Question

Calculate the variance for the data set. Round your answer to the nearest hundredth. Show all of your steps. {10, 19, 21, 28, 12, 20, 16}
simple

Answer 1

To calculate the variance for the data set \(\{10, 19, 21, 28, 12, 20, 16\}\), follow these steps:

### Step 1: Find the mean
First, we need to calculate the mean (average) of the data set.

\[
\text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}}
\]

Calculating the sum:

\[
10 + 19 + 21 + 28 + 12 + 20 + 16 = 126
\]

Counting the number of data points, we have 7 data points.

Now, calculate the mean:

\[
\text{Mean} = \frac{126}{7} = 18
\]

### Step 2: Calculate each data point's deviation from the mean
Next, we will calculate the deviation of each data point from the mean and then square it:

- \( (10 - 18)^2 = (-8)^2 = 64 \)
- \( (19 - 18)^2 = (1)^2 = 1 \)
- \( (21 - 18)^2 = (3)^2 = 9 \)
- \( (28 - 18)^2 = (10)^2 = 100 \)
- \( (12 - 18)^2 = (-6)^2 = 36 \)
- \( (20 - 18)^2 = (2)^2 = 4 \)
- \( (16 - 18)^2 = (-2)^2 = 4 \)

### Step 3: Sum of squared deviations
Now, sum these squared deviations:

\[
64 + 1 + 9 + 100 + 36 + 4 + 4 = 218
\]

### Step 4: Calculate the variance
Finally, the variance for a sample is calculated using the formula:

\[
\text{Variance} = \frac{\text{Sum of squared deviations}}{N}
\]

Where \(N\) is the number of data points. In this case, \(N = 7\).

\[
\text{Variance} = \frac{218}{7} \approx 31.14
\]

### Final Answer
Rounding this to the nearest hundredth, the variance of the data set \(\{10, 19, 21, 28, 12, 20, 16\}\) is:

\[
\text{Variance} \approx 31.14
\]