Math / Data & Statistics

QuestionThe weights (in pounds) of nine players from a college football team are recorded as follows. $\begin{array}{lllllllll} 204 & 219 & 305 & 291 & 265 & 286 & 303 & 253 & 261 \end{array}$ Send data to Excel
Find the range, variance, and standard deviation. Round the variance to one decimal place and the standard deviation to two decimal places, if necessary. Round intermediate calculations to one decimal place.
Part: $0 / 3$
Part 1 of 3
The range is $\square$ pounds.

Studdy Solution

STEP 1

What is this asking? We need to find how spread out these football player weights are, using range, variance, and standard deviation. Watch out! Don't mix up variance and standard deviation!
Variance is a squared measure of spread, while standard deviation is the square root of the variance, giving us a more interpretable measure in the original units (pounds).
Also, remember to round correctly at each step!

STEP 2

1. Calculate the range
2. Calculate the mean
3. Calculate the variance
4. Calculate the standard deviation

STEP 3

The maximum weight is $\text{max}\{204, 219, 305, 291, 265, 286, 303, 253, 261\} = \textbf{305}$ pounds.
Woohoo, big guy!

STEP 4

The minimum weight is $\text{min}\{204, 219, 305, 291, 265, 286, 303, 253, 261\} = \textbf{204}$ pounds.
Still impressive!

STEP 5

The range is the difference between the maximum and minimum values.
So, $\textbf{Range} = 305 - 204 = \textbf{101}$ pounds.
That's a pretty big difference in weight!

STEP 6

Add up all the weights: $204 + 219 + 305 + 291 + 265 + 286 + 303 + 253 + 261 = \textbf{2387}$ pounds.
That's a whole lot of football player!

STEP 7

We have $\textbf{9}$ players, so the mean weight is $\frac{2387}{9} \approx \textbf{265.2}$ pounds.
Remember, we're rounding to one decimal place for intermediate calculations!

STEP 8

For each weight, subtract the mean ( $265.2$ ) and square the result.
Let's do it: $(204 - 265.2)^2 \approx 3733.0$ $(219 - 265.2)^2 \approx 2134.4$ $(305 - 265.2)^2 \approx 1584.0$ $(291 - 265.2)^2 \approx 665.5$ $(265 - 265.2)^2 \approx 0.0$ $(286 - 265.2)^2 \approx 432.6$ $(303 - 265.2)^2 \approx 1421.3$ $(253 - 265.2)^2 \approx 148.8$ $(261 - 265.2)^2 \approx 17.6$

STEP 9

Now, add up all those squared differences: $3733.0 + 2134.4 + 1584.0 + 665.5 + 0.0 + 432.6 + 1421.3 + 148.8 + 17.6 \approx \textbf{10137.2}$ .

STEP 10

Divide the sum of squared differences by $9 - 1 = 8$ to get the variance: $\frac{10137.2}{8} \approx \textbf{1267.2}$ pounds squared.

STEP 11

The standard deviation is just the square root of the variance: $\sqrt{1267.2} \approx \textbf{35.60}$ pounds.
Rounded to two decimal places!

STEP 12

The range is $\textbf{101}$ pounds.
The variance is $\textbf{1267.2}$ pounds squared.
The standard deviation is $\textbf{35.60}$ pounds.

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Studdy Solution

STEP 1

STEP 2

1. Calculate the range
2. Calculate the mean
3. Calculate the variance
4. Calculate the standard deviation

STEP 3

The maximum weight is $\text{max}\{204, 219, 305, 291, 265, 286, 303, 253, 261\} = \textbf{305}$ pounds.
Woohoo, big guy!

STEP 4

The minimum weight is $\text{min}\{204, 219, 305, 291, 265, 286, 303, 253, 261\} = \textbf{204}$ pounds.
Still impressive!

STEP 5

The range is the difference between the maximum and minimum values.
So, $\textbf{Range} = 305 - 204 = \textbf{101}$ pounds.
That's a pretty big difference in weight!

STEP 6

Add up all the weights: $204 + 219 + 305 + 291 + 265 + 286 + 303 + 253 + 261 = \textbf{2387}$ pounds.
That's a whole lot of football player!

STEP 7

We have $\textbf{9}$ players, so the mean weight is $\frac{2387}{9} \approx \textbf{265.2}$ pounds.
Remember, we're rounding to one decimal place for intermediate calculations!

STEP 8

STEP 9

Now, add up all those squared differences: $3733.0 + 2134.4 + 1584.0 + 665.5 + 0.0 + 432.6 + 1421.3 + 148.8 + 17.6 \approx \textbf{10137.2}$ .

STEP 10

Divide the sum of squared differences by $9 - 1 = 8$ to get the variance: $\frac{10137.2}{8} \approx \textbf{1267.2}$ pounds squared.

STEP 11

The standard deviation is just the square root of the variance: $\sqrt{1267.2} \approx \textbf{35.60}$ pounds.
Rounded to two decimal places!

STEP 12

The range is $\textbf{101}$ pounds.
The variance is $\textbf{1267.2}$ pounds squared.
The standard deviation is $\textbf{35.60}$ pounds.

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Studdy Solution

STEP 1

STEP 2

1. Calculate the range2. Calculate the mean3. Calculate the variance4. Calculate the standard deviation

STEP 3

The maximum weight is max{204,219,305,291,265,286,303,253,261}=305\text{max}\{204, 219, 305, 291, 265, 286, 303, 253, 261\} = \textbf{305}max{204,219,305,291,265,286,303,253,261}=305 pounds.Woohoo, big guy!

STEP 4

The minimum weight is min{204,219,305,291,265,286,303,253,261}=204\text{min}\{204, 219, 305, 291, 265, 286, 303, 253, 261\} = \textbf{204}min{204,219,305,291,265,286,303,253,261}=204 pounds.Still impressive!

STEP 5

The range is the difference between the **maximum** and **minimum** values.So, Range=305−204=101\textbf{Range} = 305 - 204 = \textbf{101}Range=305−204=101 pounds.That's a pretty big difference in weight!

STEP 6

Add up all the weights: 204+219+305+291+265+286+303+253+261=2387204 + 219 + 305 + 291 + 265 + 286 + 303 + 253 + 261 = \textbf{2387}204+219+305+291+265+286+303+253+261=2387 pounds.That's a whole lot of football player!

STEP 7

We have 9\textbf{9}9 players, so the mean weight is 23879≈265.2\frac{2387}{9} \approx \textbf{265.2}92387​≈265.2 pounds.Remember, we're rounding to one decimal place for intermediate calculations!

STEP 8

STEP 9

Now, add up all those squared differences: 3733.0+2134.4+1584.0+665.5+0.0+432.6+1421.3+148.8+17.6≈10137.23733.0 + 2134.4 + 1584.0 + 665.5 + 0.0 + 432.6 + 1421.3 + 148.8 + 17.6 \approx \textbf{10137.2}3733.0+2134.4+1584.0+665.5+0.0+432.6+1421.3+148.8+17.6≈10137.2.

STEP 10

Divide the sum of squared differences by 9−1=89 - 1 = 89−1=8 to get the variance: 10137.28≈1267.2\frac{10137.2}{8} \approx \textbf{1267.2}810137.2​≈1267.2 pounds squared.

STEP 11

The standard deviation is just the square root of the variance: 1267.2≈35.60\sqrt{1267.2} \approx \textbf{35.60}1267.2​≈35.60 pounds.Rounded to two decimal places!

STEP 12

The range is 101\textbf{101}101 pounds.The variance is 1267.2\textbf{1267.2}1267.2 pounds squared.The standard deviation is 35.60\textbf{35.60}35.60 pounds.

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1. Calculate the range
2. Calculate the mean
3. Calculate the variance
4. Calculate the standard deviation

The maximum weight is $\text{max}\{204, 219, 305, 291, 265, 286, 303, 253, 261\} = \textbf{305}$ pounds.
Woohoo, big guy!

The minimum weight is $\text{min}\{204, 219, 305, 291, 265, 286, 303, 253, 261\} = \textbf{204}$ pounds.
Still impressive!

The range is the difference between the maximum and minimum values.
So, $\textbf{Range} = 305 - 204 = \textbf{101}$ pounds.
That's a pretty big difference in weight!

Add up all the weights: $204 + 219 + 305 + 291 + 265 + 286 + 303 + 253 + 261 = \textbf{2387}$ pounds.
That's a whole lot of football player!

We have $\textbf{9}$ players, so the mean weight is $\frac{2387}{9} \approx \textbf{265.2}$ pounds.
Remember, we're rounding to one decimal place for intermediate calculations!

Now, add up all those squared differences: $3733.0 + 2134.4 + 1584.0 + 665.5 + 0.0 + 432.6 + 1421.3 + 148.8 + 17.6 \approx \textbf{10137.2}$ .

Divide the sum of squared differences by $9 - 1 = 8$ to get the variance: $\frac{10137.2}{8} \approx \textbf{1267.2}$ pounds squared.

The standard deviation is just the square root of the variance: $\sqrt{1267.2} \approx \textbf{35.60}$ pounds.
Rounded to two decimal places!

The range is $\textbf{101}$ pounds.
The variance is $\textbf{1267.2}$ pounds squared.
The standard deviation is $\textbf{35.60}$ pounds.