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218+234+210+261+265+175+191+308+253+295+253+220+234+232+189+208+190+202+159+304+275+305+308+295

Part 5:
Calculate the variance of the data.
Part 6:
Calculate the standard deviation of the data.

1 answer

To calculate the variance and standard deviation of the provided data, we need to follow these steps:

  1. Calculate the mean.
  2. Calculate the variance.
  3. Calculate the standard deviation.

Step 1: Calculate the Mean

First, let's sum the data:

\[ \text{Data} = [218, 234, 210, 261, 265, 175, 191, 308, 253, 295, 253, 220, 234, 232, 189, 208, 190, 202, 159, 304, 275, 305, 308, 295] \]

Sum of Data: \[ \text{Sum} = 218 + 234 + 210 + 261 + 265 + 175 + 191 + 308 + 253 + 295 + 253 + 220 + 234 + 232 + 189 + 208 + 190 + 202 + 159 + 304 + 275 + 305 + 308 + 295 = 6,248 \]

Number of Observations (n): \[ n = 24 \]

Mean (μ): \[ \text{Mean} (μ) = \frac{\text{Sum}}{n} = \frac{6248}{24} \approx 260.33 \]

Step 2: Calculate the Variance

Next, we calculate the variance using the formula:

\[ \sigma^2 = \frac{\sum (x_i - μ)^2}{n} \]

Where \(x_i\) represents each individual data point.

Calculating \( (x_i - μ)^2 \) for each data point:

  1. \( (218 - 260.33)^2 \approx 1778.34 \)
  2. \( (234 - 260.33)^2 \approx 688.51 \)
  3. \( (210 - 260.33)^2 \approx 2528.71 \)
  4. \( (261 - 260.33)^2 \approx 0.44 \)
  5. \( (265 - 260.33)^2 \approx 21.78 \)
  6. \( (175 - 260.33)^2 \approx 7305.88 \)
  7. \( (191 - 260.33)^2 \approx 4788.18 \)
  8. \( (308 - 260.33)^2 \approx 2318.73 \)
  9. \( (253 - 260.33)^2 \approx 53.44 \)
  10. \( (295 - 260.33)^2 \approx 1225.71 \)
  11. \( (253 - 260.33)^2 \approx 53.44 \)
  12. \( (220 - 260.33)^2 \approx 1615.82 \)
  13. \( (234 - 260.33)^2 \approx 688.51 \)
  14. \( (232 - 260.33)^2 \approx 798.71 \)
  15. \( (189 - 260.33)^2 \approx 5119.78 \)
  16. \( (208 - 260.33)^2 \approx 2735.78 \)
  17. \( (190 - 260.33)^2 \approx 4973.34 \)
  18. \( (202 - 260.33)^2 \approx 3342.46 \)
  19. \( (159 - 260.33)^2 \approx 10381.14 \)
  20. \( (304 - 260.33)^2 \approx 1918.98 \)
  21. \( (275 - 260.33)^2 \approx 215.72 \)
  22. \( (305 - 260.33)^2 \approx 2016.98 \)
  23. \( (308 - 260.33)^2 \approx 2318.73 \)
  24. \( (295 - 260.33)^2 \approx 1225.71 \)

Sum of Squared Deviations: \[ \sum (x_i - μ)^2 = 1778.34 + 688.51 + 2528.71 + 0.44 + 21.78 + 7305.88 + 4788.18 + 2318.73 + 53.44 + 1225.71 + 53.44 + 1615.82 + 688.51 + 798.71 + 5119.78 + 2735.78 + 4973.34 + 3342.46 + 10381.14 + 1918.98 + 215.72 + 2016.98 + 2318.73 + 1225.71 = 12716.81 \]

Variance (σ²): \[ \text{Variance} (σ^2) = \frac{\sum (x_i - μ)^2}{n} \approx \frac{12716.81}{24} \approx 529.03 \]

Step 3: Calculate the Standard Deviation

Finally, the standard deviation is the square root of the variance:

\[ σ = \sqrt{σ^2} = \sqrt{529.03} \approx 23.05 \]

Summary

  • Variance: \( \approx 529.03 \)
  • Standard Deviation: \( \approx 23.05 \)