Question

the ages (in years) and heights (in inches) of all pitchers for a baseball team are listed. find the coefficient of variation for each of the two data sets. then compare the results. click the icon to view the data sets. cv_heights = % (round to one decimal place as needed.)

Explanation:

Step1: Calculate the mean of heights

The formula for the mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$.
For heights \(x=\{71,75,71,80,79,72,76,72,73,71,72,74\}\), \(n = 12\).
\(\sum_{i=1}^{12}x_{i}=71 + 75+71+80+79+72+76+72+73+71+72+74=876\)
\(\bar{x}_{heights}=\frac{876}{12}=73\)

Step2: Calculate the standard deviation of heights

The formula for the population standard deviation \(\sigma=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}}\)
\((x_{1}-\bar{x})^{2}=(71 - 73)^{2}=4\), \((x_{2}-\bar{x})^{2}=(75 - 73)^{2}=4\), \((x_{3}-\bar{x})^{2}=(71 - 73)^{2}=4\), \((x_{4}-\bar{x})^{2}=(80 - 73)^{2}=49\), \((x_{5}-\bar{x})^{2}=(79 - 73)^{2}=36\), \((x_{6}-\bar{x})^{2}=(72 - 73)^{2}=1\), \((x_{7}-\bar{x})^{2}=(76 - 73)^{2}=9\), \((x_{8}-\bar{x})^{2}=(72 - 73)^{2}=1\), \((x_{9}-\bar{x})^{2}=(73 - 73)^{2}=0\), \((x_{10}-\bar{x})^{2}=(71 - 73)^{2}=4\), \((x_{11}-\bar{x})^{2}=(72 - 73)^{2}=1\), \((x_{12}-\bar{x})^{2}=(74 - 73)^{2}=1\)
\(\sum_{i = 1}^{12}(x_{i}-\bar{x})^{2}=4+4 + 4+49+36+1+9+1+0+4+1+1=115\)
\(\sigma_{heights}=\sqrt{\frac{115}{12}}\approx3.1\)

Step3: Calculate the coefficient of variation for heights

The formula for the coefficient of variation \(CV=\frac{\sigma}{\bar{x}}\times100\%\)
\(CV_{heights}=\frac{3.1}{73}\times 100\%\approx4.3\%\)

Answer:

\(4.3\)