by hand, find the population variance and standard deviation or the sample variance and standard deviation as indicated. Sample: 6, 52, 13, 49, 35, 25, 31, 29, 31, 29

We will calculate the **sample variance** and the **sample standard deviation** for the given data set since "sample" is indicated.

The steps to calculate sample variance and standard deviation are as follows:

### 1. Find the sample mean (\(\bar{x}\)):

\[
\bar{x} = \frac{\sum x_i}{n}
\]

Where:
- \(\sum x_i\) is the sum of the sample values
- \(n\) is the number of sample points

Given sample: 6, 52, 13, 49, 35, 25, 31, 29, 31, 29
\[
\sum x_i = 6 + 52 + 13 + 49 + 35 + 25 + 31 + 29 + 31 + 29 = 300
\]
\[
n = 10
\]
\[
\bar{x} = \frac{300}{10} = 30
\]

### 2. Calculate the squared deviations from the mean:

For each data point \(x_i\), calculate \((x_i - \bar{x})^2\):

\[
(6 - 30)^2 = (-24)^2 = 576
\]
\[
(52 - 30)^2 = (22)^2 = 484
\]
\[
(13 - 30)^2 = (-17)^2 = 289
\]
\[
(49 - 30)^2 = (19)^2 = 361
\]
\[
(35 - 30)^2 = (5)^2 = 25
\]
\[
(25 - 30)^2 = (-5)^2 = 25
\]
\[
(31 - 30)^2 = (1)^2 = 1
\]
\[
(29 - 30)^2 = (-1)^2 = 1
\]
\[
(31 - 30)^2 = (1)^2 = 1
\]
\[
(29 - 30)^2 = (-1)^2 = 1
\]

### 3. Sum of squared deviations:
\[
\sum (x_i - \bar{x})^2 = 576 + 484 + 289 + 361 + 25 + 25 + 1 + 1 + 1 + 1 = 1764
\]

### 4. Compute the sample variance (\(s^2\)):

\[
s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}
\]
\[
s^2 = \frac{1764}{10 - 1} = \frac{1764}{9} = 196
\]

### 5. Compute the sample standard deviation (\(s\)):

\[
s = \sqrt{s^2} = \sqrt{196} = 14
\]

### Summary:
- **Sample variance (\(s^2\))**: 196
- **Sample standard deviation (\(s\))**: 14

Would you like more details or have any questions?

---

Here are 5 related questions to expand on this topic:

1. How do we calculate population variance and standard deviation?
2. What is the difference between population and sample variance?
3. Why do we divide by \(n-1\) instead of \(n\) in the sample variance formula?
4. How does the variance change if an outlier is added to the dataset?
5. What are the applications of standard deviation in real-world scenarios?

**Tip**: The standard deviation gives insight into how spread out the values in a dataset are. A smaller standard deviation indicates that data points tend to be closer to the mean.

By hand, find the population variance and standard deviation or the sample variance and standard deviation as indicated. Sample: 6, 52, 13, 49, 35, 25, 31, 29, 31, 29

Learn how to calculate the sample variance and standard deviation for the data set: 6, 52, 13, 49, 35, 25, 31, 29, 31, 29. This step-by-step guide covers key statistical concepts and formulas, suitable for students in Grades 10-12.

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