PertengahanFundamentals·9 min

Standard Deviation Formula Explained: Step-by-Step Guide

Master the standard deviation formula with our step-by-step guide. Learn the difference between population and sample formulas, calculations, and applications.

By Standard Deviation Calculator Team · Data Science Team·Published 16 April 2026

What is the Standard Deviation Formula?

The standard deviation formula is the mathematical equation used to quantify the amount of variation or dispersion in a set of data values. A low standard deviation indicates that the data points tend to be close to the mean (μ or x̄), while a high standard deviation indicates that the data points are spread out over a wider range of values.

In statistics, the formula you use depends on whether you are working with an entire population or a sample drawn from that population. The core concept involves calculating the average of the squared deviations from the mean, known as the variance (σ²), and then taking the square root to return the measurement to the original units.

Population Standard Deviation

σ = √[ Σ (xi - μ)² / N ]

σ (sigma): Population standard deviation
Σ (sigma): Sum of...
xi: Each individual value in the dataset
μ (mu): Population mean
N: Total number of data points in the population

Population vs. Sample Standard Deviation

In real-world data analysis, it is rare to have data for an entire population. Most of the time, we collect a sample to make inferences about the larger population. Because a sample only estimates the population mean, calculating standard deviation using the population formula on a sample consistently underestimates the true variability. To correct this bias, we use the sample standard deviation formula.

Sample Standard Deviation

s = √[ Σ (xi - x̄)² / (n - 1) ]

Don't mix up your formulas!

Using 'N' for a sample or 'n-1' for a population will result in an incorrect measure of spread. The sample formula with n-1 is known as Bessel's correction and is strictly required for unbiased estimation of population variance.

Step-by-Step Calculation of the Formula

Calculating standard deviation by hand requires a systematic approach. By following these steps, you can accurately compute either the population or sample standard deviation for any dataset.

Calculate the Mean

Sum all the data points (Σxi) and divide by the total number of points (N or n) to find the mean (μ or x̄).

Find the Deviations

Subtract the mean from each individual data point to find the deviation: (xi - mean).

Square the Deviations

Square each of the deviations calculated in the previous step: (xi - mean)². This ensures all values are positive.

Sum the Squared Deviations

Add up all the squared deviations to find the sum of squares: Σ(xi - mean)².

Divide by N or n-1

For a population, divide by N. For a sample, divide by (n - 1). This gives you the variance (σ² or s²).

Take the Square Root

Take the square root of the variance to find the standard deviation (σ or s).

Why Does the Sample Formula Divide by n-1?

Dividing by n-1 instead of n is a concept known as Bessel's correction. Because the sample mean (x̄) is calculated from the sample data itself, the deviations (xi - x̄) are mathematically constrained to sum to zero. This means the data points are slightly closer to the sample mean than they are to the true population mean (μ).

By dividing by n-1 (the degrees of freedom), we inflate the variance just enough to compensate for this underestimation, providing an unbiased estimator of the population variance.

Sources

References and further authoritative reading used in preparing this article.

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How to Read This Article

A statistics tutorial is a practical interpretation guide, not just a formula dump. It refers to the assumptions, notation, and reporting language that analysts need when they explain a result to a teacher, manager, client, or reviewer. The article body covers the specific topic, while the sections below create a common interpretation frame that readers can reuse across related metrics.

Reading goal	What to focus on	Common mistake
Definition	What the metric is and what quantity it summarizes	Treating the formula as self-explanatory
Formula choice	Sample versus population assumptions and notation	Using n when n-1 is required or vice versa
Interpretation	Whether the result indicates concentration, spread, or risk	Calling a large value good or bad without context

Frequently Asked Questions

How should I interpret a high standard deviation?

A high standard deviation means the observations are spread farther from the mean on average. Whether that spread is acceptable depends on the context: wide dispersion might signal risk in finance, instability in manufacturing, or genuine natural variation in scientific data.

Why do some articles mention n while others mention n-1?

The denominator reflects the difference between population and sample formulas. Population variance and population standard deviation use N because the full dataset is known. Sample variance and sample standard deviation often use n-1 because Bessel’s correction reduces bias when estimating population spread from a sample.

What is a statistical interpretation guide?

A statistical interpretation guide is a page that moves beyond arithmetic and explains meaning. It tells you what a metric is, when the formula applies, and how to describe the result in plain English without overstating certainty.

Can I cite this article in a report?

You should cite the underlying authoritative reference for formal work whenever possible. This page is best used as an explanatory bridge that helps you understand the concept before quoting the original standard or handbook.

Why include direct citations on every article page?

Direct citations give readers a route to verify the definition, notation, and assumptions. That improves trust and reduces the chance that a simplified explanation is mistaken for the entire technical standard.

Authoritative References

These sources define the concepts referenced most often across our articles. Bessel's correction is a sample adjustment, variance is a squared measure of spread, and standard deviation is the square root of variance expressed in the same units as the data.