Standard error of the mean (SEM) calculator

--- --- Methodology

The standard error of the mean measures how much a sample mean would vary if you drew the sample again. Paste your data below and it is calculated alongside the standard deviation it derives from.

Enter your data

Separate numbers with commas, spaces or new lines, or paste a spreadsheet column. Decimals and negatives are fine; write 10:3 for a value that occurs 3 times.

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Calculation type

Sample (n − 1) Part of a larger group Population (n) The entire group

Not sure which? How to choose sample or population

Standard error of the mean

1.07633

If you drew another sample of 8 from the same population, its mean would typically land within about 1.08 of this one's (15.88). The individual values vary more than that: their standard deviation is 3.044.

Count (n): 8
Mean (x̄): 15.875
Standard deviation (s): 3.04432
Variance (s²): 9.26786

Minimum: 12
Q1 (25%): 13.75
Median: 15.5
Q3 (75%): 17.5
Maximum: 21
Range: 9

More statistics (5)

Relative SD (%RSD): 19.1768%
Coefficient of variation: 0.191768
Sum (Σx): 127
Sum of squares, Σ(x − x̄)²: 64.875
IQR (Q3 − Q1): 3.75

The wide, faint curve is the spread of the individual values (s = 3.044). The narrow curve is the spread of the sample mean (SE = 1.076), √8 times tighter, because averaging cancels noise.

Show the working, step by step

Standard error formula

SE = s / √n

where s is the sample standard deviation and n the sample size. That is the whole calculation; the difficulty is not arithmetic but knowing which of SD and SE answers the question in front of you.

Standard deviation or standard error?

	Standard deviation	Standard error
Describes	Spread of the observations	Precision of the mean
Answers	"How different are individuals?"	"How well do I know the average?"
As n grows	Settles on the population's true spread	Shrinks towards zero
Formula	√[Σ(x − x̄)²/(n − 1)]	s / √n
Use it for	Describing your data	Confidence intervals, significance tests

A concrete case. Adult heights have a standard deviation of roughly 7 cm, and that is a fact about people: measuring a million of them will not change it. But the standard error of the mean height from a sample of 100 is 7/√100 = 0.7 cm, and from a sample of 10,000 it is 0.07 cm. More data does not make people more alike; it makes your estimate of their average sharper.

Papers usually report both. The standard deviation and standard error calculator gives s and s/√n from the same data, ready to write as mean ± SD or mean ± SEM.

How sample size drives it

n	√n	SE when s = 10
5	2.24	4.47
25	5.00	2.00
100	10.00	1.00
400	20.00	0.50
1600	40.00	0.25

Each halving of the standard error costs four times the data. This diminishing return is the central constraint in study design: it is why sample sizes are chosen against a target precision rather than picked for roundness, and why very precise estimates are so expensive. The central limit theorem calculator draws the sampling distribution of the mean against the population for any n, and the standard deviation of the sample mean page adds the finite population correction for samples drawn from a small population.

Recover a standard deviation from a published standard error

Papers frequently report "mean ± SE" without giving the standard deviation, which is a problem if you want to pool their result with others. Multiplying back by √n recovers it exactly: no approximation is involved, because SE = s/√n is a definition.

Standard error (SE)

Sample size (n)

Standard deviation (s) 4.21746

Standard deviation4.21746

Variance17.787

Standard error given0.77

√n5.47723

The wide, faint curve is the spread of the individual values (s = 4.217). The narrow curve is the spread of the sample mean (SE = 0.77), √30 times tighter, because averaging cancels noise.

Show the working, step by step

If the paper reported a confidence interval instead of a standard error, the standard deviation can still be recovered; see the confidence interval page for the relationship, or use the Cochrane Handbook's method of dividing the interval width by 2t*.

Standard deviation calculator

The spread of the data itself.
Confidence interval

Turns a standard error into a range for the mean.
Variance calculator

The squared spread underneath both.
Z-score calculator

Standardising a single value rather than a mean.

Standard error calculator: the worked example on this page, with its result and chart — Standard error calculator: the worked example above, at a glance.

Common questions

What is the difference between standard deviation and standard error?

The standard deviation describes how spread out your data is. The standard error describes how much your sample mean would move around if you repeated the study. They answer different questions and are not interchangeable.

The standard error is always the smaller of the two, by a factor of √n, and it keeps shrinking as you collect more data. The standard deviation does not; it converges on the real spread of the population, which more data cannot change.

Should error bars show SD or SE?

Show the standard deviation if you want the reader to see how variable the individual observations are. Show the standard error or a confidence interval if you want them to judge how precisely the mean is known.

SE bars are always narrower, which is why they are sometimes chosen for the wrong reason. Whichever you use, state it in the caption: a bar with no label is uninterpretable.

Why does the standard error shrink with a larger sample?

Because averaging cancels noise. Individual values scatter by σ no matter how many you collect, but their average becomes more stable as the high and low draws increasingly offset one another. The rate of that stabilisation is exactly 1/√n.

The square root is the reason large studies get expensive: halving the standard error requires four times the data, and halving it again requires sixteen times.

Can I get the standard deviation back from a published standard error?

Yes, exactly: s = SE × √n. That recovery is used routinely in meta-analysis to reconstruct SDs from papers that only reported standard errors. The second calculator on this page does it.