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 can follow these steps:

1. **Find the mean (average)** of the data set.
2. **Subtract the mean from each data point** and square the result (this is called the squared deviation).
3. **Find the average of these squared deviations**.
4. The result is the variance.

Let's perform the calculations step by step for the data set: {4, 16, 21, 32, 11, 12}.

### Step 1: Calculate the Mean

The mean is calculated as follows:
\[
\text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}}
\]

Calculating the sum of the data points:
\[
4 + 16 + 21 + 32 + 11 + 12 = 96
\]

Now, dividing by the number of data points (which is 6):
\[
\text{Mean} = \frac{96}{6} = 16
\]

### Step 2: Calculate the Squared Deviations

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

- 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 compile the squared deviations:
\[
144, 0, 25, 256, 25, 16
\]

### Step 3: Calculate the Average of the Squared Deviations

Now, we find the mean of these squared deviations:
\[
\text{Sum of squared deviations} = 144 + 0 + 25 + 256 + 25 + 16 = 466
\]
\[
\text{Variance} = \frac{\text{Sum of squared deviations}}{\text{Number of data points}}
\]

**Since this is a sample variance**, we need to divide by \(n - 1\) where \(n\) is the number of data points. Here \(n = 6\).

Calculating the variance:
\[
\text{Variance} = \frac{466}{6 - 1} = \frac{466}{5} = 93.2
\]

### Final Result

Thus, the variance of the data set {4, 16, 21, 32, 11, 12} is:
\[
\boxed{93.2}
\]