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Chapter 3: Problem 5

Find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as "minutes") in your results. (The same data were used in Section 3-1, where we found measures of center. Here we find measures of variation.) Then answer the given questions. Listed below are the jersey numbers of 11 players randomly selected from the roster of the Seattle Seahawks when they won Super Bowl XLVIII. What do the results tell us? \(\begin{array}{llllllllllll}89 & 91 & 55 & 7 & 20 & 99 & 25 & 81 & 19 & 82 & 60\end{array}\)

Short Answer

Expert verified
Range: 92Variance: 883.19Standard Deviation: 29.72The jersey numbers have wide variations.

Step by step solution

01

- List the Data Set

The data set containing the jersey numbers is: 89, 91, 55, 7, 20, 99, 25, 81, 19, 82, 60.
02

- Calculate the Range

To find the range, subtract the smallest number in the data set from the largest number. Range = 99 - 7 = 92.
03

- Calculate the Mean

The mean is calculated by summing all data points and dividing by the number of data points. Mean, \(\bar{x}\) = \(\frac{89 + 91 + 55 + 7 + 20 + 99 + 25 + 81 + 19 + 82 + 60}{11}\) \(\bar{x} = \frac{628}{11} \approx 57.09\)
04

- Calculate the Variance

Variance is the average of the squared deviations from the mean. First, find each deviation from the mean: \(x - \bar{x}\). Then square each deviation, sum those squares, and divide by \(n - 1\) (for a sample). \begin{align*} \text{Variance}, \(s^2\) &= \frac{(89 - 57.09)^2 + (91 - 57.09)^2 + (55 - 57.09)^2 + (7 - 57.09)^2 + (20 - 57.09)^2 + (99 - 57.09)^2 + (25 - 57.09)^2 + (81 - 57.09)^2 + (19 - 57.09)^2 + (82 - 57.09)^2 + (60 - 57.09)^2}{10} \&= \frac{8831.91}{10} \&= 883.19 \text{So, the variance is } 883.19. \end{align*}
05

- Calculate the Standard Deviation

The standard deviation is the square root of the variance. \[ \text{Standard Deviation}, \(s\) = \sqrt{883.19} \approx 29.72 \]
06

- Summary of Results

The calculated measures of variation for the jersey numbers are as follows: Range = 92 Variance = 883.19 Standard Deviation \( \approx \) 29.72These results show the spread of jersey numbers among the selected players. A large range and standard deviation suggest that the jersey numbers vary widely.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

range in statistics
In statistics, the range is a simple measure of variation that tells us how spread apart the values in a data set are. It is calculated by subtracting the smallest value from the largest value in the data set. In our exercise, the jersey numbers range from 7 to 99. This gives us a range of \(99 - 7 = 92\). So, the jersey numbers spread across 92 units. This large range indicates significant variability among the jersey numbers. It is important to use the range to get an initial sense of how much the data points differ from one another.
variance calculation
The variance is a more detailed measure of how much the data values differ from the mean. It considers all data points and quantifies their spread by calculating the average of the squared deviations from the mean.

To calculate variance:
  • First, find the mean of the data set. For the jersey numbers, the mean is approximately 57.09.
  • Next, subtract the mean from each data point to get the deviations.
  • Square each deviation to eliminate negative values and emphasize larger differences.
  • Sum all the squared deviations.
  • Finally, divide this sum by the number of data points minus one (for a sample), that is, \(n - 1\).
For our data, the variance formula looks like this: \begin{align*} \text{Variance}, \(s^2\) &= \frac{(89 - 57.09)^2 + (91 - 57.09)^2 + (55 - 57.09)^2 + (7 - 57.09)^2 + (20 - 57.09)^2 + (99 - 57.09)^2 + (25 - 57.09)^2 + (81 - 57.09)^2 + (19 - 57.09)^2 + (82 - 57.09)^2 + (60 - 57.09)^2}{10} \ &= 883.19 \text{So, the variance is } 883.19.
standard deviation
The standard deviation is another crucial measure of variation. It tells us how much the individual data points typically differ from the mean. It is the square root of the variance, making it easier to understand as it brings the measure back to the same units as the original data.

To calculate the standard deviation, you simply take the square root of the variance. For our jersey numbers, the variance is approximately 883.19, so the standard deviation \(s\) is: \[ \text{Standard Deviation}, \(s\) = \sqrt{883.19} \approx 29.72 \] This means, on average, each player's jersey number differs from the mean by approximately 29.72. A higher standard deviation indicates that the data points are more spread out. In our case, a standard deviation of ~29.72 is quite large, showing a substantial spread in jersey numbers among the selected players. Understanding standard deviation helps in assessing the consistency and reliability of the data.

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Most popular questions from this chapter

The 20 brain volumes \(\left(\mathrm{cm}^{3}\right)\) from Data Set 8 "IQ and Brain Size" in Appendix B vary from a low of \(963 \mathrm{~cm}^{3}\) to a high of \(1439 \mathrm{~cm}^{3}\). Use the range rule of thumb to estimate the standard deviation \(s\) and compare the result to the exact standard deviation of \(124.9 \mathrm{~cm}^{3}\).

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Watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, \((b)\) median, (c) mode, (d) midrange, and then answer the given question. Listed below are the highest amounts of net worth (in millions of dollars) of celebrities. The celebrities are Tom Cruise, Will Smith, Robert De Niro, Drew Carey, George Clooney, John Travolta, Samuel L. Jackson, Larry King, Demi Moore, and Bruce Willis. What do the results tell us about the population of all celebrities? Based on the nature of the amounts, what can be inferred about their precision? $$ \begin{array}{llllllllll} 250 & 200 & 185 & 165 & 160 & 160 & 150 & 150 & 150 & 150 \end{array} $$

Find the mean and median for each of the two samples, then compare the two sets of results. Waiting times (in seconds) of customers at the Madison Savings Bank are recorded with two configurations: single customer line; individual customer lines. Carefully examine the data to determine whether there is a difference between the two data sets that is not apparent from a comparison of the measures of center. If so, what is it? \(\begin{array}{lllllllllll}\text { Single Line } & 390 & 396 & 402 & 408 & 426 & 438 & 444 & 462 & 462 & 462 \\ \text { Individual Lines } & 252 & 324 & 348 & 372 & 402 & 462 & 462 & 510 & 558 & 600\end{array}\)

The geometric mean is often used in business and economics for finding average rates of change, average rates of growth, or average ratios. To find the geometric mean of \(n\) values (all of which are positive), first multiply the values, then find the \(n\) th root of the product. For a 6-year period, money deposited in annual certificates of deposit had annual interest rates of \(5.154 \%, 2.730 \%, 0.488 \%, 0.319 \%, 0.313 \%\), and \(0.268 \%\). Identify the single percentage growth rate that is the same as the five consecutive growth rates by computing the geometric mean of \(1.05154,1.02730,1.00488,1.00319,1.00313\), and \(1.00268\).
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