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To solve the problem of finding the standard deviation, let’s go through the steps of completing the table based on the values provided.

### 1. Calculate \( n \), the number of data points:
The data points given are: \(-1, -5, -3, -5, -2\).
So, \( n = 5 \).

### 2. Calculate \( \sum x \), the sum of all the data points:
\[
\sum x = (-1) + (-5) + (-3) + (-5) + (-2) = -16
\]

### 3. Calculate \( \bar{x} \) (mean of the data), which is \( \frac{\sum x}{n} \):
\[
\bar{x} = \frac{-16}{5} = -3.2
\]

### 4. Populate the table:

| \( x \)  | \( x - \bar{x} \)   | \( (x - \bar{x})^2 \) |
|:--------:|:--------------------:|:---------------------:|
|   -1     | \( -1 - (-3.2) = 2.2 \)   | \( 2.2^2 = 4.84 \)  |
|   -5     | \( -5 - (-3.2) = -1.8 \)  | \( (-1.8)^2 = 3.24 \)  |
|   -3     | \( -3 - (-3.2) = 0.2 \)   | \( 0.2^2 = 0.04 \)  |
|   -5     | \( -5 - (-3.2) = -1.8 \)  | \( (-1.8)^2 = 3.24 \)  |
|   -2     | \( -2 - (-3.2) = 1.2 \)   | \( 1.2^2 = 1.44 \)  |

### 5. Calculate \( \sum (x - \bar{x})^2 \), the sum of squared deviations:
\[
\sum (x - \bar{x})^2 = 4.84 + 3.24 + 0.04 + 3.24 + 1.44 = 12.8
\]

### 6. Calculate \( \frac{\sum (x - \bar{x})^2}{n} \) to get the variance:
\[
\text{Variance} = \frac{12.8}{5} = 2.56
\]

### 7. Calculate the standard deviation \( \sigma \), which is the square root of the variance:
\[
\sigma = \sqrt{2.56} = 1.6
\]

### Summary of Results:
- \( n = 5 \)
- \( \sum x = -16 \)
- \( \bar{x} = -3.2 \)
- \( \sum (x - \bar{x})^2 = 12.8 \)
- Variance \( = 2.56 \)
- Standard deviation \( \sigma = 1.6 \)

Would you like further details on any step?

Here are 5 related questions:
1. How would the standard deviation change if one of the data points was different?
2. What is the difference between population and sample standard deviation?
3. Can you explain why we square the deviations when calculating standard deviation?
4. How is the mean affected by extreme data points in this set?
5. What are the implications of a higher standard deviation in data analysis?

**Tip:** The standard deviation is a measure of how spread out the numbers in a data set are. A smaller standard deviation means the data points tend to be close to the mean.

Find the standard deviation by completing the table for the data points: -1, -5, -3, -5, -2.

This problem walks through how to find the standard deviation for the data points -1, -5, -3, -5, and -2. Learn how to calculate the mean, variance, and standard deviation step-by-step. Suitable for Grades 10-12.

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