The following data are the distances between 20 retail stores and a large distribution center. The distances are in miles.  29; 37; 38; 40; 58; 67; 68; 69; 76; 86; 87; 95; 96; 96; 99; 106; 112; 127; 145; 150

 

What is the variance for this distribution, please report to 2 decimal places.

To calculate the variance of the given data set, we can follow these steps:

1. **Calculate the mean (average) of the data.**
2. **Subtract the mean from each data point and square the result.**
3. **Sum up all the squared differences.**
4. **Divide the sum by the number of data points (since this is a sample variance, we'll use \( n - 1 \) in the denominator).**

Let's start with these calculations.

### Step 1: Calculate the Mean
The mean (\(\mu\)) is calculated by summing all the values and dividing by the total number of values.

\[
\mu = \frac{\sum x}{n} = \frac{29 + 37 + 38 + 40 + 58 + 67 + 68 + 69 + 76 + 86 + 87 + 95 + 96 + 96 + 99 + 106 + 112 + 127 + 145 + 150}{20}
\]

### Step 2: Find the Squared Differences and Sum Them

We will subtract the mean from each data point, square the difference, and then sum up all the squared differences.

### Step 3: Calculate the Variance

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

Let's compute this step-by-step.The variance of the distribution is approximately \( 1191.42 \), rounded to two decimal places.

Would you like further details, or do you have any questions? Here are some related questions:

1. How do you interpret variance in a data set?
2. What is the standard deviation, and how is it related to variance?
3. How does variance change with different data points?
4. What is the significance of calculating variance in real-world scenarios?
5. How does the variance calculation differ between a sample and a population?

**Tip:** Variance is a measure of how spread out the data points are around the mean; a higher variance indicates more variability.

The following data are the distances between 20 retail stores and a large distribution center. The distances are in miles. 29; 37; 38; 40; 58; 67; 68; 69; 76; 86; 87; 95; 96; 96; 99; 106; 112; 127; 145; 150. What is the variance for this distribution, please report to 2 decimal places.

This problem involves calculating the variance of distances (in miles) between 20 retail stores and a large distribution center. Using descriptive statistics, the variance is computed to 2 decimal places, providing insights into the spread of the data points. Suitable for students in Grades 10-12.

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