To calculate the variance of the data set {10, 19, 21, 28, 12, 20, 16}, we will follow these steps:
Step 1: Find the mean (average) of the data set.
The mean is calculated by summing all the values and dividing by the number of values.
\[ \text{Mean} = \frac{10 + 19 + 21 + 28 + 12 + 20 + 16}{7} \]
Calculating the sum:
\[ 10 + 19 + 21 + 28 + 12 + 20 + 16 = 126 \]
Now divide by the number of values (7):
\[ \text{Mean} = \frac{126}{7} = 18 \]
Step 2: Calculate each data point's deviation from the mean, then square each deviation.
Now we find the deviations and their squares:
- \( 10 - 18 = -8 \) → \( (-8)^2 = 64 \)
- \( 19 - 18 = 1 \) → \( (1)^2 = 1 \)
- \( 21 - 18 = 3 \) → \( (3)^2 = 9 \)
- \( 28 - 18 = 10 \) → \( (10)^2 = 100 \)
- \( 12 - 18 = -6 \) → \( (-6)^2 = 36 \)
- \( 20 - 18 = 2 \) → \( (2)^2 = 4 \)
- \( 16 - 18 = -2 \) → \( (-2)^2 = 4 \)
Step 3: Sum the squared deviations.
Now we add all the squared deviations:
\[ 64 + 1 + 9 + 100 + 36 + 4 + 4 = 218 \]
Step 4: Calculate the variance.
To find the variance, we divide the sum of the squared deviations by the number of data points (n = 7):
\[ \text{Variance} = \frac{218}{7} \approx 31.14 \]
Step 5: Round the result.
Finally, rounding to the nearest hundredth, the variance is:
\[ \text{Variance} \approx 31.14 \]
Thus, the variance of the data set {10, 19, 21, 28, 12, 20, 16} is 31.14.
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