To find the variance and standard deviation for the given data sets, we will follow these steps:
### Part (i): Data: 6, 7, 10, 12, 13, 4, 8, 12
**Step 1: Calculate the Mean (x̄)**
Mean is calculated using the formula:
\[ \text{Mean} (x̄) = \frac{\sum x_i}{n} \]
Where \( n \) is the number of observations.
Calculating:
\[ x̄ = \frac{6 + 7 + 10 + 12 + 13 + 4 + 8 + 12}{8} = \frac{72}{8} = 9 \]
**Step 2: Calculate the Variance (σ²)**
Variance is calculated using the formula:
\[ \sigma^2 = \frac{\sum (x_i - x̄)^2}{n} \]
Calculating:
\[
\sigma^2 = \frac{(6 - 9)^2 + (7 - 9)^2 + (10 - 9)^2 + (12 - 9)^2 + (13 - 9)^2 + (4 - 9)^2 + (8 - 9)^2 + (12 - 9)^2}{8}
\]
\[
= \frac{(-3)^2 + (-2)^2 + (1)^2 + (3)^2 + (4)^2 + (-5)^2 + (-1)^2 + (3)^2}{8}
\]
\[
= \frac{9 + 4 + 1 + 9 + 16 + 25 + 1 + 9}{8}
\]
\[
= \frac{74}{8} = 9.25
\]
**Step 3: Calculate the Standard Deviation (σ)**
Standard deviation is the square root of variance:
\[ \sigma = \sqrt{\sigma^2} = \sqrt{9.25} \approx 3.04 \]
### Part (ii): Data: 5, 12, 3, 18, 6, 8, 2, 10
**Step 1: Calculate the Mean (x̄)**
Calculating:
\[ x̄ = \frac{5 + 12 + 3 + 18 + 6 + 8 + 2 + 10}{8} = \frac{64}{8} = 8 \]
**Step 2: Calculate the Variance (σ²)**
Calculating:
\[
\sigma^2 = \frac{(5 - 8)^2 + (12 - 8)^2 + (3 - 8)^2 + (18 - 8)^2 + (6 - 8)^2 + (8 - 8)^2 + (2 - 8)^2 + (10 - 8)^2}{8}
\]
\[
= \frac{(-3)^2 + (4)^2 + (-5)^2 + (10)^2 + (-2)^2 + (0)^2 + (-6)^2 + (2)^2}{8}
\]
\[
= \frac{9 + 16 + 25 + 100 + 4 + 0 + 36 + 4}{8}
\]
\[
= \frac{194}{8} = 24.25
\]
**Step 3: Calculate the Standard Deviation (σ)**
Calculating:
\[ \sigma = \sqrt{24.25} \approx 4.92 \]
### Part (iii): Data: 350, 361, 370, 373, 376, 379, 385, 387, 394, 395
**Step 1: Calculate the Mean (x̄)**
Calculating:
\[ x̄ = \frac{350 + 361 + 370 + 373 + 376 + 379 + 385 + 387 + 394 + 395}{10} = \frac{3770}{10} = 377 \]
**Step 2: Calculate the Variance (σ²)**
Calculating:
\[
\sigma^2 = \frac{(350 - 377)^2 + (361 - 377)^2 + (370 - 377)^2 + (373 - 377)^2 + (376 - 377)^2 + (379 - 377)^2 + (385 - 377)^2 + (387 - 377)^2 + (394 - 377)^2 + (395 - 377)^2}{10}
\]
\[
= \frac{(-27)^2 + (-16)^2 + (-7)^2 + (-4)^2 + (-1)^2 + (2)^2 + (8)^2 + (10)^2 + (17)^2 + (18)^2}{10}
\]
\[
= \frac{729 + 256 + 49 + 16 + 1 + 4 + 64 + 100 + 289 + 324}{10}
\]
\[
= \frac{1832}{10} = 183.2
\]
**Step 3: Calculate the Standard Deviation (σ)**
Calculating:
\[ \sigma = \sqrt{183.2} \approx 13.54 \]
### Summary of Results:
- For (i): Variance = 9.25, Standard Deviation ≈ 3.04
- For (ii): Variance = 24.25, Standard Deviation ≈ 4.92
- For (iii): Variance = 183.2, Standard Deviation ≈ 13.54
