Copyright © 2014 Pearson Education. All rights reserved Copyright © 2014 Pearson Education, Inc. 5.2 Properties of the Normal Distribution LEARNING GOAL Know how to interpret the normal distribution in terms of the rule, standard scores, and percentiles.
Copyright © 2014 Pearson Education. All rights reserved Slide Copyright © 2014 Pearson Education, Inc. A simple rule, called the 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 rule.
Copyright © 2014 Pearson Education. All rights reserved Slide Copyright © 2014 Pearson Education, Inc. 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 Rule for a Normal Distribution
Copyright © 2014 Pearson Education. All rights reserved Slide Copyright © 2014 Pearson Education, Inc. 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 = Estimate the percentage of students having test scores between ? 2. Estimate the percentage of students having test scores between ? 3.Estimate the percentage of students having test scores between ? 4. Estimate the percentage of students having test scores above 500? 5. Estimate the percentage of students having test scores between ? EXAMPLE 1 SAT Scores
Copyright © 2014 Pearson Education. All rights reserved Slide Copyright © 2014 Pearson Education, Inc. 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
Copyright © 2014 Pearson Education. All rights reserved Slide Copyright © 2014 Pearson Education, Inc. 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
Copyright © 2014 Pearson Education. All rights reserved Slide Copyright © 2014 Pearson Education, Inc. 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 – mean standard deviation
Copyright © 2014 Pearson Education. All rights reserved Slide Copyright © 2014 Pearson Education, Inc. 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
Copyright © 2014 Pearson Education. All rights reserved Slide Copyright © 2014 Pearson Education, Inc. 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 – data value – mean standard deviation
Copyright © 2014 Pearson Education. All rights reserved Slide Copyright © 2014 Pearson Education, Inc. 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