Question

Calculate the variance for the data set
10,19,21,28,12,20,16

Answer 1

To calculate the variance of a data set, you can follow these steps:

1. **Calculate the mean (average)** of the data set.
2. **Subtract the mean** from each data point and square the result.
3. **Calculate the average of those squared differences**. This value will be the variance.

Here’s how to do it for the data set: 10, 19, 21, 28, 12, 20, 16.

### Step 1: Calculate the mean

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

\[
= \frac{10 + 19 + 21 + 28 + 12 + 20 + 16}{7} = \frac{126}{7} = 18
\]

### Step 2: Calculate the squared differences from the mean

Now, we subtract the mean from each data point and square the result:

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

### Step 3: Calculate the average of those squared differences

Now, we find the sum of the squared differences and then divide by the number of data points (7 in this case since we are calculating the population variance):

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

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

Thus, the variance of the data set is approximately **31.14**.