Consider a sample with data values of 10, 20, 12, 17, and 16. Compute the variance and standarddeviation.

1. Break Down the Problem

To compute the variance and standard deviation, we will:

1. Calculate the mean of the data.
2. Determine the deviations of each data point from the mean.
3. Compute the squared deviations.
4. Calculate the variance.
5. Determine the standard deviation as the square root of the variance.

2. Relevant Concepts

• Mean: $\mu = \frac{\sum x_i}{N}$
• Variance: $\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}$
• Standard Deviation: $\sigma = \sqrt{\sigma^2}$

3. Analysis and Detail

Given data values: $10, 20, 12, 17, 16$

Step 1: Calculate the mean (μ) $\mu = \frac{10 + 20 + 12 + 17 + 16}{5} = \frac{75}{5} = 15$

Step 2: Determine the squared deviations from the mean

• For $x_1 = 10$: $(10 - 15)^2 = (-5)^2 = 25$
• For $x_2 = 20$: $(20 - 15)^2 = (5)^2 = 25$
• For $x_3 = 12$: $(12 - 15)^2 = (-3)^2 = 9$
• For $x_4 = 17$: $(17 - 15)^2 = (2)^2 = 4$
• For $x_5 = 16$: $(16 - 15)^2 = (1)^2 = 1$

Step 3: Compute the variance $\sigma^2 = \frac{(25 + 25 + 9 + 4 + 1)}{5} = \frac{64}{5} = 12.8$

Step 4: Compute the standard deviation $\sigma = \sqrt{12.8} \approx 3.58$

4. Verify and Summarize

The calculations for the mean, squared deviations, variance, and standard deviation have been verified step by step.

Final Answer

• Variance: $\sigma^2 = 12.8$
• Standard Deviation: $\sigma \approx 3.58$