810 CHAPTER 12 Statistics In this book, the notation zx represents the z-score, or standard score, of the data value x. For example, z110 represents the z-score, or standard score, of the data value 110. If a normal distribution has a mean of 86 with a standard deviation of 12, a data value of 110 has a z-score or standard score of = − = = z 110 86 12 24 12 2 110 Therefore, a data value of 110 in this distribution has a z-score of 2 and is two standard deviations above the mean. Data values below the mean will always have negative z-scores; data values above the mean will always have positive z-scores. The mean will always have a z-score of 0. MATHEMATICS TODAY Six Sigma VDB Photos/Shutterstock Many companies use a process called Six Sigma, a qualitycontrol strategy, to help the company improve quality and reduce errors. Six Sigma refers to an interval in a normal distribution from six standard deviations below the mean to six standard deviations above the mean. As 99.9997% of a normal distribution is within six standard deviations of the mean, Six Sigma means the company’s goal is to produce error-free products 99.9997% of the time. Companies such as General Electric (GE), Amazon, and Caterpillar have all reported success after implementing Six Sigma. GE estimates that by using Six Sigma, it was able to save approximately $30 billion during the first 6 years of implementation. Thousands of companies worldwide use Six Sigma. Their employees can earn Six Sigma certification online indicating they have achieved a comprehensive level of knowledge in Six Sigma principles and methodology. Why This Is Important Many companies use statistics to improve quality control and reduce costs. z-Scores or Standard Scores The formula for determining z -scores or standard scores is = − z value of the piece of data mean standard deviation Example 1 Determining z-Scores A normal distribution has a mean of 80 and a standard deviation of 14. Determine z-scores for the following data values. a) 94 b) 115 c) 80 d) 59 Solution a) = − z value mean standard deviation = − = = z 94 80 14 14 14 1 94 A data value of 94 has a z-score of 1. Therefore, a data value of 94 is one standard deviation above the mean. b) = − = = z 115 80 14 35 14 2.5 115 A data value of 115 has a z-score of 2.5, and is 2.5 standard deviations above the mean. c) = − = = z 80 80 14 0 14 0 80 A data value of 80 has a z-score of 0. The mean always has a z-score of 0. d) = − = − = − z 59 80 14 21 14 1.5 59 A data value of 59 has a z-score of −1.5 and is 1.5 standard deviations below the mean. 7 Now try Exercise 15 Exploring z -scores
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