Find the population variance and standard deviation.
10​, 13​, 17​, 19​, 21

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.

Find the population variance and standard deviation for the data set 10, 13, 17, 19, 21.

Learn how to calculate the population variance and standard deviation for the data set 10, 13, 17, 19, 21. This tutorial provides a detailed step-by-step solution covering key statistical concepts such as the mean, variance, and standard deviation. Suitable for students in Grades 9-12.

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