Question

Calculate the standard deviation of the following data.
3, 4, 5, 6, 2, 3, 12, 79, 5

26.3

616.9

694.2

24.8

Answer 1

To calculate the standard deviation of the given data set \(3, 4, 5, 6, 2, 3, 12, 79, 5\), we can follow these steps:

1. **Calculate the mean (average)** of the data set.
2. **Calculate the variance** by finding the average of the squared differences from the mean.
3. **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 (\( \sigma^2 \)):
1. Find the difference from the mean, square it, and take the average of those squared differences.
2. 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**.