Given 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 calculate the variance for the data mean 241 variance and Standard Deviation?

1 answer

To calculate the variance and standard deviation for the given dataset, we can follow these steps:

  1. Find the Mean (μ): You have provided the mean as 241.

  2. 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.

  3. 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:

  • \( (159 - 241)^2 = (−82)^2 = 6724 \)
  • \( (175 - 241)^2 = (−66)^2 = 4356 \)
  • \( (189 - 241)^2 = (−52)^2 = 2704 \)
  • \( (190 - 241)^2 = (−51)^2 = 2601 \)
  • \( (191 - 241)^2 = (−50)^2 = 2500 \)
  • \( (202 - 241)^2 = (−39)^2 = 1521 \)
  • \( (208 - 241)^2 = (−33)^2 = 1089 \)
  • \( (210 - 241)^2 = (−31)^2 = 961 \)
  • \( (218 - 241)^2 = (−23)^2 = 529 \)
  • \( (220 - 241)^2 = (−21)^2 = 441 \)
  • \( (232 - 241)^2 = (−9)^2 = 81 \)
  • \( (234 - 241)^2 = (−7)^2 = 49 \)
  • \( (234 - 241)^2 = (−7)^2 = 49 \)
  • \( (253 - 241)^2 = (12)^2 = 144 \)
  • \( (253 - 241)^2 = (12)^2 = 144 \)
  • \( (261 - 241)^2 = (20)^2 = 400 \)
  • \( (265 - 241)^2 = (24)^2 = 576 \)
  • \( (275 - 241)^2 = (34)^2 = 1156 \)
  • \( (295 - 241)^2 = (54)^2 = 2916 \)
  • \( (295 - 241)^2 = (54)^2 = 2916 \)
  • \( (304 - 241)^2 = (63)^2 = 3969 \)
  • \( (305 - 241)^2 = (64)^2 = 4096 \)
  • \( (308 - 241)^2 = (67)^2 = 4489 \)
  • \( (308 - 241)^2 = (67)^2 = 4489 \)
  1. Sum of Squared Differences: \[ 6724 + 4356 + 2704 + 2601 + 2500 + 1521 + 1089 + 961 + 529 + 441 + 81 + 49 + 49 + 144 + 144 + 400 + 576 + 1156 + 2916 + 2916 + 3969 + 4096 + 4489 + 4489 = 68858 \]

  2. Count of Data Points (n): There are 24 data points.

  3. Calculate Variance: \[ \sigma^2 = \frac{68858}{24} \approx 2869.08 \]

  4. 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