relative to
Group A?

29. The Normal Curve Table
Z Area Above Z
3.529 .000208
4.00 .0000317
Answer: About 6.57 to 1.