Copyright © 2014 Pearson Education. All rights reserved. 5.2-1 Copyright © 2014 Pearson Education, Inc. 5.2 Properties of the Normal Distribution LEARNING.

Copyright © 2014 Pearson Education. All rights reserved. 5.2-1Copyright © 2014 Pearson Education, Inc.

5.2 Properties of the Normal Distribution

LEARNING GOAL

Know how to interpret the normal distribution in terms of the 68-95-99.7 rule, standard scores, and percentiles.

Copyright © 2014 Pearson Education. All rights reserved. 5.2-2Slide 5.2- 2Copyright © 2014 Pearson Education, Inc.

A simple rule, called the 68-95-99.7 rule, gives precise guidelines for the percentage of data values that lie within 1, 2, and 3 standard deviations of the mean for any normal distribution.

Figure 5.17 Normal distribution illustrating the 68-95-99.7 rule.


• About 68% (more precisely, 68.3%), or just over two-thirds, of the data points fall within 1 standard deviation of the mean.

• About 95% (more precisely, 95.4%) of the data points fall within 2 standard deviations of the mean.

• About 99.7% of the data points fall within 3 standard deviations of the mean.

The 68-95-99.7 Rule for a Normal Distribution


The tests that make up the verbal (critical reading) and mathematics SAT (and the GRE, LSAT, and GMAT) are designed so that their scores are normally distributed with a mean of = 500 and a standard deviation of = 100. 1. Estimate the percentage of students having test scores between 400-500?2. Estimate the percentage of students having test scores between 300-700?

3.Estimate the percentage of students having test scores between 200-800?

4. Estimate the percentage of students having test scores above 500?

5. Estimate the percentage of students having test scores between 300-800?

EXAMPLE 1 SAT Scores


You measure your resting heart rate at noon every day for a year and record the data. You discover that the data have a normal distribution with a mean of 66 and a standard deviation of 4. On how many days was your heart rate below 58 beats per minute?

EXAMPLE 4 Normal Heart Rate


On a visit to the doctor’s office, your fourth-grade daughter is told that her height is 1 standard deviation above the mean for her age and sex. What is her percentile for height? Assume that heights of fourth-grade girls are normally distributed.

EXAMPLE 5 Finding a Percentile


Standard Scores

Computing Standard Scores

The number of standard deviations a data value lies above or below the mean is called its standard score (or z-score), defined by

z = standard score =

The standard score is positive for data values above the mean and negative for data values below the mean.

data value – meanstandard deviation


The Stanford-Binet IQ test is scaled so that scores have a mean of 100 and a standard deviation of 16. Find the standard scores for IQs of 85, 100, and 125.

EXAMPLE 6 Finding Standard Scores


Cholesterol levels in men 18 to 24 years of age are normally distributed with a mean of 178 and a standard deviation of 41.

a. What is the percentile for a 20-year-old man with a cholesterol level of 190?

EXAMPLE 7 Cholesterol Levels

Solution: a.The standard score for a cholesterol level of 190 is

z = standard score = = ≈ 0.29

Table 5.1 shows that a standard score of 0.29 corresponds to about the 61st percentile.

190 – 17841

data value – mean standard deviation


Cholesterol levels in men 18 to 24 years of age are normally distributed with a mean of 178 and a standard deviation of 41.

b. What cholesterol level corresponds to the 90th percentile?

EXAMPLE 7 Cholesterol Levels

Copyright © 2014 Pearson Education. All rights reserved. 5.2-1 Copyright © 2014 Pearson Education, Inc. 5.2 Properties of the Normal Distribution LEARNING.

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