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To find the standard deviation of the given data set \[4, 2, 11, 7, 5, 3, 14, 9, 8\], follow these steps:

### Step 1: Find the mean
The mean (\(\mu\)) is calculated as the sum of all the numbers divided by the count of numbers.

\[
\mu = \frac{4 + 2 + 11 + 7 + 5 + 3 + 14 + 9 + 8}{9} = \frac{63}{9} = 7
\]

### Step 2: Calculate each data point's deviation from the mean, and square it
Now, subtract the mean from each number and square the result:
\[
(4 - 7)^2 = (-3)^2 = 9
\]
\[
(2 - 7)^2 = (-5)^2 = 25
\]
\[
(11 - 7)^2 = (4)^2 = 16
\]
\[
(7 - 7)^2 = (0)^2 = 0
\]
\[
(5 - 7)^2 = (-2)^2 = 4
\]
\[
(3 - 7)^2 = (-4)^2 = 16
\]
\[
(14 - 7)^2 = (7)^2 = 49
\]
\[
(9 - 7)^2 = (2)^2 = 4
\]
\[
(8 - 7)^2 = (1)^2 = 1
\]

### Step 3: Find the average of these squared deviations
Sum the squared deviations and divide by the number of data points:

\[
\frac{9 + 25 + 16 + 0 + 4 + 16 + 49 + 4 + 1}{9} = \frac{124}{9} \approx 13.78
\]

### Step 4: Take the square root of the result
Now, take the square root of the variance to find the standard deviation:

\[
\sqrt{13.78} \approx 3.713
\]

### Conclusion:
The standard deviation of the data set is approximately **3.713**, rounded to the nearest thousandth.

Would you like any additional details or have questions?

Here are five related questions you could explore:
1. What is the difference between variance and standard deviation?
2. How does standard deviation change if one value in the set is significantly altered?
3. Can standard deviation be negative?
4. How does sample standard deviation differ from population standard deviation?
5. What does a larger standard deviation indicate about a data set?

**Tip:** In statistics, a smaller standard deviation means the data points are closer to the mean, indicating less variability.

Find the standard deviation of the set of numbers: 4, 2, 11, 7, 5, 3, 14, 9, 8.

Learn how to calculate the standard deviation of the data set [4, 2, 11, 7, 5, 3, 14, 9, 8] with a detailed step-by-step solution. This problem demonstrates key statistical concepts like mean, variance, and standard deviation. Suitable for students in Grades 9-12.

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