3.4 Measures of Position3.4 Measures of PositionLEARNING OBJECTIVES FOR THIS SECTION1Determine and Interpret z-Scores2Interpret Percentiles3Determine and Interpret Quartiles4Determine and Interpret the Interquartile Range5Check a Set of Data for OutliersThis section discusses measures of position, which describe the relative positionof a certain data value within the entire set of data.OBJECTIVE 1 Determine and Interpret zz-ScoresFor the 2013 baseball season, the Boston Red Sox led the American League with 853853 runs scored, whereas the St. Louis Cardinals led the National League with 783783 runs scored. It appears that the Red Sox were the better run-producing team. However, this comparison is unfair because the teams play in different leagues. The Red Sox play in the American League, where the designated hitter bats for the pitcher, whereas the Cardinals play in the National League, where the pitcher must bat (pitchers are typically poor hitters). To compare the teams' runs scored, we must determine their relative standings within their respective leagues. We can do this using a zz-score. The videoexplains the formula.DEFINITION

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The z-score represents the distance that a data value is from the mean in terms of the number of standard deviations. We find it by subtracting the mean from the data value and dividing this result by the standard deviation.Population zz-scorez=x−μσz=x−μσSz=The zz-score is unitless. It has mean 00 and standard deviation 1.If a data value is larger than the mean, the zz-score is positive. If a data value is smaller than the mean, the zz-score is negative. If the data value equals the mean, the zz-score is zero. A zz-score measures the number of standard deviations an observation is above or below the mean. For example, a zz-score of 1.241.24means the data value is 1.241.24 standarddeviations above the mean. A z-score of −2.31-2.31 means the data value is 2.312.31 standard deviations below the mean.NOTERound zz-scores to the nearest hundredth.ProblemDetermine whether the Boston Red Sox or the St. Louis Cardinals had a relatively better run-producing season. The Red Sox scored 853853 runs and play in the American League, where the mean number of runs scored was μ=701.7μ=701.7 and the standard deviation was σ=60.5 runsσ=60.5 runs. The Cardinals scored 783783 runs and play in the National League, where the mean number of runs scored was μ=648.7μ=648.7 and the standard deviation was σ=75.7 runs.σ=75.7 runs.ApproachThe word "relatively" suggests we are comparing two data sets. Therefore, compute each team's z-score. The team with the higher zz-score had the

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better season. Because we know the values of the population parameters, compute the population zz-score.SolutionCompute each team’s zz-score, rounded to two decimal places.Red Sox's z-score=x−μσ=853−701.760.5=2.50 Red S Cardinals' s z-score=x−μσ=783−648.775.7=1.77Cardinals' s zSo the Red Sox had a run production 2.502.50 standard deviations above the mean, whereas the Cardinals had a run production 1.771.77 standard