A z score indicates how far above or below the mean a data value is in terms of the standard deviation. The z score is defined as
Problem 16 from Section 14.4 uses z score to decide what is more “impressive”.
Problem 16 In 1949, Jackie Robinson hit .342 for the Brooklyn Dodgers. In 1973, Rod Carew hit .350 for the Minnesota Twins. In the 1940s, the mean batting average was .267 and the standard deviation was .0326. In the 1970s, the mean batting average was .261 and the standard deviation was .0317. Determine which batting average was more impressive. Explain/show how you determined your solution.
Solution To determine which average is more impressive, we need to discover which batting average is higher than the mean. To do this we need to consider the standard deviation which indicates how spread out the distribution of scores is. A z score helps us do this.
For Jackie Robinson, we use the mean and standard deviation for his decade to get
This means Jackie Robinson’s batting average is about 2.3 standard deviations above the mean.
For Rod Carew, the z score is
Rod Carew’s batting average is 2.81 standard deviations above average. Since he is farther above average than Jackie Robinson’s batting average, Rod Carew’s average is more impressive.