The standard deviation is calculated by first finding the mean of the data set, then determining how each data point differs from the mean, squaring these differences, summing them, dividing by the number of data points minus one (for a sample), and finally taking the square root of that result.

Steps to Calculate Standard Deviation

1. Calculate the mean (average) of the data.
2. Subtract the mean from each data value to find the deviation of each value.
3. Square each deviation.
4. Sum all the squared deviations.
5. Divide this sum by n−1n-1n−1 if the data represents a sample (where nnn is the number of data points).
6. Take the square root of this quotient to get the sample standard deviation.

Formula for Sample Standard Deviation

s=∑i=1n(xi−xˉ)2n−1s=\sqrt{\frac{\sum_{i=1}^n(x_i-\bar{x})^2}{n-1}}s=n−1∑i=1n​(xi​−xˉ)2​​

where:

• sss is the sample standard deviation,
• xix_ixi​ are the data points,
• xˉ\bar{x}xˉ is the sample mean,
• nnn is the number of data points.

If the data represents an entire population, divide by NNN instead of n−1n-1n−1:

σ=∑i=1N(xi−μ)2N\sigma =\sqrt{\frac{\sum_{i=1}^N(x_i-\mu)^2}{N}}σ=N∑i=1N​(xi​−μ)2​​

where:

• σ\sigma σ is the population standard deviation,
• μ\mu μ is the population mean,
• NNN is the population size.

This process quantifies the amount of variation or dispersion in a set of data points.