The following sample data are the distances between nine retail stores and a large distribution center.

The distances are in miles. 30; 45; 58; 76; 87; 95; 106; 127; 150.

What is the variance? *Round to the nearest whole number without a comma.* 

What is the standard deviation? *Round to two decimal places.*
 
***Make sure to complete all the steps using a calculator.*** 

To calculate the **variance** and **standard deviation**, we'll follow these steps:

### Step 1: Calculate the Mean (Average)
First, sum up all the distances and divide by the number of data points (in this case, 9).

\[
\text{Mean} = \frac{30 + 45 + 58 + 76 + 87 + 95 + 106 + 127 + 150}{9}
\]

\[
\text{Mean} = \frac{774}{9} = 86
\]

### Step 2: Find the Squared Differences from the Mean
Next, subtract the mean from each data point, square the result, and list them:

\[
(30 - 86)^2 = (-56)^2 = 3136
\]
\[
(45 - 86)^2 = (-41)^2 = 1681
\]
\[
(58 - 86)^2 = (-28)^2 = 784
\]
\[
(76 - 86)^2 = (-10)^2 = 100
\]
\[
(87 - 86)^2 = (1)^2 = 1
\]
\[
(95 - 86)^2 = (9)^2 = 81
\]
\[
(106 - 86)^2 = (20)^2 = 400
\]
\[
(127 - 86)^2 = (41)^2 = 1681
\]
\[
(150 - 86)^2 = (64)^2 = 4096
\]

### Step 3: Calculate the Variance
Now, add all these squared differences together and divide by the number of data points (9):

\[
\text{Variance} = \frac{3136 + 1681 + 784 + 100 + 1 + 81 + 400 + 1681 + 4096}{9} = \frac{11960}{9} \approx 1339
\]

**Variance** (rounded to the nearest whole number) = **1339**

### Step 4: Calculate the Standard Deviation
The standard deviation is the square root of the variance:

\[
\text{Standard Deviation} = \sqrt{1339} \approx 36.6
\]

**Standard Deviation** (rounded to two decimal places) = **36.60**

---

### Final Answers:
- **Variance**: 1339
- **Standard Deviation**: 36.60

Do you have any questions or need further details?

Here are some related questions you might find helpful:
1. What are the steps to calculate the population standard deviation?
2. How do variance and standard deviation differ conceptually?
3. How would the variance change if we added a store further from the distribution center?
4. What is the difference between population and sample variance?
5. How do outliers affect standard deviation?

**Tip:** Standard deviation is a great way to measure the spread of data. The larger the standard deviation, the more spread out the data points are.

The following sample data are the distances between nine retail stores and a large distribution center. The distances are in miles. 30; 45; 58; 76; 87; 95; 106; 127; 150. What is the variance? Round to the nearest whole number without a comma. What is the standard deviation? Round to two decimal places.

Learn how to calculate the variance and standard deviation for the distances between nine retail stores and a large distribution center. This detailed step-by-step solution covers important concepts such as mean, variance, and standard deviation, suitable for Grades 7-9.

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