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

1 answer

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.