To calculate the variance of a data set, we will follow these steps:
- Calculate the mean (average) of the data.
- Subtract the mean from each data point and square the result (these are the squared deviations).
- Calculate the average of those squared deviations to find the variance.
Let's show the work step-by-step for the data set: 10, 19, 21, 28, 12, 20, 16.
Step 1: Calculate the Mean
\[ \text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}} = \frac{10 + 19 + 21 + 28 + 12 + 20 + 16}{7} \] \[ \text{Mean} = \frac{126}{7} = 18 \]
Step 2: Calculate the Squared Deviations
Next, we will find the squared deviations 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 \(16\): \((16 - 18)^2 = (-2)^2 = 4\)
Step 3: Calculate the Variance
Now, we sum all the squared deviations:
\[ \text{Sum of squared deviations} = 64 + 1 + 9 + 100 + 36 + 4 + 4 = 218 \]
Since we are working with a sample, we will divide by \(n - 1\) (where \(n\) is the number of data points, which is 7):
\[ \text{Variance} = \frac{218}{n - 1} = \frac{218}{7 - 1} = \frac{218}{6} \approx 36.33 \]
Step 4: Round to the Nearest Hundredth
When rounding 36.33 to the nearest hundredth, we find:
\[ \text{Variance} \approx 36.33 \]
Therefore, the variance of the data set (rounded to the nearest hundredth) is 36.33.
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