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Z SCORESMM3D3
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
• 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
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. z < 2.01
2. z < 3.39
3. z < 0.08
4. z < -1.53
5. z < -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• 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
• z < -.4
• 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
• z > .2
• Use the z score table• .5793 fall below • 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
• -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%
¿150−11025¿1.6
¿50−11025¿−2.4
PRACTICE WORKSHEET