How to Calculate Standard Deviation | Formula and Explanation for Population and Sample

Standard Deviation Formula

The standard deviation formula is a mathematical formula used to calculate the measure of how spread out the values in a data set are, or in other words, it measures the variability or dispersion of data points from the mean (average) of the data set

The standard deviation formula is a mathematical formula used to calculate the measure of how spread out the values in a data set are, or in other words, it measures the variability or dispersion of data points from the mean (average) of the data set. The standard deviation is denoted by the symbol σ (sigma) for a population, and s for a sample.

The formula for the standard deviation of a population is:

σ = √( Σ(x – μ)² / N )

where:
σ = standard deviation of the population
Σ = sum of ( )
x = individual data point
μ = mean (average) of the population
N = total number of data points in the population

On the other hand, for a sample, we use the following formula:

s = √( Σ(x – x̄)² / (n – 1) )

where:
s = standard deviation of the sample
Σ = sum of ( )
x = individual data point
x̄ = mean (average) of the sample
n = total number of data points in the sample

Both formulas are quite similar, but the denominator differs by either N or (n – 1) to account for whether we are calculating the standard deviation for a population or a sample. The reason we subtract 1 in the sample formula is to account for the degrees of freedom, which ensures a more accurate estimation of the true standard deviation of the population.

Once you have calculated the standard deviation using the given formula, the result gives you a numerical value that represents how closely data points are clustered around the mean. A lower standard deviation indicates that the data points are tightly grouped around the mean, while a higher standard deviation suggests that the data points are more spread out.

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