Standard Error Calculator

SE = s / √n s = √( Σ(xᵢ − x̄)² / (n − 1) ) s = √(p(1 − p)) × 100

SE

The standard error — the standard deviation of the estimate, and the page's main result. It answers a question about the estimate rather than about the data: how far a sample mean would typically sit from the population mean if the sampling were repeated. It is the number a confidence interval is built from and a test statistic divides by, which is why it is worth computing on its own

s

The standard deviation of the sample, with the n − 1 denominator. It is what the standard error is derived from in the mean mode, and it is printed as its own row so the division can be checked. The n − 1 rather than n is what makes it an unbiased estimate of the population spread — using n would understate the variability slightly, and the understatement is largest exactly where samples are smallest

n

How many observations the sample holds — the count, also printed as its own row. It appears under a square root, which is why the standard error shrinks slowly: ten times the data buys a standard error about three times smaller. In the proportion mode this is the sample size field rather than a count derived from a list, because a percentage carries no list with it

x̄

The sample mean, printed so that the standard deviation's arithmetic has a visible centre. It plays no part in the standard error itself — the standard error is a property of the spread and the sample size, not of where the data sits — which is why a dataset can be shifted by any constant and leave the standard error unchanged

p

The sample proportion, as a fraction in the formula and a percentage on the panel. In the proportion mode it replaces the standard deviation entirely: for a yes/no question the variability is a function of the percentage, at its maximum when the split is even and shrinking to nothing as the percentage approaches 0 or 100. The × 100 converts the result back to percentage points so that it can be read alongside the percentage it came from

Use it when you have the data in hand and want the standard error of an estimate — to build a confidence interval by hand, to check a number in a report, or to see how much of a result's width comes from the spread of the data and how much from the sample size. Printing the standard deviation and the count alongside it makes that division of labour visible: change the data and the standard deviation row moves, change only the sample size and the standard error row moves while the standard deviation stays put. The two modes exist because the two most common estimates need different inputs. A mean needs the sample, since nothing predicts how spread out the measurements will be; a proportion does not, since its spread follows from its own value, which is why surveys can quote a margin of error from the sample size alone. Neither mode checks whether the sample is large enough for the estimate to be roughly normal. That depends on how skewed the underlying distribution is, and for the proportion mode there is a more specific rule of thumb about n·p and n·(1 − p) that the page does not enforce.

1. The default: eight numbers with a standard deviation of about 2.14

   1. Count: eight numbers. Mean: (2 + 4 + 4 + 4 + 5 + 5 + 7 + 9) / 8 = 40 / 8 = 5
   2. Squared deviations from 5 sum to 32, so the sample variance is 32 / 7 = 4.5714
   3. Standard deviation: √4.5714 = 2.1381
   4. Standard error: 2.1381 / √8 = 2.1381 / 2.8284 = 0.7559

   The numbers are spread either side of 5 without any of them being far out, which is what makes this a useful default: the mean is a round number and the standard deviation is not, so the last two rows cannot be confused with each other. Reading down the panel is the point — 2.1381 is a fact about the data, and 0.7559 is a fact about the estimate, and the only thing between them is the division by √8. The sample size and percentage fields sit unused in this mode.

2. Nine numbers chosen to make the standard error exactly 1

   1. Mean: (3 × 4 + (−3) × 4 + 0) / 9 = 0
   2. Squared deviations: four 9s, four 9s and a 0, summing to 72
   3. Sample variance: 72 / 8 = 9, so the standard deviation is 3
   4. Standard error: 3 / √9 = 3 / 3 = 1

   Nine observations is small enough that the n − 1 denominator visibly matters: dividing by 9 instead of 8 would give a variance of 8 and a standard deviation of 2.8284, and the standard error would come out at 0.9428 rather than 1. That is the correction doing its job — with a small sample the observed spread is a slight underestimate of the population's, and dividing by one less than the count corrects for it. The n = 9 and s = 3 pair is the same one the confidence interval page uses, and its interval comes out at exactly ±2.306 because the t value at 8 degrees of freedom is 2.306.

3. A proportion needs no data list

   1. Proportion as a fraction: 60% = 0.6
   2. Standard deviation in percentage points: √(0.6 × 0.4) × 100 = 0.4899 × 100 = 48.9898
   3. Standard error: 48.9898 / √100 = 48.9898 / 10 = 4.899
   4. The data list is ignored entirely — the count row reports the sample size field instead

   The data field is still on screen and still holds a list, and nothing in the answer uses it: in this mode the variability comes from the percentage, and the sample size comes from its own field. That is why the standard deviation row reads 48.99 — an unfamiliar-looking number for a binary question. It is the standard deviation of a yes/no variable coded as 0 and 100, and its only purpose here is to be divided by √n. The same structure appears on the sample size page, where the worst case p = 0.5 is used to plan a study before any percentage is known.

4. Two observations, where the n − 1 denominator is doing the most work

   1. Mean: 5
   2. Squared deviations: 25 and 25, summing to 50
   3. Sample variance: 50 / 1 = 50, so the standard deviation is √50 = 7.0711
   4. Standard error: 7.0711 / √2 = 7.0711 / 1.4142 = 5

   With two observations the standard deviation is simply half the distance between them — the n − 1 denominator leaves a single degree of freedom, and the formula reduces to something that can be seen rather than computed. The standard error is then the standard deviation divided by √2, which is 5 here. This is the smallest sample the page accepts in the mean mode, and it is worth trying once to see how wide an estimate from two numbers really is: the standard error comes out at half the range of the data.

The page reports the standard error of the estimate and stops before turning it into an interval or a test. That boundary is deliberate: an interval needs a critical value, which depends on a confidence level and on whether the spread had to be estimated from the sample, and a test needs a second statistic to compare against. Both are on the neighbouring pages, and the arithmetic here is the input to both rather than a substitute for either. The mean mode assumes the observations are independent, which fails for repeated measurements on the same subject, for matched pairs, and for clustered data such as students within schools; in all three the effective sample size is smaller than the count and the standard error comes out too small. It also assumes the list contains every observation you meant to include — the standard error is computed from whatever is typed, and dropping an outlier by accident narrows it. The proportion mode carries a separate caveat: its formula is a large-sample approximation that behaves badly when the count of successes or failures is small, so a percentage near 0 or 100 from a modest sample is better served by an exact interval than by this standard error. Neither mode judges whether the estimate is biased, which no standard error can detect.

What is the difference between the standard deviation and the standard error?

The standard deviation describes how far individual values sit from their mean; the standard error describes how far a sample mean sits from the population mean. The second is the first divided by √n, and that division is the whole relationship — which is why both are printed. The standard deviation is a property of the data and does not change as you collect more of it, while the standard error keeps shrinking as the sample grows. A useful check on the panel: alter the numbers and both rows move; change only the count and the standard deviation stays where it was while the standard error falls.

Why does the page divide by n − 1 instead of n?

Because the sample's own mean is used as the centre when measuring the spread, and that mean sits closer to the data than the true population mean does — so the naive average squared deviation is systematically a little too small. Dividing by one less than the count corrects for it. The effect is largest for small samples: with two observations it doubles the variance, and by a hundred observations it is a 1% adjustment. Dividing by n is not wrong for describing the sample you have; it is wrong for estimating the spread of the population it came from, which is what a standard error is for.

How do I calculate the standard error of a proportion without any data?

The proportion supplies its own variability, so no list is needed — only the sample size and the percentage. Code the two outcomes as 0 and 100 and the standard deviation works out to √(p(1 − p)) × 100, which is at its largest when the split is even and shrinks towards zero as the percentage approaches either end. For 60% of 100 people that is √(0.6 × 0.4) × 100 = 48.9898, and dividing by √100 gives a standard error of 4.899 percentage points. This is the same input pair the margin of error page uses, where the standard error is multiplied by a critical value to give a plus-or-minus.

Does the standard error get smaller as I add more data?

Yes, but only in proportion to the square root of the sample size. Going from 100 observations to 400 halves the standard error; getting another halving costs 1600. That is why the standard error is the number to watch when planning a study, and why the improvement from a very large sample over a merely large one is easy to overestimate. The standard deviation row, by contrast, stays roughly where it is — more data does not make the underlying population less variable, it only makes the estimate of its centre more precise.

What does a standard error of 0 mean?

Either that every observation in the sample was identical, or that the proportion was 0% or 100% — in both cases the formula finds no variation to work with and returns zero. Read literally, a standard error of zero says the estimate is exact, and that is a claim the data cannot support: a sample of four identical readings is evidence that the spread is small, not proof that it is zero. The page returns the zero rather than an error, because the arithmetic is correct and the reason is visible on the other rows, but a standard error of exactly zero should be treated as a sign that the sample is too small or too uniform to say anything about variability.

Is this the same as the central limit theorem calculator?

They compute the same quantity from different inputs, and which one you want depends on what you have. This page takes the sample — a list of numbers, or a percentage and a size — and is what you use once the data is in hand. The central limit theorem page takes the population standard deviation and a sample size, the values that exist before anything is collected, and is the one to use when planning a study or working out how the standard error responds to n. The two differ by exactly the estimation step in between, which is why their results agree whenever the sample standard deviation happens to equal the population value.

• 1.3.5.6. Measures of Scale — e-Handbook of Statistical Methods (the sample standard deviation and the n − 1 denominator the standard error is built from) — National Institute of Standards and Technology (NIST)
• 1.3.6.1. What is a Probability Distribution — e-Handbook of Statistical Methods (the sampling distribution of a statistic, whose spread is what the standard error measures) — National Institute of Standards and Technology (NIST)
• 1.3.5.1. Measures of Location — e-Handbook of Statistical Methods (the sample mean printed alongside the spread, and why the two are reported together) — National Institute of Standards and Technology (NIST)