To find the population standard deviation, we'll follow these steps:

1. Find the mean (average) of the data set:

$\text{Mean} = \frac{\sum \text{(all data points)}}{\text{Number of data points}}$

$\text{Mean} = \frac{21 + 37 + 22 + 34 + 40}{5} = \frac{154}{5} = 30.8$

2. Calculate each data point's deviation from the mean and square it:

• $(21 - 30.8)^2 = (-9.8)^2 = 96.04$
• $(37 - 30.8)^2 = (6.2)^2 = 38.44$
• $(22 - 30.8)^2 = (-8.8)^2 = 77.44$
• $(34 - 30.8)^2 = (3.2)^2 = 10.24$
• $(40 - 30.8)^2 = (9.2)^2 = 84.64$

3. Find the average of these squared deviations (this gives the variance):

$\text{Variance} = \frac{\sum (\text{Squared deviations})}{\text{Number of data points}}$

$\text{Variance} = \frac{96.04 + 38.44 + 77.44 + 10.24 + 84.64}{5} = \frac{306.8}{5} = 61.36$

4. Take the square root of the variance to find the standard deviation:

$\text{Standard Deviation} = \sqrt{61.36} \approx 7.83$

Rounding to the hundredths place, the population standard deviation is 7.83.

Would you like more details, or have any questions?

Relative Questions:

1. How does the population standard deviation differ from the sample standard deviation?
2. Why is it important to round your answers to a specific decimal place?
3. How would adding an extreme value to this data set affect the standard deviation?
4. What are some examples of when you would need to calculate population standard deviation in real life?
5. How does the standard deviation help in understanding the distribution of data?

Tip: Always use the population formula when you are working with the entire data set, as opposed to a sample.