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 work

Answer 1

To calculate the variance of the data set {10, 19, 21, 28, 12, 20, 16}, we will follow these steps:

### Step 1: Find the mean (average) of the data set.
The mean is calculated by summing all the values and dividing by the number of values.

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

Calculating the sum:

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

Now divide by the number of values (7):

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

### Step 2: Calculate each data point's deviation from the mean, then square each deviation.
Now we find the deviations and their squares:

1. \( 10 - 18 = -8 \) → \( (-8)^2 = 64 \)
2. \( 19 - 18 = 1 \) → \( (1)^2 = 1 \)
3. \( 21 - 18 = 3 \) → \( (3)^2 = 9 \)
4. \( 28 - 18 = 10 \) → \( (10)^2 = 100 \)
5. \( 12 - 18 = -6 \) → \( (-6)^2 = 36 \)
6. \( 20 - 18 = 2 \) → \( (2)^2 = 4 \)
7. \( 16 - 18 = -2 \) → \( (-2)^2 = 4 \)

### Step 3: Sum the squared deviations.
Now we add all the squared deviations:

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

### Step 4: Calculate the variance.
To find the variance, we divide the sum of the squared deviations by the number of data points (n = 7):

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

### Step 5: Round the result.
Finally, rounding to the nearest hundredth, the variance is:

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

Thus, the variance of the data set {10, 19, 21, 28, 12, 20, 16} is **31.14**.