122 CHAPTER 3 Describing, Exploring, and Comparing Data 2. z scores are expressed as numbers with no units of measurement. 3. Using the same range rule of thumb introduced in the preceding section, a data value is significantly low if its z score is less than or equal to -2 or the value is significantly high if its z score is greater than or equal to +2. 4. If an individual data value is less than the mean, its corresponding z score is a negative number. Using z Scores to Identify Significant Values In Section 3-2 we used the range rule of thumb to conclude that a value is significantly low or significantly high if it is at least 2 standard deviations away from the mean. It follows that significantly low values have z scores less than or equal to -2 and significantly high values have z scores greater than or equal to +2, as illustrated in Figure 3-6. Using this criterion with the two individual values used in Example 1, we see that the weight of the quarter is significantly high because its z score is greater than +2, but the body temperature is neither significantly low nor significantly high because its z score is between -2 and +2. EXAMPLE 1 Comparing Adult Body Temperature and Weight of a Quarter Which of the following two data values is more extreme relative to the data set from which it came? ■ The 99°F temperature of an adult (among 106 adults with sample mean x = 98.20°F and sample standard deviation s = 0.62°F) ■ The 5.7790 g weight of a quarter (among 40 quarters with sample mean x = 5.63930 g and sample standard deviation s = 0.06194 g) SOLUTION The 99°F body temperature and the 5.7790 g weight of a quarter can be standardized by converting each of them to z scores as shown below. 99°F body temperature: z = x - x s = 99oF - 98.20oF 0.62 oF = 1.29 5.7790 g weight of a quarter: z = x - x s = 5.7790 g - 5.63930 g 0.06194 g = 2.26 INTERPRETATION The z scores show that the 99°F body temperature is 1.29 standard deviations above the mean, and the 5.7790 g weight of the quarter is 2.26 standard deviations above the mean. Because the weight of the quarter is farther above the mean, it is the more extreme value. A weight of 5.7790 g of a quarter is more extreme than a 99°F body temperature. YOUR TURN. Do Exercise 13 “Tallest and Shortest Men.”
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