To calculate the standard deviation for the data set provided $(4, 21, 11, 6, 8)$, we can follow these steps:

Step 1: Find the Mean

The mean $\mu$ is the sum of all values divided by the number of values.

$\mu = \frac{4 + 21 + 11 + 6 + 8}{5} = \frac{50}{5} = 10$

Step 2: Find the Variance

The variance is the average of the squared differences between each value and the mean.

1. Calculate each difference from the mean, square it, and then find the average:

   • $(4 - 10)^2 = (-6)^2 = 36$
   • $(21 - 10)^2 = (11)^2 = 121$
   • $(11 - 10)^2 = (1)^2 = 1$
   • $(6 - 10)^2 = (-4)^2 = 16$
   • $(8 - 10)^2 = (-2)^2 = 4$

2. Sum of squared differences: $36 + 121 + 1 + 16 + 4 = 178$

3. Divide by the number of data points to get the variance: $\text{Variance} = \frac{178}{5} = 35.6$

Step 3: Calculate the Standard Deviation

The standard deviation is the square root of the variance.

$\text{Standard Deviation} = \sqrt{35.6} \approx 5.97$

Rounded to the second decimal place, the standard deviation is approximately 5.97.

Would you like further details on any step, or do you have any questions?

Here are five related questions:

1. How would the standard deviation change if an additional player scored?
2. What does a high standard deviation indicate in this context?
3. How would we calculate the variance without rounding?
4. Why do we square the differences in standard deviation calculations?
5. How would this calculation differ for a population vs. a sample?

Tip: Standard deviation is a key measure in statistics for understanding data spread around the mean.