To calculate the variance of the given data, we need to follow a few steps. We'll first find the mean, but since you provided the mean (241), we will use it in our variance calculation.
Steps to Calculate Variance:
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List of Data Points: We have the data points as follows: \[ 159, 175, 189, 190, 191, 202, 208, 210, 218, 220, 232, 234, 234, 253, 253, 261, 265, 275, 295, 295, 304, 305, 308, 308 \]
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Count the Number of Data Points (n): There are 24 data points.
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Calculate the Squared Differences from the Mean: We will calculate \((x_i - \text{mean})^2\) for each data point \(x_i\).
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Sum of Squared Differences: We will sum up all the squared differences.
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Calculate Variance: The variance is given by: \[ \text{Variance} = \frac{\text{Sum of Squared Differences}}{n} \]
Calculating the Squared Differences:
\[ \begin{align*} (159 - 241)^2 & = ( -82 )^2 = 6724 \ (175 - 241)^2 & = ( -66 )^2 = 4356 \ (189 - 241)^2 & = ( -52 )^2 = 2704 \ (190 - 241)^2 & = ( -51 )^2 = 2601 \ (191 - 241)^2 & = ( -50 )^2 = 2500 \ (202 - 241)^2 & = ( -39 )^2 = 1521 \ (208 - 241)^2 & = ( -33 )^2 = 1089 \ (210 - 241)^2 & = ( -31 )^2 = 961 \ (218 - 241)^2 & = ( -23 )^2 = 529 \ (220 - 241)^2 & = ( -21 )^2 = 441 \ (232 - 241)^2 & = ( -9 )^2 = 81 \ (234 - 241)^2 & = ( -7 )^2 = 49 \ (234 - 241)^2 & = ( -7 )^2 = 49 \ (253 - 241)^2 & = ( 12 )^2 = 144 \ (253 - 241)^2 & = ( 12 )^2 = 144 \ (261 - 241)^2 & = ( 20 )^2 = 400 \ (265 - 241)^2 & = ( 24 )^2 = 576 \ (275 - 241)^2 & = ( 34 )^2 = 1156 \ (295 - 241)^2 & = ( 54 )^2 = 2916 \ (295 - 241)^2 & = ( 54 )^2 = 2916 \ (304 - 241)^2 & = ( 63 )^2 = 3969 \ (305 - 241)^2 & = ( 64 )^2 = 4096 \ (308 - 241)^2 & = ( 67 )^2 = 4489 \ (308 - 241)^2 & = ( 67 )^2 = 4489 \ \end{align*} \]
Sum of Squared Differences:
Now let's sum these squared differences: \[ 6724 + 4356 + 2704 + 2601 + 2500 + 1521 + 1089 + 961 + 529 + 441 + 81 + 49 + 49 + 144 + 144 + 400 + 576 + 1156 + 2916 + 2916 + 3969 + 4096 + 4489 + 4489 = 49607 \]
Calculate Variance:
Now we can calculate the variance. \[ \text{Variance} = \frac{49607}{24} \approx 2067.79 \]
Thus, the variance of the data is approximately 2067.79.