Standard Deviation Calculator

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population standard deviation

Population Standard Deviation

Population standard deviation divides the sum of squared deviations by N, the total count, because every member of the group has already been measured and μ is known exactly, with no correction needed. The formula is σ = √(Σ(xᵢ−μ)²/N).

When to use population standard deviation

Use σ when your dataset is 100% of the group you want to describe, not part of it: the grades of all 8 students in a class, for example, or the ages of all 4 dogs at a shelter. For the grades 2, 4, 4, 4, 5, 5, 7, 9 (N=8), the mean is 5, the sum of squared deviations is 32, the population variance is 32/8=4, and σ=√4=2. Since μ is the population's real mean, not an estimate, there is no bias to correct, so the divisor stays at N.
The contrast with sample standard deviation shows up most clearly in small datasets. For the 5 values 2, 4, 4, 4, 5, taken here as the entire population, the squared deviations add up to 4.8; dividing by N=5, the population variance is 0.96 and σ≈0.9798. Treating those same 5 numbers as a sample drawn from a larger group instead divides by N−1=4, giving variance 1.2 and sample standard deviation ≈1.0954, about 12% higher. Picking the wrong divisor, N instead of N−1 or vice versa, changes the result noticeably exactly when N is small.
In practice, research and experimental data are rarely the entire population, so σ shows up less often day-to-day than the sample standard deviation s. σ fits best when the phenomenon itself is closed by definition, such as the temperature variation recorded across all 30 days of one specific month, or the weight spread of all 4 units of a discontinued product line.

Examples

Grades of 8 students (whole class)

Input: 2, 4, 4, 4, 5, 5, 7, 9
Expected output: σ = 2

Squared deviations add up to 32; dividing by N=8 gives variance 4 and σ=√4=2.

5 values as a closed population

Input: 2, 4, 4, 4, 5
Expected output: σ ≈ 0.98

Population variance = 4.8/5 = 0.96; σ = √0.96 ≈ 0.9798, about 12% smaller than the sample standard deviation of the same numbers.

Age of 4 shelter dogs

Input: 3, 5, 6, 10
Expected output: σ ≈ 2.55

Mean 6; squared deviations sum to 26; population variance 26/4=6.5; σ=√6.5≈2.5495.

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

Should I use σ or the sample standard deviation if my data came from surveying part of a larger group?

Use the sample standard deviation (s), which divides by N−1. σ is only correct when the data covers 100% of the group under study, not a sample of it.

For N=8, is the difference between dividing by N and N−1 large?

Not very: for grades 2,4,4,4,5,5,7,9, σ=2 (÷8) against s≈2.1381 (÷7), a difference of about 7%. The larger N gets, the smaller the impact of Bessel's correction.

Does the calculator automatically detect whether my data is a population or a sample?

No; you choose the type before calculating. The formula only changes in the divisor (N for population, N−1 for sample), so the same set of numbers produces two different results depending on the option selected.

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population standard deviation

Population Standard Deviation

When to use population standard deviation

Use σ when your dataset is 100% of the group you want to describe, not part of it: the grades of all 8 students in a class, for example, or the ages of all 4 dogs at a shelter. For the grades 2, 4, 4, 4, 5, 5, 7, 9 (N=8), the mean is 5, the sum of squared deviations is 32, the population variance is 32/8=4, and σ=√4=2. Since μ is the population's real mean, not an estimate, there is no bias to correct, so the divisor stays at N.

The contrast with sample standard deviation shows up most clearly in small datasets. For the 5 values 2, 4, 4, 4, 5, taken here as the entire population, the squared deviations add up to 4.8; dividing by N=5, the population variance is 0.96 and σ≈0.9798. Treating those same 5 numbers as a sample drawn from a larger group instead divides by N−1=4, giving variance 1.2 and sample standard deviation ≈1.0954, about 12% higher. Picking the wrong divisor, N instead of N−1 or vice versa, changes the result noticeably exactly when N is small.

In practice, research and experimental data are rarely the entire population, so σ shows up less often day-to-day than the sample standard deviation s. σ fits best when the phenomenon itself is closed by definition, such as the temperature variation recorded across all 30 days of one specific month, or the weight spread of all 4 units of a discontinued product line.

Examples

Grades of 8 students (whole class)

Input: 2, 4, 4, 4, 5, 5, 7, 9
Expected output: σ = 2

Squared deviations add up to 32; dividing by N=8 gives variance 4 and σ=√4=2.

5 values as a closed population

Input: 2, 4, 4, 4, 5
Expected output: σ ≈ 0.98

Population variance = 4.8/5 = 0.96; σ = √0.96 ≈ 0.9798, about 12% smaller than the sample standard deviation of the same numbers.

Age of 4 shelter dogs

Input: 3, 5, 6, 10
Expected output: σ ≈ 2.55

Mean 6; squared deviations sum to 26; population variance 26/4=6.5; σ=√6.5≈2.5495.

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

Should I use σ or the sample standard deviation if my data came from surveying part of a larger group?

Use the sample standard deviation (s), which divides by N−1. σ is only correct when the data covers 100% of the group under study, not a sample of it.

For N=8, is the difference between dividing by N and N−1 large?

Not very: for grades 2,4,4,4,5,5,7,9, σ=2 (÷8) against s≈2.1381 (÷7), a difference of about 7%. The larger N gets, the smaller the impact of Bessel's correction.