Listed below are the amounts​ (dollars) it costs for marriage proposal packages at different baseball stadiums. Find the​ range, variance, and standard deviation for the given sample data. Include appropriate units in the results. Are there any​ outliers, and are they likely to have much of an effect on the measures of​ variation?
36

50

50

60

65

95

95

115

190

217

250

325

400

1750

2500

Let's solve this problem step by step:

### 1. **Range Calculation**
The **range** is the difference between the maximum and minimum values.

- Maximum value: 2500
- Minimum value: 36

\[
\text{Range} = 2500 - 36 = 2464 \, \text{dollars}
\]

### 2. **Variance Calculation**
The **variance** measures the spread of the data around the mean. For a sample, the formula is:

\[
s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2
\]

where \( n \) is the sample size, \( x_i \) are the data points, and \( \bar{x} \) is the sample mean.

#### Steps:
- **Sample size (n):** 15
- **Mean (\(\bar{x}\)):** Sum all the values, then divide by 15.

\[
\bar{x} = \frac{36 + 50 + 50 + 60 + 65 + 95 + 95 + 115 + 190 + 217 + 250 + 325 + 400 + 1750 + 2500}{15}
\]

After finding the mean, we will use it to compute the squared differences from the mean and get the variance.

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

\[
s = \sqrt{s^2}
\]

### 4. **Outliers**
Outliers can be identified using the IQR (Interquartile Range) method or based on how far they deviate from the mean (e.g., more than 3 standard deviations). We will compute these metrics after calculating the variance and standard deviation to check if the largest values (like 1750 and 2500) significantly affect the variation.

Let me calculate the required quantities for you.

Listed below are the amounts​ (dollars) it costs for marriage proposal packages at different baseball stadiums. Find the​ range, variance, and standard deviation for the given sample data. Include appropriate units in the results. Are there any​ outliers, and are they likely to have much of an effect on the measures of​ variation?
36, 50, 50, 60, 65, 95, 95, 115, 190, 217, 250, 325, 400, 1750, 2500

Find the range, variance, and standard deviation of marriage proposal costs at baseball stadiums, with detailed step-by-step solutions. This problem covers key concepts like descriptive statistics, variance formulas, and outlier detection, ideal for Grades 10-12.

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