Let's solve the problem step by step. We are asked to find the sample variance and the standard deviation for the data set:

$21, 14, 3, 10, 9$

Step 1: Calculate the sample mean (average)

The formula for the mean is:

$\bar{x} = \frac{\sum x_i}{n}$

Where:

• $x_i$ are the data points.
• $n$ is the number of data points.

$\bar{x} = \frac{21 + 14 + 3 + 10 + 9}{5} = \frac{57}{5} = 11.4$

Step 2: Calculate the squared differences from the mean

For each data point, subtract the mean and square the result:

$(21 - 11.4)^2 = 92.16$ $(14 - 11.4)^2 = 6.76$ $(3 - 11.4)^2 = 70.56$ $(10 - 11.4)^2 = 1.96$ $(9 - 11.4)^2 = 5.76$

Step 3: Calculate the sample variance

The sample variance $s^2$ is the average of the squared differences, but since it is a sample variance, we divide by $n - 1$ (degrees of freedom):

$s^2 = \frac{92.16 + 6.76 + 70.56 + 1.96 + 5.76}{5 - 1} = \frac{177.2}{4} = 44.3$

So the sample variance $s^2 = 44.3$, which matches the answer provided in the image.

Step 4: Calculate the sample standard deviation

The standard deviation $s$ is the square root of the sample variance:

$s = \sqrt{44.3} \approx 6.7$

Final Answers:

• Sample variance $s^2 = 44.3$ (already marked correctly in the image).
• Sample standard deviation $s \approx 6.7$.

Would you like to go into more details or have any questions?

Additional Questions:

1. How is population variance different from sample variance?
2. Why do we divide by $n - 1$ in the sample variance calculation?
3. Can you explain why variance is squared and not just the absolute difference?
4. How does the sample size affect the standard deviation?
5. What is the interpretation of standard deviation in real-world data?

Tip: The standard deviation provides a measure of how spread out the data is around the mean.