SECTION 2.5 Measures of Position 107 Finding a Percentile For the data set in Example 2, find the percentile that corresponds to $57,000. SOLUTION The costs of tuition and fees are in thousands of dollars, so $57,000 is the data entry 57. Begin by ordering the data. 19 30 39 44 45 46 48 48 48 48 55 55 55 55 56 57 57 58 58 59 59 59 60 61 61 There are 15 data entries less than 57 and the total number of data entries is 25. Percentile of 57 = Number of data entries less than 57 Total number of entries # 100 = 15 25 # 100 = 60 The cost of $57,000 corresponds to the 60th percentile. Interpretation The cost of $57,000 is greater than 60% of the other tuition costs. TRY IT YOURSELF 6 For the data set in Try It Yourself 2, find the percentile that corresponds to $31,000, which is the data entry 31. Answer: Page A37 The Standard Score When you know the mean and standard deviation of a data set, you can measure the position of an entry in the data set with a standard score, or z-score. The standard score, or z-score, represents the number of standard deviations a value x lies from the mean m. To find the z-score for a value, use the formula z = Value - Mean Standard deviation = x - m s . DEFINITION A z-score can be negative, positive, or zero. When z is negative, the corresponding x-value is less than the mean. When z is positive, the corresponding x-value is greater than the mean. For z = 0, the corresponding x-value is equal to the mean. A z-score can be used to identify an unusual value of a data set that is approximately bell-shaped. When a distribution is approximately bell-shaped, you know from the Empirical Rule that about 95% of the data lie within 2 standard deviations of the mean. So, when this distribution’s values are transformed to z-scores, about 95% of the z-scores should fall between -2 and 2. A z-score outside of this range will occur about 5% of the time and would be considered unusual. So, according to the Empirical Rule, a z-score less than -3 or greater than 3 would be very unusual, with such a score occurring about 0.3% of the time. EXAMPLE 6 −3 −2 −1 0 1 2 3 Usual scores Unusual scores z–score Very unusual scores
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