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}

Answer 1

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

1. **Calculate the mean (average)** of the data set.
2. **Find the squared differences** from the mean for each data point.
3. **Calculate the average of those squared differences** to find the variance.

### Step 1: Calculate the Mean

First, we find the mean of the data set.

\[
\text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}} = \frac{10 + 19 + 21 + 28 + 12 + 20 + 16}{7}
\]

Calculating the sum:

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

Now, we divide by the number of data points (which is 7):

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

### Step 2: Find the Squared Differences

Next, we calculate the squared differences from the mean for each data point:

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

Now, we list the squared differences:

\[
\{64, 1, 9, 100, 36, 4, 4\}
\]

### Step 3: Calculate the Variance

Now, we find the average of the squared differences (this gives us the variance).

First, compute the sum of the squared differences:

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

Now, we divide by the number of data points (7) to find the variance:

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

### Final Result

Thus, the variance of the data set \(\{10, 19, 21, 28, 12, 20, 16\}\) rounded to the nearest hundredth is:

\[
\boxed{31.14}
\]