To calculate the standard deviation of the given data set \(3, 4, 5, 6, 2, 3, 12, 79, 5\), we can follow these steps:
- Calculate the mean (average) of the data set.
- Calculate the variance by finding the average of the squared differences from the mean.
- Take the square root of the variance to get the standard deviation.
Step 1: Calculate the Mean
Mean (\( \mu \)) is calculated as: \[ \mu = \frac{\text{Sum of all values}}{\text{Total number of values}} \] \[ \mu = \frac{3 + 4 + 5 + 6 + 2 + 3 + 12 + 79 + 5}{9} = \frac{119}{9} \approx 13.22 \]
Step 2: Calculate the Variance
Next, we calculate the variance (\( \sigma^2 \)):
- Find the difference from the mean, square it, and take the average of those squared differences.
- The squared differences are calculated as follows:
\[ (3 - 13.22)^2 = (-10.22)^2 \approx 104.4484 \] \[ (4 - 13.22)^2 = (-9.22)^2 \approx 84.9284 \] \[ (5 - 13.22)^2 = (-8.22)^2 \approx 67.4884 \] \[ (6 - 13.22)^2 = (-7.22)^2 \approx 52.3684 \] \[ (2 - 13.22)^2 = (-11.22)^2 \approx 125.0884 \] \[ (3 - 13.22)^2 = (-10.22)^2 \approx 104.4484 \] \[ (12 - 13.22)^2 = (-1.22)^2 \approx 1.4884 \] \[ (79 - 13.22)^2 = (65.78)^2 \approx 4326.0484 \] \[ (5 - 13.22)^2 = (-8.22)^2 \approx 67.4884 \]
Now sum them up: \[ 104.4484 + 84.9284 + 67.4884 + 52.3684 + 125.0884 + 104.4484 + 1.4884 + 4326.0484 + 67.4884 \approx 4670.96 \]
Now divide by \(n - 1\) (since this is a sample standard deviation): \[ \sigma^2 = \frac{4670.96}{9 - 1} = \frac{4670.96}{8} \approx 583.87 \]
Step 3: Calculate the Standard Deviation
Finally, take the square root of the variance to find the standard deviation (\( \sigma \)): \[ \sigma \approx \sqrt{583.87} \approx 24.17 \]
Conclusion
After calculating, the closest options for the standard deviation from those provided would be:
- 24.8
Thus, the correct answer is 24.8.
1 answer