3-3 Measures of Relative Standing and Boxplots 123 Values not significant Significantly low values Significantly high values z −2 −1 2 −3 3 0 1 FIGURE 3-6 Interpreting z Scores Using the Range Rule of Thumb • Significantly low values: z … -2 • Significantly high values: z Ú 2 • Values not significant: -2 6 z 6 2 A z score is a measure of position, in the sense that it describes the location of a value (in terms of standard deviations) relative to the mean. Percentiles and quartiles are other measures of position useful for comparing values within the same data set or between different sets of data. Percentiles Beyond the importance of their use in ordinary situations, percentiles play an important role in many resampling procedures, some of which are included in this book. It is therefore important to develop an ability to find percentile values. Percentiles are one type of quantiles—or fractiles—which partition data into groups with roughly the same number of values in each group. DEFINITION Percentiles are measures of location, denoted P1, P2, c,P99, which divide a set of data into 100 groups with about 1% of the values in each group. Cost of Laughing Index There really is a Cost of Laughing Index (CLI), which tracks costs of such items as rubber chickens, Groucho Marx glasses, admissions to comedy clubs, and 13 other leading humor indicators. This is the same basic approach used in developing the Consumer Price Index (CPI), which is based on a weighted average of goods and services purchased by typical consumers. While standard scores and percentiles allow us to compare different values, they ignore any element of time. Index numbers, such as the CLI and CPI, allow us to compare the value of some variable to its value at some base time period. The value of an index number is the current value, divided by the base value, multiplied by 100. EXAMPLE 2 Is an Earthquake Magnitude of 4.01 Significantly High? Among the earthquakes listed in Data Set 24 “Earthquakes,” one of the stronger earthquakes had a magnitude of 4.01. The magnitudes are measured on the Richter scale, and only earthquakes of magnitude 1.00 or higher are included. The 600 magnitudes in the data set have a mean of 2.572 and a standard deviation of 0.651. For this data set, is the magnitude of 4.01 significantly high? SOLUTION The magnitude of 4.01 is converted to a z score as shown below: z = x - x s = 4.01 - 2.572 0.651 = 2.21 INTERPRETATION The magnitude of 4.01 converts to the z score of 2.21. Because the z score of 2.21 is greater than or equal to +2, that magnitude is significantly high. (Refer to Figure 3-6.) YOUR TURN. Do Exercise 9 “ACT.”
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