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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
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
Example
• What percent of the population scores lower than 85? 16%
99.7%
95%
68%
35
60
85
110
135
160
185
Example
• What percent of the population scores lower than 100?
99.7%
95%
68%
35
60
85
100 110
135
160
185
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:
𝑥−𝜇
𝑧=
𝜎
Practice:
Convert the following IQ Score N(110, 25) to z scores:
1. 100
1. -.4
2. 125
2. .6
3. 75
3. -1.4
4. 140
4. 1.2
5. 45
5. -2.6
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
0
1
2
3
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%
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%
Practice:
Use your z score table to find the percent of the
population that fall below the following z scores:
1. z < 2.01
1. 97.78%
2. z < 3.39
2. 99.97%
3. z < 0.08
3. 53.19%
4. z < -1.53
4. 6.30%
5. z < -3.47
5. .03%
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
• 1-.9952
• .0048 or .48%
Find the percent of the population that is above a z score of -1.91
• z > -1.91
• 1-.0281
• .9719 or 97.19%
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
• .9099-.6064
• .3035 or 30.35%
Find the percent of the population that is between -2.01 and 1.89
• -2.01 < z < 1.89
• .9706-.0222
• .9484 or 94.84%
PRACTICE WORKSHEET
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
𝑥−𝜇
100 − 110
•𝑧=
= −.4
=
𝜎
25
• z < -.4
• Use the z score table
• .3446 or 34.46%
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
115 − 110
=
𝜎
25
• z > .2
• Use the z score table
• .5793 fall below
• 1-.5793
• .4207 or 42.07%
= .2
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
𝑥−𝜇
150 − 110
• 𝑧= 𝜎
= 1.6
=
25
50 − 110
𝑥−𝜇
= −2.4
=
• 𝑧= 𝜎
25
• -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.
• .9452-.0082
• .9370 or 93.7%
PRACTICE WORKSHEET