To calculate the variance and standard deviation of the given data set, we can follow the steps below.
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List the data: \[ 159, 175, 189, 190, 191, 202, 208, 210, 218, 220, 232, 234, 234, 253, 253, 261, 265, 275, 295, 295, 304, 305, 308, 308 \]
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Calculate the mean (\(\mu\)): \[ \text{Mean} = \frac{\sum x_i}{n} \] where \( n \) is the total number of data points.
- Total number of data points \( n = 24 \).
- Sum of the data points: \[ 159 + 175 + 189 + 190 + 191 + 202 + 208 + 210 + 218 + 220 + 232 + 234 + 234 + 253 + 253 + 261 + 265 + 275 + 295 + 295 + 304 + 305 + 308 + 308 = 6351 \]
- Mean: \[ \mu = \frac{6351}{24} \approx 264.625 \]
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Calculate the variance (\(\sigma^2\)): \[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n} \] Now we subtract the mean from each data point, square the result, and sum those squared differences: \[ \begin{align*} (159 - 264.625)^2 & \approx 11127.390625 \ (175 - 264.625)^2 & \approx 8027.390625 \ (189 - 264.625)^2 & \approx 5704.890625 \ (190 - 264.625)^2 & \approx 5523.890625 \ (191 - 264.625)^2 & \approx 5436.890625 \ (202 - 264.625)^2 & \approx 3903.390625 \ (208 - 264.625)^2 & \approx 3244.890625 \ (210 - 264.625)^2 & \approx 2988.890625 \ (218 - 264.625)^2 & \approx 2197.890625 \ (220 - 264.625)^2 & \approx 1979.890625 \ (232 - 264.625)^2 & \approx 1068.890625 \ (234 - 264.625)^2 & \approx 929.390625 \ (234 - 264.625)^2 & \approx 929.390625 \ (253 - 264.625)^2 & \approx 136.390625 \ (253 - 264.625)^2 & \approx 136.390625 \ (261 - 264.625)^2 & \approx 12.390625 \ (265 - 264.625)^2 & \approx 0.140625 \ (275 - 264.625)^2 & \approx 105.390625 \ (295 - 264.625)^2 & \approx 917.390625 \ (295 - 264.625)^2 & \approx 917.390625 \ (304 - 264.625)^2 & \approx 1532.890625 \ (305 - 264.625)^2 & \approx 1605.890625 \ (308 - 264.625)^2 & \approx 1909.390625 \ (308 - 264.625)^2 & \approx 1909.390625 \ \end{align*} \]
Now sum all the squared differences: \[ \sum (x_i - \mu)^2 \approx 11127.390625 + 8027.390625 + 5704.890625 + 5523.890625 + 5436.890625 + 3903.390625 + 3244.890625 + 2988.890625 + 2197.890625 + 1979.890625 + 1068.890625 + 929.390625 + 929.390625 + 136.390625 + 136.390625 + 12.390625 + 0.140625 + 105.390625 + 917.390625 + 917.390625 + 1532.890625 + 1605.890625 + 1909.390625 + 1909.390625 \approx 34684.0 \]
Thus, the variance: \[ \sigma^2 \approx \frac{34684.0}{24} \approx 1445.16667 \]
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Calculate the standard deviation (\(\sigma\)): \[ \sigma = \sqrt{\sigma^2} \approx \sqrt{1445.16667} \approx 38.025 \]
Summary:
- Variance: Approximately 1445.17
- Standard Deviation: Approximately 38.03