Question

Calculate the variance for the sample data set. Round your answer to the nearest hundredth.

{10, 19, 21, 28, 12, 20, 16}

(2 points)
Variance =

Answer 1

To calculate the variance for the sample data set, we can follow these steps:

1. Calculate the mean (average) of the sample.
2. Subtract the mean from each data point and square the result (this gives us the squared differences).
3. Sum all the squared differences.
4. Divide the sum by \( n - 1 \), where \( n \) is the number of data points in the sample (this is because we're calculating sample variance).

Let's calculate it step-by-step:

### Step 1: Calculate the mean
\[
\text{Mean} = \frac{\text{Sum of data points}}{n} = \frac{10 + 19 + 21 + 28 + 12 + 20 + 16}{7}
\]
\[
\text{Mean} = \frac{126}{7} = 18
\]

### Step 2: Calculate squared differences from the mean
- \( (10 - 18)^2 = (-8)^2 = 64 \)
- \( (19 - 18)^2 = (1)^2 = 1 \)
- \( (21 - 18)^2 = (3)^2 = 9 \)
- \( (28 - 18)^2 = (10)^2 = 100 \)
- \( (12 - 18)^2 = (-6)^2 = 36 \)
- \( (20 - 18)^2 = (2)^2 = 4 \)
- \( (16 - 18)^2 = (-2)^2 = 4 \)

### Step 3: Sum of squared differences
\[
\text{Sum} = 64 + 1 + 9 + 100 + 36 + 4 + 4 = 218
\]

### Step 4: Calculate the sample variance
Since this is a sample variance, we divide by \( n - 1 \) where \( n = 7 \):
\[
\text{Sample Variance} = \frac{\text{Sum of squared differences}}{n - 1} = \frac{218}{7 - 1} = \frac{218}{6} \approx 36.33
\]

### Final Result
The sample variance rounded to the nearest hundredth is:
\[
\text{Variance} \approx 36.33
\]

So, the final answer is:
\[
\text{Variance} = 36.33
\]