1. Empirical Rule
MM3D3

2. Normal Distributions Normal distributions are based on two parameters
Mean
If you have population data use 𝜇
If you have sample data use 𝑥
Standard Deviation
If you have population data use 𝜎
If you have sample data use s
When a distribution is normal we use shorthand to show the mean and standard deviation
Sample: N 𝑥 , 𝑠
Population: N 𝜇, 𝜎

3. Normal Curves
Use the parameters to find the inflection points on the curve.
Where the curve changes concavity

4. Normal Curves The mean is in the exact middle
Add and subtract the standard deviation to find the inflection points
𝑥 −3𝑠
𝑥 −2𝑠
𝑥 −𝑠 𝑥
𝑥 +𝑠
𝑥 +2𝑠
𝑥 +3𝑠

5. Normal Curves
IQ scores are normally distributed with a mean of 110 and standard deviation 25.
𝑥 −3𝑠
𝑥 −2𝑠
𝑥 −𝑠 𝑥
𝑥 +𝑠
𝑥 +2𝑠
𝑥 +3𝑠 35 60 85
110
135
160
185

6. Empirical Rule
68% of the data is within one standard deviation of the mean
95% of the data is within two standard deviations of the mean
99.7% of the data is within three standard deviations of the mean
99.7%
95%
68%
𝑥 −3𝑠
𝑥 −2𝑠
𝑥 −𝑠 𝑥
𝑥 +𝑠
𝑥 +2𝑠
𝑥 +3𝑠

7. Applying the Empirical Rule
Often the empirical rule is used to determine the percent of data that falls above or below a point of inflection.
For example what percent of people score lower than 110 on the IQ test
It is helpful to know the percent of the curve that is represented by each section of the distribution.

8. The middle 68 % ? ?
34%
34%

9. The middle 95 %
34%
34%
13.5% ? ?
13.5%

10. The middle 99.7 %
34%
34%
13.5%
13.5% ?
2.35%
2.35% ?

11. The Tails
34%
34%
13.5%
13.5%
0.15% ? ?
0.15%
2.35%
2.35%

12. IQ Scores What is the 68% range? 85-135 N (110, 25) 35 60 85 110 135
160
185
What is the 68% range?
85-135

13. IQ Scores What is the 95% range? 60-160 N (110, 25) 35 60 85 110 135
185
What is the 95% range?
60-160

14. IQ Scores What is the 99.7% range? 35-185 N (110, 25) 35 60 85 110 135
160
185
What is the 99.7% range?
35-185

15. IQ Scores What percent falls between 85 and 160? 81.5 N (110, 25) 35
135
160
185
What percent falls between 85 and 160?
81.5

16. IQ Scores What percent falls between 35 and 135? 83.85 N (110, 25) 3560 85
110
135
160
185
What percent falls between 35 and 135?
83.85

17. IQ Scores What percent falls below 85? 16 N (110, 25) 35 60 85 110 135
160
185
What percent falls below 85? 16

18. IQ Scores What percent falls below 185? 99.85 N (110, 25) 35 60 85 110
135
160
185
What percent falls below 185?
99.85

19. IQ Scores What percent is above 185? 0.15 N (110, 25) 35 60 85 110 135
160
185
What percent is above 185?
0.15

20. IQ Scores What percent is above 60? 97.5 N (110, 25) 35 60 85 110 135
160
185
What percent is above 60?
97.5