standard deviation of the set 6, 2, 8, 1, 1, 7, 3, and 4

To calculate the standard deviation of the data set \( \{6, 2, 8, 1, 1, 7, 3, 4\} \), follow these steps:

### Step 1: Find the Mean
The mean \( \mu \) of the data set is given by:
\[
\mu = \frac{\sum x_i}{n}
\]
where \( x_i \) are the data points and \( n \) is the number of data points.

\[
\mu = \frac{6 + 2 + 8 + 1 + 1 + 7 + 3 + 4}{8} = \frac{32}{8} = 4
\]

### Step 2: Calculate the Variance
The variance \( \sigma^2 \) is the average of the squared differences between each data point and the mean:
\[
\sigma^2 = \frac{\sum (x_i - \mu)^2}{n}
\]

\[
\begin{align*}
(6 - 4)^2 &= 4, \\
(2 - 4)^2 &= 4, \\
(8 - 4)^2 &= 16, \\
(1 - 4)^2 &= 9, \\
(1 - 4)^2 &= 9, \\
(7 - 4)^2 &= 9, \\
(3 - 4)^2 &= 1, \\
(4 - 4)^2 &= 0.
\end{align*}
\]

Now, sum these squared differences:
\[
4 + 4 + 16 + 9 + 9 + 9 + 1 + 0 = 52
\]

Then, divide by the number of data points \( n = 8 \):
\[
\sigma^2 = \frac{52}{8} = 6.5
\]

### Step 3: Calculate the Standard Deviation
The standard deviation \( \sigma \) is the square root of the variance:
\[
\sigma = \sqrt{6.5} \approx 2.55
\]

Thus, the standard deviation of the data set is approximately \( 2.55 \).

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Do you want more details or have any questions?

Here are 5 related questions for practice:
1. What is the variance of the set \( \{3, 5, 7, 9, 11\} \)?
2. How does changing one data point affect the standard deviation?
3. Find the standard deviation for the set \( \{10, 15, 20, 25, 30\} \).
4. How is population standard deviation different from sample standard deviation?
5. How do outliers affect the standard deviation of a data set?

**Tip:** Standard deviation measures the spread of a data set; larger values mean more spread out data!

Learn how to calculate the standard deviation of the data set {6, 2, 8, 1, 1, 7, 3, 4}. This guide provides a step-by-step solution, covering concepts such as mean, variance, and standard deviation. Ideal for students in Grades 9-12 looking to strengthen their understanding of statistical measures.

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