Question

Calculate the variance for the data set. Show all of your steps.
{4, 16, 21, 32, 11, 12}

Answer 1

To calculate the variance of a data set, we need to follow these steps:

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

Let's go through these steps for the data set {4, 16, 21, 32, 11, 12}.

### Step 1: Calculate the Mean
To find the mean, sum all the numbers in the data set and divide by the number of data points.

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

\[
\text{Mean} = \frac{4 + 16 + 21 + 32 + 11 + 12}{6} = \frac{96}{6} = 16
\]

### Step 2: Calculate the Squared Differences from the Mean
Now we'll find the difference between each data point and the mean, then square that difference:

- For 4: \((4 - 16)^2 = (-12)^2 = 144\)
- For 16: \((16 - 16)^2 = 0^2 = 0\)
- For 21: \((21 - 16)^2 = 5^2 = 25\)
- For 32: \((32 - 16)^2 = 16^2 = 256\)
- For 11: \((11 - 16)^2 = (-5)^2 = 25\)
- For 12: \((12 - 16)^2 = (-4)^2 = 16\)

Now, we can list the squared differences:
- For 4: \(144\)
- For 16: \(0\)
- For 21: \(25\)
- For 32: \(256\)
- For 11: \(25\)
- For 12: \(16\)

### Step 3: Calculate the Variance
To calculate the variance, sum the squared differences and divide by the number of data points.

\[
\text{Sum of squared differences} = 144 + 0 + 25 + 256 + 25 + 16 = 466
\]

Now, divide by the number of data points (6):

\[
\text{Variance} = \frac{\text{Sum of squared differences}}{\text{Number of data points}} = \frac{466}{6} \approx 77.67
\]

Thus, the variance of the data set \{4, 16, 21, 32, 11, 12\} is approximately **77.67**.