To calculate the variance and standard deviation for the given dataset, we can follow these steps:
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Find the Mean (μ): You have provided the mean as 241.
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Calculate the Variance (σ²): The formula for variance is: \[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n} \] where \(x_i\) are the data points, \(\mu\) is the mean, and \(n\) is the number of data points.
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Calculate the Sum of Squared Differences from the Mean:
Given 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
Calculating the squared differences from the mean for each data point:
- \( (159 - 241)^2 = (−82)^2 = 6724 \)
- \( (175 - 241)^2 = (−66)^2 = 4356 \)
- \( (189 - 241)^2 = (−52)^2 = 2704 \)
- \( (190 - 241)^2 = (−51)^2 = 2601 \)
- \( (191 - 241)^2 = (−50)^2 = 2500 \)
- \( (202 - 241)^2 = (−39)^2 = 1521 \)
- \( (208 - 241)^2 = (−33)^2 = 1089 \)
- \( (210 - 241)^2 = (−31)^2 = 961 \)
- \( (218 - 241)^2 = (−23)^2 = 529 \)
- \( (220 - 241)^2 = (−21)^2 = 441 \)
- \( (232 - 241)^2 = (−9)^2 = 81 \)
- \( (234 - 241)^2 = (−7)^2 = 49 \)
- \( (234 - 241)^2 = (−7)^2 = 49 \)
- \( (253 - 241)^2 = (12)^2 = 144 \)
- \( (253 - 241)^2 = (12)^2 = 144 \)
- \( (261 - 241)^2 = (20)^2 = 400 \)
- \( (265 - 241)^2 = (24)^2 = 576 \)
- \( (275 - 241)^2 = (34)^2 = 1156 \)
- \( (295 - 241)^2 = (54)^2 = 2916 \)
- \( (295 - 241)^2 = (54)^2 = 2916 \)
- \( (304 - 241)^2 = (63)^2 = 3969 \)
- \( (305 - 241)^2 = (64)^2 = 4096 \)
- \( (308 - 241)^2 = (67)^2 = 4489 \)
- \( (308 - 241)^2 = (67)^2 = 4489 \)
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Sum of Squared Differences: \[ 6724 + 4356 + 2704 + 2601 + 2500 + 1521 + 1089 + 961 + 529 + 441 + 81 + 49 + 49 + 144 + 144 + 400 + 576 + 1156 + 2916 + 2916 + 3969 + 4096 + 4489 + 4489 = 68858 \]
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Count of Data Points (n): There are 24 data points.
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Calculate Variance: \[ \sigma^2 = \frac{68858}{24} \approx 2869.08 \]
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Calculate Standard Deviation (σ): \[ \sigma = \sqrt{\sigma^2} \approx \sqrt{2869.08} \approx 53.67 \]
Final Results:
- Variance (σ²): ≈ 2869.08
- Standard Deviation (σ): ≈ 53.67