To calculate the standard deviation of a data set, we follow these steps:
- Calculate the mean (average) of the data set.
- Subtract the mean from each number to find the deviation from the mean.
- Square each of the deviations.
- Find the average of these squared deviations.
- Take the square root of that average to get the standard deviation.
Let's go step-by-step using the given weights:
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Calculate the mean:
Sum of weights: \[ 217 + 235 + 212 + 253 + 233 + 171 + 184 + 301 + 256 + 297 + 247 + 226 + 237 + 241 + 195 + 204 + 197 + 214 + 166 + 309 + 281 + 303 + 309 + 299 = 5493 \]
Number of weights: 24
Mean: \[ \text{Mean} = \frac{\text{Total Sum}}{\text{Number of Items}} = \frac{5493}{24} \approx 228.875 \]
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Calculate the deviations from the mean:
Deviations: \[ \begin{align*} 217 - 228.875 & = -11.875 \ 235 - 228.875 & = 6.125 \ 212 - 228.875 & = -16.875 \ 253 - 228.875 & = 24.125 \ 233 - 228.875 & = 4.125 \ 171 - 228.875 & = -57.875 \ 184 - 228.875 & = -44.875 \ 301 - 228.875 & = 72.125 \ 256 - 228.875 & = 27.125 \ 297 - 228.875 & = 68.125 \ 247 - 228.875 & = 18.125 \ 226 - 228.875 & = -2.875 \ 237 - 228.875 & = 8.125 \ 241 - 228.875 & = 12.125 \ 195 - 228.875 & = -33.875 \ 204 - 228.875 & = -24.875 \ 197 - 228.875 & = -31.875 \ 214 - 228.875 & = -14.875 \ 166 - 228.875 & = -62.875 \ 309 - 228.875 & = 80.125 \ 281 - 228.875 & = 52.125 \ 303 - 228.875 & = 74.125 \ 309 - 228.875 & = 80.125 \ 299 - 228.875 & = 70.125 \ \end{align*} \]
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Square the deviations:
Squared deviations: \[ \begin{align*} (-11.875)^2 & = 141.266 \ (6.125)^2 & = 37.515625 \ (-16.875)^2 & = 284.390625 \ (24.125)^2 & = 581.640625 \ (4.125)^2 & = 17.015625 \ (-57.875)^2 & = 3355.640625 \ (-44.875)^2 & = 2015.515625 \ (72.125)^2 & = 5200.640625 \ (27.125)^2 & = 735.265625 \ (68.125)^2 & = 4636.640625 \ (18.125)^2 & = 328.515625 \ (-2.875)^2 & = 8.265625 \ (8.125)^2 & = 66.015625 \ (12.125)^2 & = 147.015625 \ (-33.875)^2 & = 1145.390625 \ (-24.875)^2 & = 620.640625 \ (-31.875)^2 & = 1012.890625 \ (-14.875)^2 & = 221.640625 \ (-62.875)^2 & = 3943.390625 \ (80.125)^2 & = 6420.015625 \ (52.125)^2 & = 2725.640625 \ (74.125)^2 & = 5497.265625 \ (80.125)^2 & = 6420.015625 \ (70.125)^2 & = 4925.015625 \ \end{align*} \]
Total squared deviations: \[ \text{Total Squared Deviations} = 141.266 + 37.515625 + 284.390625 + 581.640625 + 17.015625 + 3355.640625 + 2015.515625 + 5200.640625 + 735.265625 + 4636.640625 + 328.515625 + 8.265625 + 66.015625 + 147.015625 + 1145.390625 + 620.640625 + 1012.890625 + 221.640625 + 3943.390625 + 6420.015625 + 2725.640625 + 5497.265625 + 6420.015625 + 4925.015625 = 318271.99 \]
To find the average of the squared deviations: \[ \text{Variance} = \frac{\text{Total Squared Deviations}}{N} = \frac{318271.99}{24} \approx 13261.33375 \]
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Calculate the standard deviation: \[ \text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{13261.33375} \approx 115.116 \]
Thus, the standard deviation of the weights of the football team is approximately 115.12 pounds.
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