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. 

1212​, 

1616​, 

2323​, 

1313​, 

1919​, 

1616​, 

2121​, 

1616​, 

1515​, 

1717 

Let's calculate the range, mean, variance, and standard deviation for the given data set of ages:

### Data Set:
12, 16, 23, 13, 19, 16, 21, 16, 15, 17

### 1. **Range**:
The range is the difference between the maximum and minimum values in the data set.

- Maximum value = 23
- Minimum value = 12

\[
\text{Range} = 23 - 12 = 11
\]

### 2. **Mean**:
The mean is the average of the data set, calculated by summing all the values and dividing by the number of values.

\[
\text{Mean} = \frac{12 + 16 + 23 + 13 + 19 + 16 + 21 + 16 + 15 + 17}{10} = \frac{168}{10} = 16.8
\]

### 3. **Variance**:
Variance measures how much the data points differ from the mean. It is calculated as the average of the squared differences from the mean.

\[
\text{Variance} = \frac{\sum{(x_i - \text{Mean})^2}}{n}
\]
Where \( n \) is the number of observations, \( x_i \) is each individual observation, and the mean is 16.8.

Let's calculate the squared differences:

\[
\begin{aligned}
(12 - 16.8)^2 & = 23.04 \\
(16 - 16.8)^2 & = 0.64 \\
(23 - 16.8)^2 & = 38.44 \\
(13 - 16.8)^2 & = 14.44 \\
(19 - 16.8)^2 & = 4.84 \\
(16 - 16.8)^2 & = 0.64 \\
(21 - 16.8)^2 & = 17.64 \\
(16 - 16.8)^2 & = 0.64 \\
(15 - 16.8)^2 & = 3.24 \\
(17 - 16.8)^2 & = 0.04 \\
\end{aligned}
\]

Sum of squared differences:

\[
23.04 + 0.64 + 38.44 + 14.44 + 4.84 + 0.64 + 17.64 + 0.64 + 3.24 + 0.04 = 103.60
\]

\[
\text{Variance} = \frac{103.60}{10} = 10.36
\]

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

\[
\text{Standard Deviation} = \sqrt{10.36} \approx 3.22
\]

### Summary:
- **Range**: 11
- **Mean**: 16.8
- **Variance**: 10.36
- **Standard Deviation**: 3.22

Would you like further details or have any questions? Here are some related questions to consider:

1. How would the mean change if the sample size increased?
2. What does the standard deviation tell us about the spread of the ages?
3. How is variance related to standard deviation?
4. How would the calculation differ if we were dealing with population data instead of sample data?
5. What does a low standard deviation indicate about the data set?

**Tip:** When analyzing data, always consider both the mean and standard deviation to get a complete picture of the data's distribution and spread.

Learn how to calculate the range, mean, variance, and standard deviation of a sample data set. Example data set: 12, 16, 23, 13, 19, 16, 21, 16, 15, 17. Understand the importance of these statistical measures in analyzing data distributions.

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