Question

Calculate the variance for the data set. Show me the steps please. 4, 16, 21, 32, 11, 12,

Answer 1

To calculate the variance of the data set \(4, 16, 21, 32, 11, 12\), we will follow these steps:

1. **Calculate 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. **Calculate the average of those squared deviations** to find the variance.

### Step 1: Calculate the Mean

The formula for the mean (\(\mu\)) is:

\[
\mu = \frac{\sum x_i}{n}
\]

where \(x_i\) is each value in the data set and \(n\) is the number of values.

Calculating the sum of the data set:

\[
4 + 16 + 21 + 32 + 11 + 12 = 96
\]

The number of values \(n\) is \(6\).

Calculating the mean:

\[
\mu = \frac{96}{6} = 16
\]

### Step 2: Calculate the Squared Deviations

Now we calculate each squared deviation from the mean:

- 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 \(12\):
\[
(12 - 16)^2 = (-4)^2 = 16
\]

### Step 3: Calculate the Variance

Now we sum the squared deviations:

\[
144 + 0 + 25 + 256 + 25 + 16 = 466
\]

To find the variance (\(\sigma^2\)), we divide the total squared deviations by the number of values \(n\) (if you are calculating the population variance) or by \(n-1\) (if you are calculating the sample variance).

Since we are not specified whether it's a sample or a population, I will calculate both:

- **Population Variance**:
\[
\sigma^2 = \frac{466}{6} \approx 77.67
\]

- **Sample Variance** (using \(n - 1\)):
\[
s^2 = \frac{466}{5} = 93.2
\]

### Final Results

- The population variance is approximately **77.67**.
- The sample variance is **93.2**.

You can use either based on whether your data set represents a complete population or a sample.