Unemployment Benefits The average weekly unemployment benefits (in dollars) for a random selection of
states are listed below. Calculate the range, variance,
and standard deviation for the data.
239 214 327 416 321 289 209 356
190 272 252 272 310 276 251


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

The given data set consists of the weekly unemployment benefits in dollars for the following 15 states:

**239, 214, 327, 416, 321, 289, 209, 356, 190, 272, 252, 272, 310, 276, 251**

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

- Maximum value = 416
- Minimum value = 190

\[
\text{Range} = 416 - 190 = 226
\]

### Step 2: Variance
To compute the variance, follow these steps:

1. **Find the mean** (average) of the data set.

\[
\text{Mean} = \frac{\sum x_i}{n} = \frac{239 + 214 + 327 + 416 + 321 + 289 + 209 + 356 + 190 + 272 + 252 + 272 + 310 + 276 + 251}{15} = \frac{4504}{15} \approx 300.27
\]

2. **Find the squared differences** from the mean for each data point.

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

Let's calculate this directly.

### Step 3: Standard Deviation
The **standard deviation** is the square root of the variance:

\[
\text{Standard Deviation} = \sqrt{\text{Variance}}
\]

I'll compute the variance and standard deviation now.### Results:

1. **Range**: 226
2. **Variance**: 3339.17 (rounded to two decimal places)
3. **Standard Deviation**: 57.79 (rounded to two decimal places)

Would you like more details on these calculations or have any further questions?

Here are 5 related questions to deepen your understanding:
1. What is the difference between sample and population variance?
2. How does the range affect the variability of a data set?
3. Can variance ever be negative? Why or why not?
4. How would adding an extreme outlier affect the standard deviation?
5. What are the implications of a high vs. low standard deviation?

**Tip**: When calculating variance, always check whether you're using the formula for a sample or the entire population to avoid errors.

Unemployment Benefits The average weekly unemployment benefits (in dollars) for a random selection of states are listed below. Calculate the range, variance, and standard deviation for the data. 239 214 327 416 321 289 209 356 190 272 252 272 310 276 251

Learn how to calculate the range, variance, and standard deviation for a set of unemployment benefits data. This problem explores key statistical concepts, including the calculation of the range, variance, and standard deviation using a given data set of weekly unemployment benefits from 15 states. Suitable for students in Grades 9-12.

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