The standard deviation formula is derived from the variance, which quantifies the dispersion of data points around the mean.

To calculate the standard deviation, you must first determine the mean (average) of your dataset. Begin by summing all the values in your dataset, and then divide this total by the number of data points, denoted as $N$. After obtaining the mean, subtract it from each individual data point to find the deviation of each point from the mean. Next, square each of these deviations to ensure they are positive, and then sum all the squared deviations.

For a population, you calculate the variance by dividing this total by the number of data points ($N$). Conversely, for a sample, you divide by one less than the number of data points ($N - 1$) to obtain the sample variance. Finally, to find the standard deviation, take the square root of the variance.

In formal terms, the standard deviation for a population, denoted by $\sigma$, is given by the formula:

In this equation, $x_i$ represents each individual data point, $\mu$ is the mean of the population, and $N$ is the total number of data points.

For a sample, the standard deviation, denoted by $s$, is calculated using the formula:

Here, $\bar{x}$ represents the sample mean, and $N - 1$ reflects the sample size, which accounts for the degrees of freedom.

Understanding standard deviation is crucial for analyzing the spread of your data. A small standard deviation indicates that the data points are clustered closely around the mean, while a large standard deviation signifies that the data points are spread over a wider range.