Variance and Standard Deviation Step by Step (Population vs Sample)

Why the mean is not enough

Class A scored 70, 75, 80, 85, 90. Class B scored 60, 70, 80, 90, 100. Both means are 80, but B is clearly more spread out. The standard deviations say so with numbers: 7.07 for A and 14.14 for B.

Deviations, variance and standard deviation

The deviation of a value is its distance from the mean, $latex x-\bar{x}$. You cannot average the deviations directly, because they always add up to 0: the positive ones cancel the negative ones. Squaring each deviation fixes the signs and gives extra weight to values far from the mean.

• The variance is the average of the squared deviations.
• The standard deviation is the square root of the variance. It comes back to the units of the data (points, centimeters, dollars), while the variance is in squared units.

For a population (every value in the group), with mean $latex \mu$ and $latex N$ values:

$latex \sigma^2=\frac{\sum (x-\mu)^2}{N} \qquad \sigma=\sqrt{\sigma^2}$

For a sample (some values taken from a larger group), with mean $latex \bar{x}$ and $latex n$ values:

$latex s^2=\frac{\sum (x-\bar{x})^2}{n-1} \qquad s=\sqrt{s^2}$

See the squares

Each blue square has a side equal to one value's distance from the mean, so its area is that deviation squared. The dashed orange square is the "average square": its area equals the variance, so its side is the standard deviation. Drag a point away from the others and watch the orange square grow. Switch to $latex n-1$ to see the sample version.

Step by step

1. Find the mean: add the values and divide by how many there are.
2. Subtract the mean from each value to get the deviations.
3. Square each deviation.
4. Add the squares. This is the sum of squares.
5. Divide by $latex N$ (population) or $latex n-1$ (sample). That is the variance.
6. Take the square root. That is the standard deviation.

Here is the whole process for the population 2, 4, 4, 4, 5, 5, 7, 9. The mean is $latex \mu=\frac{40}{8}=5$.

x	x − 5	(x − 5)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
Total	0	32

The sum of squares is $latex 32$. Divide by $latex N=8$: $latex \sigma^2=\frac{32}{8}=4$. Then $latex \sigma=\sqrt{4}=2$.

Population or sample: N or n − 1?

Use $latex N$ when your data is everyone you care about, such as all 30 students in a class. Use $latex n-1$ when the data is a sample used to estimate a larger group, such as 30 voters chosen from a whole city.

The reason for $latex n-1$ is that the sample mean is calculated from the same data, so it sits closer to those values than the true mean would. Deviations measured from $latex \bar{x}$ come out a little too small on average, and dividing by a slightly smaller number corrects that. Another way to see it: once the mean is fixed, only $latex n-1$ of the deviations are free to vary, because the last one must make them add up to 0.

With large samples the difference fades. For $latex n=100$ the two standard deviations differ by about 0.5%. Calculators usually show both, labeled σ and s. The Standard deviation calculator reports both, and the Mean, median and mode calculator gives the mean and the range.

A shortcut formula

Expanding the square gives a version that skips the deviations:

$latex \sigma^2=\frac{\sum x^2}{N}-\mu^2$

For 2, 4, 4, 4, 5, 5, 7, 9: $latex \sum x^2=232$, so $latex \sigma^2=\frac{232}{8}-5^2=29-25=4$. Same answer. It is handy for exams, but it subtracts two big similar numbers, so keep every digit to avoid rounding errors.

How to read a standard deviation

• It uses the same units as the data. A standard deviation of 10 points means a typical score is about 10 points from the mean.
• A bigger value means more spread. It is 0 only when every value is equal, and it can never be negative.
• Adding the same number to every value does not change it. Multiplying every value by $latex k$ multiplies it by $latex |k|$.
• Outliers weigh a lot, because their distance is squared. For bell-shaped data, about 68% of the values fall within one standard deviation of the mean and about 95% within two.

Worked examples

Common mistakes

• Dividing by the wrong number. Population: $latex N$. Sample: $latex n-1$. Check which one your problem needs.
• Adding the deviations without squaring. They always sum to 0.
• Forgetting the square root. The variance is in squared units. The standard deviation is the root.
• Rounding the mean too early. Use the exact mean when subtracting.
• Squaring the sum instead of summing the squares. $latex (\sum x)^2$ is not $latex \sum x^2$.
• Expecting a negative result. A negative variance or standard deviation means a mistake.

Solved exercises

Frequently asked questions

Why square the deviations and not use absolute values?

Absolute values give a valid measure too (the mean absolute deviation). Squares are used because they behave well in algebra and calculus, and they underlie many other formulas in statistics.

Can the standard deviation be 0?

Yes, when all the values are equal. Then there is no spread at all.

Is a large standard deviation bad?

It depends on the context. It only says the data is spread out. A standard deviation of 15 minutes is large for a train schedule and tiny for a marathon.

Which one should I use on a test: N or n − 1?

Read the wording. If the data is the whole group, divide by N. If it is a sample meant to estimate a larger group, divide by n − 1. When a problem does not say, textbooks usually treat a list of data as a sample.

Related calculators

• Standard deviation calculator
• Mean, median and mode calculator
• Weighted average calculator
• Percentage calculator

More tools are in the calculators hub.

See also

• Square Roots - Examples and Practice Problems
• Absolute Value - Definition and Applications
• Series and Sigma Notation - Examples with Answers
• Order of Operations in Math with Examples
• Frequency tables and histograms: how to build and read them
• Basic probability: sample space, events and examples