The ages (in years) of all of Mrs. Wooten's grandchildren are provided in the data set below. What is the standard deviation of their ages?

5,2,10,14,10

Round the result to one more place value than the data values in the data set.

Provide your answer below:

The population standard deviation,  $\sigma$σ​ , is  years.

To calculate the population standard deviation $\sigma$, we follow these steps:

### Step 1: Calculate the mean ($\mu$) of the data set.
The data set is: 5, 2, 10, 14, 10.

\[
\mu = \frac{5 + 2 + 10 + 14 + 10}{5} = \frac{41}{5} = 8.2 \, \text{years}
\]

### Step 2: Subtract the mean from each data point and square the result.
\[
(5 - 8.2)^2 = (-3.2)^2 = 10.24
\]
\[
(2 - 8.2)^2 = (-6.2)^2 = 38.44
\]
\[
(10 - 8.2)^2 = (1.8)^2 = 3.24
\]
\[
(14 - 8.2)^2 = (5.8)^2 = 33.64
\]
\[
(10 - 8.2)^2 = (1.8)^2 = 3.24
\]

### Step 3: Sum these squared differences.
\[
\sum (x_i - \mu)^2 = 10.24 + 38.44 + 3.24 + 33.64 + 3.24 = 88.8
\]

### Step 4: Divide by $n$ (where $n$ is the total number of observations, which is 5) to calculate the variance.
\[
\text{Variance} = \frac{88.8}{5} = 17.76
\]

### Step 5: Take the square root of the variance to find the standard deviation.
\[
\sigma = \sqrt{17.76} \approx 4.21 \, \text{years}
\]

So, the population standard deviation $\sigma$ is **4.21 years**.

Would you like more details or have any questions?

Here are 5 related questions for further practice:

1. How would the standard deviation change if one of the ages was much higher, say 25 years?
2. How is the population standard deviation different from the sample standard deviation?
3. Calculate the variance for the same data set.
4. What is the effect of removing the youngest age on the standard deviation?
5. How do you interpret the population standard deviation in the context of these ages?

**Tip:** When working with the entire population, use $n$ for the number of observations in the formula for variance. This differentiates it from the sample standard deviation, where you use $n-1$.

Learn how to calculate the population standard deviation for a given set of ages (5, 2, 10, 14, 10). Follow step-by-step calculations to find the standard deviation, rounding to one more decimal place than the data values. Suitable for advanced high school students.

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