To calculate the standard deviation of the data set \( 2, 9, 8, 2, 6, 3, 5 \), we can follow these steps:
- Calculate the mean (average) of the data set.
- Calculate the squared deviations from the mean.
- Calculate the variance (which is the average of the squared deviations).
- 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: \[ \boxed{2.6} \]