Z SCORESMA2D3

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

68%

𝑥 𝑥+𝑠𝑥−3𝑠 𝑥+3𝑠𝑥−2𝑠 𝑥+2𝑠𝑥−𝑠

95%

99.7%

Example• IQ Scores are Normally Distributed with N(110, 25)• Complete the axis for the curve

68%

95%

99.7%

110 135 160 185856035

Example• What percent of the population scores lower than 85?

68%

95%

99.7%

110 135 160 185856035

16%

Example• What percent of the population scores lower than 100?

68%

95%

99.7%

110 135 160 185856035 100

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:

0 1 2 3-1-2-3

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. 2.01

2. 3.39

3. 0.08

4. -1.53

5. -3.47

1. 97.78%

2. 99.97%

3. 53.19%

4. 6.30%

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• 1-.9952• .0048 or .48%

Find the percent of the population that is above a z score of -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• .9099-.6064• .3035 or 30.35%

Find the percent of the population that is between -2.01 and 1.89• .9706-.0222• .9484 or 94.84%

PRACTICE WORKSHEET

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:

Practice:Convert the following IQ Score N(110, 25) to z scores:

1. 100

2. 125

3. 75

4. 140

5. 45

1. -.4

2. .6

3. -1.4

4. 1.2

5. -2.6

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

• Use the z score table• .3446 or 34.46%

¿100−11025¿− .4

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

• Use the z score table• .5793 fall below • This question is asking for above, so you have to subtract

from 1.• 1-.5793• .4207 or 42.07%

¿115−11025¿ .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

• 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%

¿150−11025¿1.6

¿50−11025¿−2.4

PRACTICE WORKSHEET