Calculate the standard deviation for the data set : 389, 397, 403, 406, 426, 436, 442, 460, 460, 460

To calculate the standard deviation, follow these steps:

### 1. **Find the Mean (Average)**:
The mean (\(\mu\)) is the sum of all values divided by the number of values.
\[
\mu = \frac{389 + 397 + 403 + 406 + 426 + 436 + 442 + 460 + 460 + 460}{10}
\]
\[
\mu = \frac{4239}{10} = 423.9
\]

### 2. **Calculate the Squared Differences**:
For each value in the data set, subtract the mean and then square the result.

- \((389 - 423.9)^2 = (-34.9)^2 = 1217.01\)
- \((397 - 423.9)^2 = (-26.9)^2 = 723.61\)
- \((403 - 423.9)^2 = (-20.9)^2 = 436.81\)
- \((406 - 423.9)^2 = (-17.9)^2 = 320.41\)
- \((426 - 423.9)^2 = (2.1)^2 = 4.41\)
- \((436 - 423.9)^2 = (12.1)^2 = 146.41\)
- \((442 - 423.9)^2 = (18.1)^2 = 327.61\)
- \((460 - 423.9)^2 = (36.1)^2 = 1303.21\)
- \((460 - 423.9)^2 = (36.1)^2 = 1303.21\)
- \((460 - 423.9)^2 = (36.1)^2 = 1303.21\)

### 3. **Find the Variance**:
The variance (\(\sigma^2\)) is the average of these squared differences. Since this is a sample, divide by \(n-1\) (9 in this case):
\[
\sigma^2 = \frac{1217.01 + 723.61 + 436.81 + 320.41 + 4.41 + 146.41 + 327.61 + 1303.21 + 1303.21 + 1303.21}{9}
\]
\[
\sigma^2 = \frac{7386.49}{9} \approx 820.72
\]

### 4. **Calculate the Standard Deviation**:
The standard deviation (\(\sigma\)) is the square root of the variance:
\[
\sigma = \sqrt{820.72} \approx 28.64
\]

### Final Answer:
The standard deviation for the data set is approximately **28.64**.

Would you like more details or have any further questions?

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Here are 5 related questions:
1. How would the standard deviation change if one more value was added to the dataset?
2. What is the difference between population and sample standard deviation?
3. Can you explain how to calculate the variance in more detail?
4. How does standard deviation help in data analysis?
5. How would outliers affect the standard deviation?

**Tip:** When computing standard deviation, ensure you distinguish between population and sample calculations to avoid errors.

Calculate the standard deviation for the data set: 389, 397, 403, 406, 426, 436, 442, 460, 460, 460

Learn how to calculate the standard deviation for the data set: 389, 397, 403, 406, 426, 436, 442, 460, 460, 460. This problem covers essential statistical concepts such as mean, variance, and standard deviation. Detailed step-by-step solution provided, suitable for students in Grades 9-12.

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