To calculate the variance for the given data set \({10, 19, 21, 28, 12, 20, 16}\), we will follow these steps:
- Calculate the mean (average) of the data set.
- Find the squared differences from the mean for each data point.
- Calculate the average of those squared differences to find the variance.
Step 1: Calculate the Mean
First, we find the mean of the data set.
\[ \text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}} = \frac{10 + 19 + 21 + 28 + 12 + 20 + 16}{7} \]
Calculating the sum:
\[ 10 + 19 + 21 + 28 + 12 + 20 + 16 = 126 \]
Now, we divide by the number of data points (which is 7):
\[ \text{Mean} = \frac{126}{7} = 18 \]
Step 2: Find the Squared Differences
Next, we calculate the squared differences from the mean for each data point:
- For \(10\): \[ (10 - 18)^2 = (-8)^2 = 64 \]
- For \(19\): \[ (19 - 18)^2 = (1)^2 = 1 \]
- For \(21\): \[ (21 - 18)^2 = (3)^2 = 9 \]
- For \(28\): \[ (28 - 18)^2 = (10)^2 = 100 \]
- For \(12\): \[ (12 - 18)^2 = (-6)^2 = 36 \]
- For \(20\): \[ (20 - 18)^2 = (2)^2 = 4 \]
- For \(16\): \[ (16 - 18)^2 = (-2)^2 = 4 \]
Now, we list the squared differences:
\[ {64, 1, 9, 100, 36, 4, 4} \]
Step 3: Calculate the Variance
Now, we find the average of the squared differences (this gives us the variance).
First, compute the sum of the squared differences:
\[ 64 + 1 + 9 + 100 + 36 + 4 + 4 = 218 \]
Now, we divide by the number of data points (7) to find the variance:
\[ \text{Variance} = \frac{218}{7} \approx 31.14 \]
Final Result
Thus, the variance of the data set \({10, 19, 21, 28, 12, 20, 16}\) rounded to the nearest hundredth is:
\[ \boxed{31.14} \]