Question

In the data set below, what is the standard deviation?
2 9 8 2 6 3 5
If the answer is a decimal, round it to the nearest tenth.

standard deviation (σ):

Answer 1

To calculate the standard deviation of the data set \( 2, 9, 8, 2, 6, 3, 5 \), we can follow these steps:

1. **Calculate the mean (average)** of the data set.
2. **Calculate the squared deviations** from the mean.
3. **Calculate the variance** (which is the average of the squared deviations).
4. **Take the square root of the variance** to find the standard deviation.

### Step 1: Calculate the mean
\[
\text{Mean} = \frac{\sum \text{data}}{n} = \frac{2 + 9 + 8 + 2 + 6 + 3 + 5}{7} = \frac{35}{7} = 5
\]

### Step 2: Calculate the squared deviations from the mean
\[
(2 - 5)^2 = (-3)^2 = 9
\]
\[
(9 - 5)^2 = (4)^2 = 16
\]
\[
(8 - 5)^2 = (3)^2 = 9
\]
\[
(2 - 5)^2 = (-3)^2 = 9
\]
\[
(6 - 5)^2 = (1)^2 = 1
\]
\[
(3 - 5)^2 = (-2)^2 = 4
\]
\[
(5 - 5)^2 = (0)^2 = 0
\]

### Step 3: Calculate the variance
Now we sum up the squared deviations:
\[
9 + 16 + 9 + 9 + 1 + 4 + 0 = 48
\]
Now divide by the number of data points (since this is a population standard deviation):
\[
\text{Variance} = \frac{48}{7} \approx 6.8571
\]

### Step 4: Calculate the standard deviation
\[
\sigma = \sqrt{\text{Variance}} = \sqrt{6.8571} \approx 2.617
\]

### Final result
Rounding to the nearest tenth, the standard deviation \( \sigma \) is approximately:
\[
\text{Standard deviation } (\sigma) \approx 2.6
\]

Thus, the final answer is:
\[
\boxed{2.6}
\]