1. Unit 5 Data Analysis

2. MM3D3
Empirical Rule

3. 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

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

5. 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

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. Empirical Rule Expanded
Sometimes, 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. 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: IQ scores are normally distributed with a mean of 110 and standard deviation 25.
What percent of people score lower than 110 on the IQ test

13. 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

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

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

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

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

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

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

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

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

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

23. Z-Scores

24. Recall: 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

25. Example IQ Scores are Normally Distributed with N(110, 25)
Complete the axis for the curve
99.7%
95%
68% 35 60 85
110
135
160
185

26. Example 16% What percent of the population scores lower than 85? 99.7%
95%
68% 35 60 85
110
135
160
185

27. Example What percent of the population scores lower than 100? 99.7%
95%
68% 35 60 85
100
110
135
160
185

28. Z Scores
Allow you to get percentages that don’t fall on the boundaries for the empirical rule
Convert observations (x’s) into standardized scores (z’s) using the formula:
𝑧= 𝑥−𝜇 𝜎

29. Z Scores
The z score tells you how many standard deviations the x value is from the mean
The axis for the Standard Normal Curve: -3 -2 -1 1 2 3

30. Z Score Table:
The table will tell you the proportion of the population that falls BELOW a given z-score.
The left column gives the ones and tenths place
The top row gives the hundredths place
What percent of the population is below .56?
.7123 or 71.23%

31. Z Score Table:
The table will tell you the proportion of the population that falls BELOW a given z-score.
The left column gives the ones and tenths place
The top row gives the hundredths place
What percent of the population is below .4?
.6554 or 65.54%

32. Using the z score table
You can also find the proportion that is above a z score
Subtract the table value from 1 or 100%
Find the percent of the population that is above a z score of 2.59
.0048 or .48%
Find the percent of the population that is above a z score of -1.91
.9719 or 97.19%

33. Using the z score table
You can also find the proportion that is between two z scores
Subtract the table values from each other
Find the percent of the population that is between .27 and 1.34
.3035 or 30.35%
Find the percent of the population that is between and 1.89
.9484 or 94.84%

34. Application 1
IQ Scores are Normally Distributed with N(110, 25)
What percent of the population scores below 100?
Convert the x value to a z score
𝑧= 𝑥−𝜇 𝜎
Use the z score table
.3446 or 34.46%
= 100−110 25
=−.4

35. Application 2 IQ Scores are Normally Distributed with N(110, 25)
What percent of the population scores above 115?
Convert the x value to a z score
𝑧= 𝑥−𝜇 𝜎
Use the z score table
.5793 fall below
This question is asking for above, so you have to subtract from 1.
.4207 or 42.07%
= 115−110 25
=.2

36. Application 3 IQ Scores are Normally Distributed with N(110, 25)
What percent of the population score between 50 and 150?
Convert the x values to z scores
𝑧= 𝑥−𝜇 𝜎
Use the z score table
.9452 and .0082
This question is asking for between, so you have to subtract from each other.
.9370 or 93.7%
= 150−110 25
=1.6
= 50−110 25
=−2.4