The standard deviation is a measure of the amount of variation or dispersion in a set of values. Here's how to calculate it:

Steps to Calculate the Standard Deviation:

1. Find the mean (average) of the dataset.

   • Formula for the mean:
     $\mu = \frac{1}{N} \sum_{i=1}^{N} x_i$ where $\mu$ is the mean, $N$ is the number of data points, and $x_i$ represents each individual data point.

2. Subtract the mean from each data point to find the deviation of each data point from the mean.

3. Square each deviation to get rid of negative values.

4. Find the average of the squared deviations (this is called the variance).

   • For a population, the variance formula is: $\sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2$
   • For a sample, the variance formula is: $s^2 = \frac{1}{N-1} \sum_{i=1}^{N} (x_i - \mu)^2$ (Note the difference in denominators: for a sample, we divide by $N-1$ instead of $N$ to correct for bias.)

5. Take the square root of the variance to get the standard deviation.

   • Formula for standard deviation: $\sigma = \sqrt{\sigma^2} \quad \text{(for population)}$ $s = \sqrt{s^2} \quad \text{(for sample)}$

Example:

Let's say you have the following dataset: $3, 7, 8, 5, 12$.

1. Find the mean: $\mu = \frac{3 + 7 + 8 + 5 + 12}{5} = \frac{35}{5} = 7$

2. Find the deviations from the mean: $3 - 7 = -4, \quad 7 - 7 = 0, \quad 8 - 7 = 1, \quad 5 - 7 = -2, \quad 12 - 7 = 5$

3. Square the deviations: $(-4)^2 = 16, \quad 0^2 = 0, \quad 1^2 = 1, \quad (-2)^2 = 4, \quad 5^2 = 25$

4. Find the average of the squared deviations (variance): $\sigma^2 = \frac{16 + 0 + 1 + 4 + 25}{5} = \frac{46}{5} = 9.2 \quad \text{(for population variance)}$

5. Take the square root of the variance (standard deviation): $\sigma = \sqrt{9.2} \approx 3.03$

So, the standard deviation for this population is approximately 3.03.

Summary:

• Standard deviation tells you how spread out the values in a dataset are.
• A small standard deviation means values are close to the mean, while a large one means they are spread out.

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Do you need further details or help with another problem? Here's a list of related questions for you:

1. What is the difference between population and sample standard deviation?
2. How does variance relate to standard deviation?
3. What is the impact of outliers on standard deviation?
4. How can standard deviation be used in real-life data analysis?
5. Can standard deviation be negative?

Tip: When using the sample standard deviation formula, remember that dividing by $N-1$ instead of $N$ helps reduce bias in the estimate of the population standard deviation.