1. Z scores
MM3D3

2. 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

3. 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

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

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

6. 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

7. Practice: Convert the following IQ Score N(110, 25) to z scores:
100
125 75
140 45
-.4 .6
-1.4
1.2
-2.6

8. 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

9. 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%

10. 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%

11. Practice: Use your z score table to find the percent of the population that fall below the following z scores:
z < 2.01
z < 3.39
z < 0.08
z < -1.53
z < -3.47
97.78%
99.97%
53.19%
6.30%
.03%

12. 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
z > 2.59
.0048 or .48%
Find the percent of the population that is above a z score of -1.91
z > -1.91
.9719 or 97.19%

13. 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
.27 < z < 1.34
.3035 or 30.35%
Find the percent of the population that is between and 1.89
-2.01 < z < 1.89
.9484 or 94.84%

14. Practice worksheet

15. 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
z < -.4
Use the z score table
.3446 or 34.46%
= 100−110 25
=−.4

16. 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
z > .2
Use the z score table
.5793 fall below
.4207 or 42.07%
= 115−110 25
=.2

17. 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
-2.4 < z < 1.6
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

18. Practice worksheet