To calculate the variance of the data set, we will follow these steps:
- Find the mean (average) of the data set.
- Calculate the squared deviations from the mean.
- 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.
1 answer