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 of the data set, we will follow these steps:

1. **Find the mean (average)** of the data set.
2. **Calculate the squared deviations** from the mean.
3. **Find the average of those squared deviations** (this is the variance).

Let's go through the calculations step by step:

### Step 1: Calculate the Mean

The mean is calculated by summing all the numbers and dividing by the count of numbers.

The data set is: \(10, 19, 21, 28, 12, 20, 16\).

**Sum of the data set:**
\[
10 + 19 + 21 + 28 + 12 + 20 + 16 = 126
\]

**Count of the data set:**
\[
7 \text{ (there are 7 numbers)}
\]

**Mean:**
\[
\text{Mean} = \frac{\text{Sum}}{\text{Count}} = \frac{126}{7} = 18
\]

### Step 2: Calculate the Squared Deviations

We subtract the mean from each number in the data set and square the result.

\[
\begin{align*}
(10 - 18)^2 & = (-8)^2 = 64 \\
(19 - 18)^2 & = (1)^2 = 1 \\
(21 - 18)^2 & = (3)^2 = 9 \\
(28 - 18)^2 & = (10)^2 = 100 \\
(12 - 18)^2 & = (-6)^2 = 36 \\
(20 - 18)^2 & = (2)^2 = 4 \\
(16 - 18)^2 & = (-2)^2 = 4 \\
\end{align*}
\]

### Step 3: Sum the Squared Deviations

Now, sum these squared deviations:

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

### Step 4: Calculate the Variance

The variance is found by averaging the squared deviations. The formula for variance (\( \sigma^2 \)) for a sample is:

\[
\text{Variance} = \frac{\text{Sum of squared deviations}}{n - 1}
\]

Where \( n \) is the number of data points. Since we have 7 data points, we use \( n - 1 = 6 \):

\[
\text{Variance} = \frac{218}{6} \approx 36.3333
\]

### Step 5: Round to the Nearest Hundredth

Now, round the variance to the nearest hundredth:

\[
\text{Variance} \approx 36.33
\]

### Final Answer

The variance of the data set is approximately **36.33**.