What is the difference between sample and population standard deviation?

Population standard deviation divides the sum of squared deviations by n, while sample standard deviation divides by n − 1. The n − 1 adjustment, known as Bessel's correction, removes the bias that appears when you estimate spread from a sample rather than measuring the entire population. Use the sample formula whenever your data is a subset of a larger group, which is most of the time.

How do you calculate standard deviation step by step?

Find the mean, subtract it from each value and square the result, add those squared deviations together, divide by n (population) or n − 1 (sample) to get the variance, then take the square root of the variance. The worksheet above lays out each value's deviation so you can follow the arithmetic.

Is variance the same as standard deviation?

No. Variance is the average of the squared deviations, so it carries squared units. Standard deviation is the square root of the variance, which returns the spread to the original units and makes it directly comparable to the data and the mean.

Why divide by n − 1 for a sample?

When you compute deviations from the sample mean rather than the true population mean, the squared deviations come out slightly too small on average. Dividing by n − 1 instead of n inflates the result just enough to correct that bias, giving an unbiased estimate of the population variance. This is Bessel's correction.

What does a standard deviation of 0 mean?

A standard deviation of zero means every value in the data set is identical — there is no spread at all. Each deviation from the mean is zero, so the sum of squared deviations and the variance are both zero, and so is their square root.

Can standard deviation be negative?

No. It is a square root of a sum of squared numbers, so it is always zero or positive. If a tool ever shows a negative standard deviation, something has gone wrong in the calculation.

What is the empirical rule (68–95–99.7)?

For data that follows a normal, bell-shaped distribution, about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. It is a fast way to judge whether a value is typical or unusual.

What is the coefficient of variation?

The coefficient of variation (CV) is the standard deviation divided by the mean, expressed as a percentage. Because it is unitless, it lets you compare the relative spread of data sets that have different scales or units — a CV of 10% is tighter than a CV of 40% regardless of what is being measured.

Statistics

Standard Deviation

Paste a data set and get its sample or population standard deviation instantly — with the mean, variance, a full summary, and every step of the working shown.

Enter your data

n = 8

Method

Data set

Separate numbers with commas, spaces, or new lines.

Sample standard deviation

s =2.1381

Widely dispersed

Mean

Variance

4.5714

Count

Min

Max

Range

Sum

Std error

0.7559

42.76%

Switch method any time.

Try an example

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The basics

What is standard deviation?

Standard deviation answers one question: how far, on average, do the numbers in a data set sit from their mean? A small value means the data huddles around the average; a large value means it is scattered. Because it is reported in the same units as your data, it reads as a plain “typical distance from the middle” — not an abstract score.

s = √[ Σ(x − x̄)² / (n − 1) ]

Spread, in the original units

It is the square root of the variance, so it lives on the same scale as the data. If your values are in centimetres, the standard deviation is in centimetres too.

Scores 2…9 → s ≈ 2.14

σ² = Σ(x − x̄)² / n

Built on squared distance

Each value's distance from the mean x̄ is squared so that positive and negative gaps both count. Averaging those squares gives the variance, and the square root brings it back to a readable size.

Σ(x − x̄)² then ÷ by n or n − 1

Method

How to calculate it by hand

Five steps turn a list of numbers into a standard deviation. The tool runs them live, but this is exactly what it is doing.

1
Find the mean
Add every value and divide by how many there are. This average x̄ is the centre you measure spread from.
x̄ = Σx / n
2
Take each deviation and square it
Subtract the mean from each value to get its deviation, then square it so negatives don't cancel positives.
(x − x̄)²
3
Sum the squared deviations
Add up all those squares. This total is the raw amount of spread in the data.
Σ(x − x̄)²
4
Divide for the variance
Divide by n for a population, or n − 1 for a sample. The result is the variance.
σ² = Σ(x − x̄)² / n · s² = Σ(x − x̄)² / (n − 1)
5
Take the square root
The square root of the variance is the standard deviation, back in the original units.
s = √(s²)

Worksheet

The deviation table, value by value

This is the table you fill in by hand: every value, how far it sits from the mean, and that distance squared. The squared column totals to Σ(x − x̄)² — divide it and take the root. Edit the data and switch the divisor to watch each row recompute.

Data set

Value (x)	Deviation (x − x̄)	Squared (x − x̄)²
2	−3	9
4	−1	1
4	−1	1
4	−1	1
5	0	0
5	0	0
7	2	4
9	4	16
Σ = 40	≈ 0	Σ = 32

Mean (x̄)

Variance (s²)

4.5714

Std deviation (s)

2.1381

Variance = 32 ÷ n − 1 (7) = 4.5714; the standard deviation is its square root.

Worked examples

See it across cases

The same five numbers as a sample, then as a population, then a set with no spread at all — each computed live so you can see exactly where the divisor changes the answer.

A sample drawn from a larger group — divide by n − 1.

Data 4, 8, 6, 5, 3 (n = 5)

Mean: x̄ = 26 ÷ 5 = 5.2

Squared deviations: Σ(x − x̄)² = (4 − 5.2)² + (8 − 5.2)² + (6 − 5.2)² + (5 − 5.2)² + (3 − 5.2)² = 14.8

Variance: s² = 14.8 ÷ n − 1 = 14.8 ÷ 4 = 3.7

Standard deviation: s = √3.7

s = 1.9235

Reference

Sample vs population

Both formulas start identically — find the mean, then sum the squared deviations. The only difference is what you divide by, and that choice changes the answer.

σ = √[ Σ(x − x̄)² / n ]

Population (divide by n)

Use this when your data is the entire group you care about — every student in a class, every product in a batch. There is no estimation, so you divide by the full count n.

Use when you measured everyone

s = √[ Σ(x − x̄)² / (n − 1) ]

Sample (divide by n − 1)

Use this when your data is a sample standing in for a larger population. Dividing by n − 1 applies Bessel's correction so the estimate isn't biased downward.

The everyday default

Reference

The empirical rule (68–95–99.7)

For data that follows a normal, bell-shaped curve, the standard deviation maps directly to how much of the data falls within reach of the mean. It is the quickest way to judge whether a value is ordinary or an outlier.

Within	Of the data	Meaning
x̄ ± 1σ	≈ 68%	One standard deviation — the bulk of the data.
x̄ ± 2σ	≈ 95%	Two standard deviations — almost everything.
x̄ ± 3σ	≈ 99.7%	Three standard deviations — virtually all of it.

Where it shows up

Why standard deviation matters

One number that quietly drives grading curves, risk models, and quality control on factory floors.

z = (x − x̄) / s

Test scores & curves

Standard deviation turns a raw score into a z-score, showing how many standard deviations above or below the class average a student landed.

volatility = σ of returns

Finance & risk

In investing, the standard deviation of returns is volatility — the headline measure of how risky an asset or portfolio is.

±3σ control limits

Quality control

Manufacturing sets control limits a few standard deviations from the target; a point beyond them flags a process that has drifted.

Worth knowing

Rules worth remembering

A handful of properties make standard deviation easier to reason about and harder to misuse.

s ≥ 0

Never negative

It is a square root of a sum of squares, so it is always zero or positive. A negative result always signals a mistake.

add c → s unchanged

Shifting doesn't change it

Add the same constant to every value and the spread is identical — only the mean moves. The standard deviation cares about distances, not position.

× c → s × |c|

Scaling scales it

Multiply every value by a constant and the standard deviation scales by the absolute value of that constant.

s² = variance

Square it for variance

Variance is just the standard deviation squared. Square to add spreads together; root to report in real units.

sensitive to outliers

Outliers dominate

Because deviations are squared, a single far-out value can inflate the result sharply — check the data before trusting a large number.

SEM = s / √n

Not the standard error

The standard error of the mean shrinks as the sample grows. It measures the mean's precision, not the spread of the data itself.

Pitfalls

Common mistakes

Most wrong answers come from one of these three slips rather than the arithmetic itself.

n vs n − 1

Wrong divisor

Using n when the data is a sample (or n − 1 for a full population) gives a slightly off answer — small data sets feel it most.

forgot to square root

Stopping at variance

The variance is an intermediate step. Reporting it as the standard deviation leaves the answer in squared units and far too large.

dropped a sign

Mishandling negatives

Deviations below the mean are negative; squaring fixes the sign, but forgetting the parentheses when squaring by hand is a classic error.

FAQ

Common questions

Standard deviation measures how spread out a set of numbers is around their mean. A small value means the data clusters tightly around the average; a large value means it is widely dispersed. It is reported in the same units as the original data, which makes it easy to interpret.

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