Standard Deviation Calculator

What the standard deviation calculator does

An average only tells you half the story. Two classes can both average 70 on a test, but in one everyone scored near 70, and in the other half the room failed and half aced it. The standard deviation is the number that tells those two apart. It measures how far the values typically sit from their average, so a small standard deviation means the numbers are bunched together, and a large one means they are spread out.

This calculator works it out from your data, and hands back the variance, the mean, the standard error, the range, and the count alongside it. It also lets you say whether your numbers are a whole population or a sample, which changes the answer, and the section below explains why.

How to use it

1. Enter your numbers. Type or paste them into the box, separated by commas, spaces, or new lines, whichever is easiest. You can mix them.
2. Choose population or sample. Pick the one that matches your data. The next section is all about this choice if you are unsure.
3. Press Calculate for the full breakdown, or Reset to clear it.

Population or sample: which to choose

This is the one setting that changes your answer, so it is worth getting right. The two options divide by a slightly different number at the very end, and which one is correct depends on what your data actually is.

Choose population when your numbers are the entire group you care about. Every student in the class. Every sale you made last month. Every part that came off the line. There is nothing outside your data, so the average you have is the true average, and the spread you get is the real spread.

Choose sample when your numbers are a smaller group standing in for a larger one. Fifty customers out of thousands. A hundred measurements meant to represent every part you will ever make. Your data is a window onto a bigger population you cannot measure in full.

The reason the sample version divides by one less, n minus 1 instead of n, is worth a sentence. When you only have a sample, your numbers sit a little closer to their own average than they would to the true average of the whole population, so the spread you measure comes out slightly too small. Dividing by one less nudges it back up and undoes that bias, giving a better estimate of the real spread. Statisticians call this Bessel's correction, and it matters most for small data sets, fading away as the data grows.

If you are not sure which you have, sample is the safer pick. Most real data is a sample of something, and erring slightly high on the spread is safer than erring low.

How the standard deviation is worked out

It sounds complicated, but it is really four small steps.

First, find the mean, the plain average of all your numbers. Second, for each number, measure how far it is from the mean and square that distance. Squaring keeps everything positive and gives the bigger gaps more weight. Third, average those squared distances, which gives you the variance. Fourth, take the square root of the variance, which brings you back to the units you started in. That is the standard deviation.

The only difference between the two types is the divide in the third step:

Population = √( Σ(x − mean)² ÷ N )

Sample = √( Σ(x − mean)² ÷ (n − 1) )

Here N and n are just how many numbers you have, and the Σ means add up the squared distances for every value.

Reading your results

The calculator gives you six numbers. Here is what each is telling you.

Standard deviation is the headline: the typical distance your values sit from the mean, in the same units as your data. Variance is the step just before it, the average squared distance. It is in squared units, which is hard to picture, so the standard deviation is the one people usually quote. Mean is the plain average, the point everything is measured against.

SEM, the standard error of the mean, is a different idea worth not mixing up. It does not describe how spread out your data is, it describes how precisely you have pinned down the mean itself. Collect more data and the SEM shrinks, because a bigger sample locates the average more tightly. Range shows your smallest and largest values, the full span of the data. Count is simply how many numbers you entered, a quick check that nothing was dropped or doubled.

A worked example: a set of eight numbers

Take the eight numbers 2, 4, 4, 4, 5, 5, 7, 9.

Their mean is 40 ÷ 8 = 5. The squared distances from 5 are 9, 1, 1, 1, 0, 0, 4, 16, which add up to 32.

As a population, you divide by 8: the variance is 32 ÷ 8 = 4, and the standard deviation is √4 = 2 exactly. As a sample, you divide by 7: the variance is 32 ÷ 7 ≈ 4.57, and the standard deviation is about 2.14.

Same numbers, two answers, because the question behind them is different. The sample figure is the larger of the two, which is Bessel's correction doing its job.

Entering your data, and the rounding

You can separate your numbers with commas, spaces, or new lines, in any mix, so however they come to you is usually fine. The order does not matter, because the standard deviation is about how spread out the numbers are, not the sequence they sit in.

You need at least two numbers for a spread to mean anything, and every entry has to be a number, so a stray letter or symbol will stop the calculation. Results are shown to four decimal places, which is more precision than almost any use calls for, and you can round them in further yourself.

Questions people ask

Should I choose sample or population?

Population if your numbers are the whole group you care about, with nothing outside them. Sample if they are a smaller set standing in for a larger group. When in doubt, choose sample, since most real data is a sample of something.

What is the difference between variance and standard deviation?

Variance is the average squared distance from the mean, so it is in squared units. The standard deviation is its square root, which puts it back in the original units. That is why the standard deviation is the one usually reported.

What does the standard deviation actually tell me?

How far your values typically fall from the average. A small standard deviation means the numbers are clustered tightly around the mean, and a large one means they are widely spread.

Why does the sample version divide by n minus 1?

Because a sample's numbers sit closer to their own average than to the true average of the whole population, so the spread comes out a touch too small. Dividing by one less corrects that bias. It is called Bessel's correction.

What is SEM?

The standard error of the mean. It measures how precisely you have estimated the mean, not how spread out the data is, and it gets smaller as you add more numbers.

Can I enter numbers separated by spaces or new lines?

Yes. Commas, spaces, and new lines all work as separators, and you can mix them in the same box.

References

A quick note on where the methods here come from. The formulas for variance and standard deviation, and the reason a sample divides by n minus 1, are set out in the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public reference for statistical methods. For a fuller plain-language treatment of the same ground, OpenIntro Statistics is a widely used free textbook.

1. NIST/SEMATECH e-Handbook of Statistical Methods (measures of spread, sample versus population). https://www.itl.nist.gov/div898/handbook/
2. Diez, Cetinkaya-Rundel, and Barr, OpenIntro Statistics (free textbook). https://www.openintro.org/book/os/