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# Standard Deviation of a Data Set

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# Variance of a Data Set - Answer Key

## 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, $x$” 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.

The answers for these will vary depending on the set of athlete’s that each student chooses. This is an example of what a student’s responses would look like.

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

1. Find the mean, $\overline{x}$. The mean is 803,800.
2. Find the median.The median is 833,000.
3.Find the mode.The mode is 833,000.
4. Find the range.The range is 334,000.
5. Complete the table by calculating the difference from the mean (deviation from the mean) and the squared difference (squared deviation).
Example Table – Highly Paid International Soccer Players.
Athletes’ Salary $x$ Deviation from mean, $x- \overline{x}$ Squared deviation, $(x- \overline{x})^2$
1,000,000 (in euros) Christian Ronaldo 196,200 38,494,440,000
875,000 Lionel Messi 71,200 5,069,440,000
833,000 Fernando Torres 29,200 852,640,000
833,000 Yaya Toure 29,200 852,640,000
833,000 Ricardo Kaka 29,200 852,640,000
833,000 Zlatan Ibrahimovic 29,200 852,640,000
791,000 Wayne Rooney -12800 163,840,000
666,000 Carlos Tevez -137,800 18,988,840,000
666,000 Samuel Eto’o -137,800 18,988,840,000
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.
The variance for this data set is 10,477,066,670. This shows that the players’ salaries are very wide spread.
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.
The standard deviation for this set is 102,357.54, meaning that on average the values are 102,357.54 units away from the mean. In this example, the salaries are on average 102,357.54 euros away from the mean salary. This shows that the data is spread out and not closely clustered to the mean.
8. What makes calculating the standard deviation different from calculating the range?
In order to calculate the range you only subtract the minimum value in the set from the maximum value, and it indicates how spread out the values are. Meanwhile the standard deviation shows how the data is clustered because it measures how close the data is to the mean.
9. What must be true about a data set if the standard deviation is 0? Explain.
If the standard deviation is zero then all the data in your set is the same. In this situation, that would mean that all the players would be paid exactly the same amount.
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