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 given data set \(\{10, 19, 21, 28, 12, 20, 16\}\), we can follow these steps:

### Step 1: Calculate the Mean
First, we need to find the mean (average) of the data set.

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

Calculating the sum:
\[
10 + 19 + 21 + 28 + 12 + 20 + 16 = 126
\]

Now, divide by the number of data points (which is 7):

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

### Step 2: Calculate Deviations from the Mean
Next, we need to find the deviations from the mean for each data point, and then square those deviations.

\[
\text{Deviations:}
\]
- \(10 - 18 = -8\) \(\Rightarrow (-8)^2 = 64\)
- \(19 - 18 = 1\) \(\Rightarrow (1)^2 = 1\)
- \(21 - 18 = 3\) \(\Rightarrow (3)^2 = 9\)
- \(28 - 18 = 10\) \(\Rightarrow (10)^2 = 100\)
- \(12 - 18 = -6\) \(\Rightarrow (-6)^2 = 36\)
- \(20 - 18 = 2\) \(\Rightarrow (2)^2 = 4\)
- \(16 - 18 = -2\) \(\Rightarrow (-2)^2 = 4\)

### Step 3: Calculate the Sum of Squared Deviations
Now, we will sum these squared deviations:

\[
\text{Sum of Squared Deviations} = 64 + 1 + 9 + 100 + 36 + 4 + 4 = 218
\]

### Step 4: Calculate the Variance
Finally, we find the variance. There are two types of variance calculations: population variance and sample variance. Since we have a complete data set (a population), we will use the population variance formula:

\[
\text{Population Variance} = \frac{\text{Sum of Squared Deviations}}{N}
\]

where \(N\) is the number of data points. Here, \(N = 7\).

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

### Step 5: Round to the Nearest Hundredth
Thus, the variance of the data set rounded to the nearest hundredth is:

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

### Final Answer
\[
\text{The variance of the data set is } 31.14.
\]