The ages​ (in years) of a random sample of shoppers at a gaming store are shown. Determine the​ range, mean,​ variance, and standard deviation of the sample data set.
12, 15,23, 13,19, 17, 21, 17, 31, 15

Let's analyze the given data set:  
Ages: \( 12, 15, 23, 13, 19, 17, 21, 17, 31, 15 \)

### 1. Range
The **range** of a data set is the difference between the largest and smallest values.
- Smallest value = \( 12 \)
- Largest value = \( 31 \)

\[
\text{Range} = 31 - 12 = 19
\]

### 2. Mean
The **mean** (average) is the sum of all values divided by the number of values.

\[
\text{Mean} = \frac{12 + 15 + 23 + 13 + 19 + 17 + 21 + 17 + 31 + 15}{10} = \frac{183}{10} = 18.3
\]

### 3. Variance
The **variance** measures how far the values are spread out from the mean. First, calculate the squared differences from the mean for each value:

\[
\text{Variance} = \frac{\sum (x_i - \mu)^2}{n - 1}
\]

where \( \mu = 18.3 \) (the mean) and \( n = 10 \) (sample size).

Let's compute each squared difference:

\[
\begin{aligned}
(12 - 18.3)^2 &= 39.69 \\
(15 - 18.3)^2 &= 10.89 \\
(23 - 18.3)^2 &= 22.09 \\
(13 - 18.3)^2 &= 28.09 \\
(19 - 18.3)^2 &= 0.49 \\
(17 - 18.3)^2 &= 1.69 \\
(21 - 18.3)^2 &= 7.29 \\
(17 - 18.3)^2 &= 1.69 \\
(31 - 18.3)^2 &= 161.29 \\
(15 - 18.3)^2 &= 10.89 \\
\end{aligned}
\]

Now, sum these squared differences:

\[
\text{Sum of squared differences} = 39.69 + 10.89 + 22.09 + 28.09 + 0.49 + 1.69 + 7.29 + 1.69 + 161.29 + 10.89 = 284.1
\]

The sample variance is:

\[
\text{Variance} = \frac{284.1}{9} = 31.57
\]

### 4. Standard Deviation
The **standard deviation** is the square root of the variance:

\[
\text{Standard Deviation} = \sqrt{31.57} \approx 5.62
\]

### Summary of Results:
- **Range**: 19
- **Mean**: 18.3
- **Variance**: 31.57
- **Standard Deviation**: 5.62

Would you like more details or have any questions?

Here are some related questions to expand on:
1. How does variance help in understanding the distribution of data?
2. How is population variance different from sample variance?
3. What is the significance of the range in a data set?
4. Can you find the mode or median of the given data set?
5. What is the difference between mean and median in interpreting data sets?

**Tip**: The larger the variance, the more spread out your data is from the mean.

The ages (in years) of a random sample of shoppers at a gaming store are shown. Determine the range, mean, variance, and standard deviation of the sample data set.
12, 15, 23, 13, 19, 17, 21, 17, 31, 15

Learn how to calculate the range, mean, variance, and standard deviation for a sample data set of shoppers' ages: 12, 15, 23, 13, 19, 17, 21, 17, 31, 15. This step-by-step solution covers key concepts in descriptive statistics, perfect for Grades 9-12 students.

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