Find the standard deviation of the given data set - 7, 47, 8, 42, 47, 95, 42, 96, 3

To find the standard deviation, we'll first calculate the mean (average) of the data, then the variance, and finally take the square root of the variance to obtain the standard deviation.

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

Given dataTo find the standard deviation, we'll first calculate the mean (average) of the data, then the variance, and finally take the square root of the variance to obtain the standard deviation.

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

Given  \(7, 47, 8, 42, 47, 95, 42, 96, 3\)

\[
\mu = \frac{7 + 47 + 8 + 42 + 47 + 95 + 42 + 96 + 3}{9} = \frac{387}{9} = 43
\]

### Step 2: Calculate Each Deviation from the Mean, Square It, and Find the Variance
The variance \(\sigma^2\) for a sample is:
\[
\sigma^2 = \frac{\sum (x - \mu)^2}{n}
\]

Calculations for each \((x - \mu)^2\):
\[
(7 - 43)^2 = 1296
\]
\[
(47 - 43)^2 = 16
\]
\[
(8 - 43)^2 = 1225
\]
\[
(42 - 43)^2 = 1
\]
\[
(47 - 43)^2 = 16
\]
\[
(95 - 43)^2 = 2704
\]
\[
(42 - 43)^2 = 1
\]
\[
(96 - 43)^2 = 2809
\]
\[
(3 - 43)^2 = 1600
\]

Now, summing these values:
\[
\sum (x - \mu)^2 = 1296 + 16 + 1225 + 1 + 16 + 2704 + 1 + 2809 + 1600 = 9668
\]

Variance:
\[
\sigma^2 = \frac{9668}{9} \approx 1074.22
\]

### Step 3: Find the Standard Deviation
The standard deviation \(\sigma\) is the square root of the variance:
\[
\sigma = \sqrt{1074.22} \approx 32.77
\]

### Final Answer
The standard deviation of the data set is approximately **32.77**.

Would you like further details, or have any questions?

---

1. How is the mean calculated for any data set?
2. What are the steps to calculate variance for different types of data (sample vs. population)?
3. Why is the square root used to find the standard deviation?
4. How does standard deviation change with outliers in the data?
5. Can we calculate standard deviation for grouped data?

**Tip:** Standard deviation is a measure of how spread out numbers are around the mean.

Learn how to find the standard deviation of the data set 7, 47, 8, 42, 47, 95, 42, 96, 3 using statistical concepts like mean, variance, and standard deviation. Suitable for students in Grades 9-12.

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