Standard Deviation Calculator

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sample standard deviation

Sample Standard Deviation

Sample standard deviation divides the sum of squared deviations by N−1 instead of N, known as Bessel's correction. The formula is s = √(Σ(xᵢ−x̄)²/(N−1)); use it whenever the data is a sample drawn from a larger group, not the whole group.

Why divide by N−1?

The correction exists because the sample mean x̄ is computed from the data itself, not from the true population; using x̄ in place of μ makes the deviations slightly smaller than they should be, so dividing by N−1 instead of N compensates for that bias, making s² an unbiased estimator of σ². For the 5 values 2, 4, 4, 4, 5, the squared deviations add up to 4.8; divided by N−1=4 (not N=5), the sample variance is 1.2 and s=√1.2≈1.0954, about 12% above the 0.9798 that dividing by N would give.
This adjustment matters more when the sample is small. For the heights of 5 people (170, 175, 160, 180, 165 cm), the mean is 170 cm, the squared deviations sum to 250 cm², and the sample variance is 250/4=62.5 cm², giving s≈7.906 cm. With N=5, the denominator N−1=4 is 20% smaller than N=5, so the correction's effect is still noticeable; with a sample of 50 people, the difference between dividing by 49 or by 50 already drops to about 2%.
The calculator rejects samples with fewer than 2 values, because N−1 would reach zero and the division would be undefined: a single number has no variation to measure. For 5 exam scores drawn from a larger class (78, 85, 90, 72, 95), the mean is 84, the squared deviations sum to 338, and the sample variance is 338/4=84.5, giving s≈9.192.

Examples

5 values, Bessel's correction

Input: 2, 4, 4, 4, 5
Expected output: s ≈ 1.0954

Squared deviations sum to 4.8, divided by N−1=4 (not N=5); the result is 12% higher than the population version of the same numbers.

Height of 5 people

Input: 170, 175, 160, 180, 165
Expected output: s ≈ 7.9057

Sample variance 250/4=62.5 cm²; the square root returns the result in the original unit, cm, not cm².

Grades from a 5-student class sample

Input: 78, 85, 90, 72, 95
Expected output: s ≈ 9.1924

Sample variance 338/4=84.5; the mean is 84, so each grade's typical deviation is close to 9 points.

Full tool FAQ

What is the difference between population and sample standard deviation?

Population (σ) divides the sum of squared differences by N. Sample (s) divides by N−1 (Bessel's correction), giving an unbiased estimate of population variance when working with a sample.

What is Bessel's correction (N−1)?

Why do I need at least two values for sample deviation?

What is variance?

What is range?

Can I use negative or decimal numbers?

How does the tool handle duplicate values?

How precise is the calculation?

Frequently asked questions

Why won't the calculator compute with just 1 number?

Because the formula divides by N−1: with N=1 the divisor would be zero, an undefined operation. At least 2 values are needed for there to be any variation to measure.

At what sample size does Bessel's correction stop making a noticeable difference?

It never fully disappears, but its effect shrinks as N grows: for N=5, N−1=4 is 20% smaller; for N=50, N−1=49 is only 2% smaller, so the result barely changes.

Should I use s if I have data for every employee in the company, not just a sample?

No; in that case use the population standard deviation (σ), which divides by N. s is specifically for when the data represents only part of a larger group.

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sample standard deviation

Sample Standard Deviation

Why divide by N−1?

The correction exists because the sample mean x̄ is computed from the data itself, not from the true population; using x̄ in place of μ makes the deviations slightly smaller than they should be, so dividing by N−1 instead of N compensates for that bias, making s² an unbiased estimator of σ². For the 5 values 2, 4, 4, 4, 5, the squared deviations add up to 4.8; divided by N−1=4 (not N=5), the sample variance is 1.2 and s=√1.2≈1.0954, about 12% above the 0.9798 that dividing by N would give.

This adjustment matters more when the sample is small. For the heights of 5 people (170, 175, 160, 180, 165 cm), the mean is 170 cm, the squared deviations sum to 250 cm², and the sample variance is 250/4=62.5 cm², giving s≈7.906 cm. With N=5, the denominator N−1=4 is 20% smaller than N=5, so the correction's effect is still noticeable; with a sample of 50 people, the difference between dividing by 49 or by 50 already drops to about 2%.

The calculator rejects samples with fewer than 2 values, because N−1 would reach zero and the division would be undefined: a single number has no variation to measure. For 5 exam scores drawn from a larger class (78, 85, 90, 72, 95), the mean is 84, the squared deviations sum to 338, and the sample variance is 338/4=84.5, giving s≈9.192.

Examples

5 values, Bessel's correction

Input: 2, 4, 4, 4, 5
Expected output: s ≈ 1.0954

Squared deviations sum to 4.8, divided by N−1=4 (not N=5); the result is 12% higher than the population version of the same numbers.

Height of 5 people

Input: 170, 175, 160, 180, 165
Expected output: s ≈ 7.9057

Sample variance 250/4=62.5 cm²; the square root returns the result in the original unit, cm, not cm².

Grades from a 5-student class sample

Input: 78, 85, 90, 72, 95
Expected output: s ≈ 9.1924

Sample variance 338/4=84.5; the mean is 84, so each grade's typical deviation is close to 9 points.

Full tool FAQ

What is the difference between population and sample standard deviation?

Population (σ) divides the sum of squared differences by N. Sample (s) divides by N−1 (Bessel's correction), giving an unbiased estimate of population variance when working with a sample.

What is Bessel's correction (N−1)?

Why do I need at least two values for sample deviation?

What is variance?

What is range?

Can I use negative or decimal numbers?

How does the tool handle duplicate values?

How precise is the calculation?

Frequently asked questions

Why won't the calculator compute with just 1 number?

Because the formula divides by N−1: with N=1 the divisor would be zero, an undefined operation. At least 2 values are needed for there to be any variation to measure.

At what sample size does Bessel's correction stop making a noticeable difference?

It never fully disappears, but its effect shrinks as N grows: for N=5, N−1=4 is 20% smaller; for N=50, N−1=49 is only 2% smaller, so the result barely changes.

Should I use s if I have data for every employee in the company, not just a sample?

No; in that case use the population standard deviation (σ), which divides by N. s is specifically for when the data represents only part of a larger group.