Standard Deviation Calculator: Sample & Population, Steps

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Formula at a glance

Standard Deviation Calculator: Sample SD — s = √( Σ(x − x̄)² / (n − 1) ). Example: Sample of 8 values → Sample standard deviation s: 5.237229366 — How the Standard Deviation Calculator works: Sample SD — s = √( Σ(x − x̄)² / (n − 1) )

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📘 About the Standard Deviation Calculator

Standard deviation measures how spread out numbers are from their mean. A small SD means the values cluster near the average; a large SD means they are widely spread. It is used for exam marks, quality control, investment risk (volatility) and scientific measurements.

Use the sample standard deviation s (dividing by n − 1) when your data are a sample from a larger group — the usual case. Use the population SD σ (dividing by n) when you have every member. The calculator gives both, with the variance, mean, standard error of the mean and a table of deviations.

🧮 Formula

Sample SD

s = √( Σ(x − x̄)² / (n − 1) )

Population SD

σ = √( Σ(x − μ)² / n )

Standard error

SE = s/√n

✏️ Worked examples

Sample of 8 values

You enter
Data	10, 12, 23, 23, 16, 23, 21, 16
Data is a	Sample (divide by n − 1)

You get
Sample standard deviation s	5.237229366
Sample variance s²	27.42857143
Population SD σ (÷ n)	4.898979486 (variance 24)
Mean	18
Count n	8
Sum	144
Sum of squared deviations	192
Standard error of the mean	1.8516402
Range	10 to 23 (13)
Coefficient of variation	29.096%
Within 1 SD of the mean	6 of 8 values (75%)

s = 5.237229366.

Classic textbook data

You enter
Data	2, 4, 4, 4, 5, 5, 7, 9
Data is a	Population (divide by n)

You get
Population standard deviation σ	2
Population variance σ²	4
Sample SD s (÷ (n − 1))	2.138089935 (variance 4.571428571)
Mean	5
Count n	8
Sum	40
Sum of squared deviations	32
Standard error of the mean	0.75592895
Range	2 to 9 (7)
Coefficient of variation	40%
Within 1 SD of the mean	6 of 8 values (75%)

σ = 2.

🧭 How to use this calculator

What you enter

DataSeparate numbers with commas, spaces or new lines.
Data is a

What you get

Sample standard deviation s
Sample variance s²
Population SD σ (÷ n)
Mean
Count n
Sum
Sum of squared deviations
Standard error of the mean
Range
Coefficient of variation
Within 1 SD of the mean

❓ Frequently asked questions

What is the difference between sample and population standard deviation?

The sample SD divides by n − 1 (Bessel’s correction) to avoid underestimating the spread of the whole population. Use it unless your data include every member.

What is a good standard deviation?

It depends on the mean and context. The coefficient of variation (SD ÷ mean) lets you compare spread between data sets.

How is standard deviation related to variance?

The variance is the square of the standard deviation. The SD is in the same units as the data, which makes it easier to interpret.

What does the 68–95–99.7 rule mean?

For normally distributed data about 68% of values lie within 1 SD of the mean, 95% within 2 SD and 99.7% within 3 SD.

σ Standard Deviation Calculator (Sample & Population)