**Standard deviation** is a measure of the amount of variation or dispersion in a set of values. It tells us how much the data points tend to deviate from the mean (average) of the dataset.

### Formula for Standard Deviation:
For a population:
\[
\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2}
\]
Where:
- \(\sigma\) is the standard deviation.
- \(N\) is the total number of data points.
- \(x_i\) represents each data point.
- \(\mu\) is the mean (average) of the data points.

For a sample:
\[
s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2}
\]
Where:
- \(s\) is the sample standard deviation.
- \(n\) is the sample size.
- \(x_i\) represents each data point.
- \(\bar{x}\) is the sample mean.

### Key Concepts:
1. **Low Standard Deviation**: Data points are close to the mean, indicating less variation.
2. **High Standard Deviation**: Data points are spread out, indicating more variation.

### Steps to Calculate Standard Deviation:
1. Find the mean of the data set.
2. Subtract the mean from each data point and square the result.
3. Find the average of the squared differences.
4. Take the square root of the average to obtain the standard deviation.

Would you like a more detailed explanation with an example?

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Here are 5 related questions:
1. How is variance related to standard deviation?
2. What is the difference between population and sample standard deviation?
3. Can standard deviation ever be negative?
4. What are the uses of standard deviation in real-life data analysis?
5. How does standard deviation differ from the range as a measure of spread?

**Tip:** The square of the standard deviation is called the variance, which is another important measure of spread in data.

Learn how to calculate standard deviation, a key concept in statistics, which measures the spread of data points from the mean. This guide covers both population and sample standard deviation formulas, explains key concepts like variance, and provides a step-by-step calculation process. Suitable for Grades 9-12 students.

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