Section 3.2 Measures of Spread 115 Explain It Again Chebyshev’s Inequality is always valid: Chebyshev’s Inequality can be used with any data set, whether or not it is bell-shaped. Chebyshev’s Inequality In any data set, the proportion of the data that will be within K standard deviations of the mean is at least 1−1/K2. Specifically, by setting K = 2 or K = 3, we obtain the following results: • At least 3/4 (75%) of the data will be within two standard deviations of the mean. • At least 8/9 (88.9%) of the data will be within three standard deviations of the mean. EXAMPLE 3.20 Using Chebyshev’s Inequality As part of a public health study, systolic blood pressure was measured for a large group of people. The mean was ¯x = 120 and the standard deviation was s = 10. What information does Chebyshev’s Inequality provide about these data? Solution We compute: ¯x − 2s = 120 − 2(10) = 100 ¯x + 2s = 120 + 2(10) = 140 ¯x − 3s = 120 − 3(10) = 90 ¯x + 3s = 120 + 3(10) = 150 We conclude: • At least 75% of the people had systolic blood pressures between 100 and 140. • At least 88.9% of the people had systolic blood pressures between 90 and 150. Comparing Chebyshev’s Inequality to the Empirical Rule Both Chebyshev’s Inequality and the Empirical Rule provide information about the proportion of a data set that is within a given number of standard deviations of the mean. An advantage of Chebyshev’s Inequality is that it applies to any data set, whereas the Empirical Rule applies only to data sets that are approximately bell-shaped. A disadvantage of Chebyshev’s Inequality is that for most data sets, it provides only a very rough approximation. Chebyshev’s Inequality produces a minimum value for the proportion of the data that will be within a given number of standard deviations of the mean. For most data sets, the actual proportions are much larger than the values given by Chebyshev’s Inequality. Check Your Understanding 6. A group of elementary school students took a standardized reading test. The mean score was 70 and the standard deviation was 10. Someone says that only 50% of the students scored between 50 and 90. Is this possible? Explain. 7. A certain type of bolt used in an aircraft must have a length between 122 and 128 millimeters in order to be acceptable. The manufacturing process produces bolts whose mean length is 125 millimeters with a standard deviation of 1 millimeter. Can you be sure that more than 85% of the bolts are acceptable? Explain. Answers are on page 123. Objective 7 Compute the coefficient of variation The Coefficient of Variation The coefficient of variation (CV for short) tells how large the standard deviation is relative to the mean. It can be used to compare the spreads of data sets whose values have different units.
navidi_monk_essential_statistics_1e_ch1_3
To see the actual publication please follow the link above