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The problem shown in the image asks for the standard deviation (\(S\)) of the minutes consumed by customers from a random sample. The data provided are:

\[
348, 355, 355, 355, 363, 364, 367, 369, 370, 378, 378, 378, 380, 386, 387, 391, 392, 399, 414, 416, 433
\]

To calculate the standard deviation \(S\), we follow these steps:

1. **Find the Mean**: Calculate the average of all the numbers.
2. **Calculate Variances**: For each data point, subtract the mean and square the result.
3. **Find the Variance**: The average of all squared differences.
4. **Standard Deviation**: The square root of the variance.

I'll compute this to find the correct standard deviation.The calculated standard deviation \(S\) for the given data set is approximately \(22.16\).

Given the options in the image:

- \(S = 17.8\)
- \(S = 516.5\)
- \(S = 22.7\)

The closest answer is \(S = 22.7\).

Would you like further details or have any other questions?

Here are 5 related questions:
1. How is the standard deviation different from variance?
2. What is the significance of using the sample standard deviation (\(S\)) versus population standard deviation (\(\sigma\))?
3. Can you explain why we use the square root of the variance to find the standard deviation?
4. How does increasing the range of data values affect the standard deviation?
5. What are the applications of standard deviation in real-world problems?

**Tip**: Always use \(n-1\) (degrees of freedom) when calculating the sample standard deviation, not \(n\) for a population.

Calculate the standard deviation of monthly minute consumption from a sample data set for a cellphone plan.

This guide provides a comprehensive explanation on calculating the standard deviation of monthly minute usage from a given data set, covering how to derive the mean, calculate variances, and find the square root of the variance.

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