# Applications of Variance and Standard Deviation

## Analyzing data using technology.

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Variance of a Data Set
Teacher Contributed

## Variance of Professional Athletes’ Salaries

### Topic

Analyzing Professional Athletes’ Salaries using Variance and Standard Deviation

### Vocabulary

• Mean
• Median
• Mode
• Range
• Difference from mean
• Squared Difference
• Variance
• Standard Deviation

### Student Exploration

#### What is your favorite sport? What do professional athletes in that sport earn?

What is your favorite sport? Pick ten athletes from your favorite sport and record all of their salaries in the table below (in the “Athlete’s Salary, x\begin{align*}x\end{align*}” column). Use the website at the end of this section to find the salaries of your ten favorite athletes, or you may conduct your own research.

Then find the following information for those ten athletes’ annual salary.

1. Find the mean, x¯¯¯\begin{align*}\overline{x}\end{align*}.
2. Find the median.
3. Find the mode.
4. Find the range.
5. Complete the table by calculating the difference from the mean (deviation from the mean) and the squared difference (squared deviation).
Sample Table
Athletes’ Salary x\begin{align*}x\end{align*} Deviation from mean, xx¯¯¯\begin{align*}x-\overline{x}\end{align*} Squared deviation, (xx¯¯¯)2\begin{align*}(x-\overline{x})^2\end{align*}
6. Calculate the variance. Again, the variance is a measure of the dispersion and its value is lower for tightly grouped data than for widely spread data. What does the variance say about your data? Explain.
7. Calculate the standard deviation. Again, the standard deviation measures how closely the data clusters around the mean. It is the square root of the variance.
8. What makes calculating the standard deviation different from calculating the range?
9. What must be true about a data set if the standard deviation is 0? Explain.
10. Use technology to check the mean, standard deviation, and variance that you found in #1, 6 and 7. For step-by-step instructions on how to use your calculator to check these revisit the “read” tab of the “Applications of Variance and Standard Deviation” chapter/concept.

Websites for Pro Athletes’ Salaries:

Major League Soccer in U.S. (MLS): http://www.mlsplayers.org/salary_info.html

International Soccer Players: http://www.theoffside.com/world-football/futebol-finances-100-best-paychecks-2011.html

WNBA: must research by individual athlete

Rugby: must research by individual athlete

Professional Golfers’ Association (PGA): http://www.pgatour.com/r/stats/info/?109

### Extension Investigation

Research a second sport and ten new professional athletes’ salaries. Complete questions 1-9 for the second set of athletes. Them compare the variance and standard deviation for the two different sports.

### Resources Cited

Major League Soccer in U.S. (MLS): http://www.mlsplayers.org/salary_info.html

International Soccer Players: http://www.theoffside.com/world-football/futebol-finances-100-best-paychecks-2011.html

WNBA: must research by individual athlete

Rugby: must research by individual athlete

Professional Golfers’ Association (PGA): http://www.pgatour.com/r/stats/info/?109

### Connections to other CK-12 Subject Areas

• Variance of a Data Set
• Standard Deviation of a Data Set
• Applications of Variance and Standard Deviation

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