Core inference · Estimation
Standard Error Calculator
The standard error is how much your estimate would jump around if you redid the study — the standard deviation of the sample mean, not of the data. For a mean it is SE = s ∕ √n; for a proportion, √[p(1−p) ∕ n]. Enter raw data, or a standard deviation and sample size, or a count of successes, and this returns the SE with a 95% confidence interval — and draws the standard error against the standard deviation, because mixing the two up is the single most common slip in applied statistics.
The standard error answers “how precisely have I pinned down the average?”, not “how spread out are the values?” — that second question is the standard deviation. The two are shown together below.
Result
In plain English
Measure eight people and the eight heights are spread out — that spread is the standard deviation. But the average of those eight is a single number, and if you grabbed a different eight you would get a slightly different average. The standard error measures how much that average would wobble from sample to sample. It is always smaller than the standard deviation, because averaging cancels out a lot of the individual noise, and it keeps shrinking as the sample grows.
- standard error (SE)
- The standard deviation of a statistic — usually the sample mean. It quantifies the uncertainty in your estimate, and is the engine of confidence intervals and t-tests.
- SE of the mean
- s ∕ √n: the sample standard deviation divided by the square root of the sample size. Bigger samples, smaller error.
- SE of a proportion
- √[p(1−p) ∕ n], where p is the observed proportion. Widest at p = 0.5, narrower as p approaches 0 or 1.
- standard error ≠ standard deviation
- The SD describes the scatter of individual data points; the SE describes the scatter of the estimate. Quote the SD to say how variable people are, the SE to say how sure you are of the mean.
- the √n law
- Because of the square root, halving the standard error takes four times the data, not twice. Precision gets expensive fast.
- confidence interval
- estimate ± (critical value) × SE. A 95% interval for a mean uses the t critical value; for a proportion, roughly ±1.96 × SE.
Frequently asked
What is the difference between standard error and standard deviation?
They answer different questions. The standard deviation measures how spread out the individual data points are — how much one person differs from the next. The standard error measures how spread out a summary statistic (usually the mean) would be if you repeated the whole study. The standard error of the mean is the standard deviation divided by √n, so it is always smaller and shrinks as the sample grows, while the standard deviation settles toward a fixed value that describes the population. Use the SD to describe variability; use the SE to describe the precision of your estimate. Reporting one when you mean the other — for instance, drawing tiny “± SE” error bars and calling them spread — is a classic and misleading mistake.
How do you calculate the standard error of the mean?
Divide the sample standard deviation by the square root of the sample size: SE = s ∕ √n. For example, eight values with a standard deviation of 2.14 give SE = 2.14 ∕ √8 = 2.14 ∕ 2.83 ≈ 0.76. From raw data, this calculator first computes s (using the n − 1 sample formula) and then divides by √n. If you already know s and n you can enter them directly. The same √n in the denominator is why larger samples estimate the mean more precisely.
Why does the standard error get smaller as the sample size grows?
Because averaging cancels noise. When you take a mean, the high and low values partly offset each other, and the more values you average, the more complete that cancellation is. Mathematically the error falls with √n, not n, so there are diminishing returns: going from 25 to 100 observations (4×) halves the standard error, and halving it again needs another 4×, up to 400. This square-root law is why huge samples can pin a mean down extremely tightly — and a warning sign too, because with enough data the standard error becomes so small that trivial, unimportant differences turn up as “statistically significant.”
Should error bars show the standard deviation or the standard error?
It depends on what you want the bars to say, and you must label which you used. Standard deviation bars show how variable the underlying data are; standard error (or, better, a confidence interval) bars show how precisely you have estimated the mean. SE bars are always narrower, so showing them makes results look more certain — which is fine if the question is about the mean, but misleading if a reader takes the small bars to mean the data themselves are tightly clustered. Because the SE depends on sample size, two studies with identical data spread can show very different SE bars purely because one had more participants. Always state “± SD” or “± SE (or 95% CI)” explicitly.