Standard deviation is a measure of the amount of variation or dispersion in a set of data values. In simpler terms, it tells you how spread out the numbers in a dataset are around the mean (average).

Here’s how to calculate the standard deviation:

Step-by-Step Process:

1. Find the mean (average) of the data: $\text{Mean} = \frac{\sum x_i}{n}$ where $\sum x_i$ is the sum of all the data points, and $n$ is the number of data points.

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

3. Square each deviation to avoid negative values and get the squared deviations.

4. Find the mean of the squared deviations (this is called the variance): $\text{Variance} = \frac{\sum (x_i - \text{mean})^2}{n}$ If you're working with a sample, use $n - 1$ in the denominator instead of $n$ (this is called Bessel's correction).

5. Take the square root of the variance to get the standard deviation: $\text{Standard Deviation} = \sqrt{\text{Variance}}$

Example:

Let’s calculate the standard deviation for this dataset:
$4, 8, 6, 5, 3$

1. Find the mean: $\text{Mean} = \frac{4 + 8 + 6 + 5 + 3}{5} = \frac{26}{5} = 5.2$

2. Subtract the mean from each data point: $(4 - 5.2) = -1.2, \quad (8 - 5.2) = 2.8, \quad (6 - 5.2) = 0.8, \quad (5 - 5.2) = -0.2, \quad (3 - 5.2) = -2.2$

3. Square each deviation: $(-1.2)^2 = 1.44, \quad (2.8)^2 = 7.84, \quad (0.8)^2 = 0.64, \quad (-0.2)^2 = 0.04, \quad (-2.2)^2 = 4.84$

4. Find the variance: $\text{Variance} = \frac{1.44 + 7.84 + 0.64 + 0.04 + 4.84}{5} = \frac{14.8}{5} = 2.96$

5. Find the standard deviation: $\text{Standard Deviation} = \sqrt{2.96} \approx 1.72$

So, the standard deviation is approximately 1.72.

---

Would you like more details on any step? Or perhaps another example?

Here are 5 related questions you might be interested in:

1. How does standard deviation relate to variance?
2. What is the difference between population and sample standard deviation?
3. How do you interpret a high vs. low standard deviation?
4. Can the standard deviation be negative?
5. How do you calculate the standard deviation for grouped data?

Tip: If you’re using a calculator or software to compute standard deviation, make sure it’s using the correct formula for either a population or a sample, depending on your dataset.