To calculate the population variance and standard deviation for the data set $10, 13, 17, 19, 21$, follow these steps:

1. Find the Mean

First, calculate the mean $\mu$ of the dataTo calculate the population variance and standard deviation for the data set $10, 13, 17, 19, 21$, follow these steps:

1. Find the Mean

First, calculate the mean $\mu$ of the

$\mu = \frac{10 + 13 + 17 + 19 + 21}{5} = \frac{80}{5} = 16$

2. Calculate the Squared Differences

For each value in the data, subtract the mean and square the result:

• $(10 - 16)^2 = (-6)^2 = 36$
• $(13 - 16)^2 = (-3)^2 = 9$
• $(17 - 16)^2 = (1)^2 = 1$
• $(19 - 16)^2 = (3)^2 = 9$
• $(21 - 16)^2 = (5)^2 = 25$

3. Find the Population Variance

Now, take the sum of these squared differences and divide by the number of data points (since this is a population variance, not a sample):

$\text{Variance} = \frac{36 + 9 + 1 + 9 + 25}{5} = \frac{80}{5} = 16$

4. Calculate the Standard Deviation

The standard deviation is the square root of the variance:

$\text{Standard Deviation} = \sqrt{16} = 4$

Final Results:

• Population Variance: $16$
• Standard Deviation: $4$

Would you like more details or have any questions?

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Here are some related questions to consider:

1. How does the formula for sample variance differ from population variance?
2. What happens to the standard deviation if all values in a data set increase by a constant?
3. How can variance help in understanding data spread?
4. Can variance ever be negative?
5. How does the standard deviation change if the data set includes an extreme outlier?

Tip: Remember, variance gives us an idea of how spread out the data is, while standard deviation gives it in the original units of the data.