Standard Error Calculator
Standard error of the mean with a confidence interval from data or summary figures.
About
Standard Error Calculator
The standard error of the mean says how precisely you know an average. It answers a different question from the standard deviation, and confusing the two is the most common error in reporting a result.
The formula
SEM = s ÷ √n
s is the sample standard deviation and n is the sample size. Because the sample size sits under a square root, precision improves slowly: quadrupling the sample halves the standard error.
A worked example
Twelve measurements: 12, 15, 11, 14, 16, 13, 12, 17, 14, 15, 13, 16.
- Mean: 14.00
- Sample standard deviation: 1.8586
- SEM: 1.8586 ÷ √12 = 0.5365
- 95% interval: 14.00 ± 2.201 × 0.5365 = 12.82 to 15.18
The values themselves spread across 11 to 17. The mean is known far more tightly than that, to within about half a unit, and the standard error is what expresses the difference.
Standard deviation against standard error
They describe different things and get swapped constantly.
| Standard deviation | Standard error | |
|---|---|---|
| Describes | Spread of the individual values | Precision of the mean |
| As n grows | Settles at the population value | Falls toward zero |
| Use when | Showing how varied the data is | Showing how sure you are of the average |
Measure the height of ten thousand people and the standard deviation stays near 7 cm, because people really do vary by that much. The standard error becomes tiny, because with that many measurements the average height is known very precisely. Reporting a standard error as though it described the spread makes data look far more uniform than it is, and it is a common fault in published error bars.
Confidence intervals
interval = mean ± t × SEM
The multiplier comes from the t distribution rather than the normal one, because s is estimated from the sample rather than known. With few degrees of freedom the t value is noticeably larger than 1.96.
| Sample size | df | t at 95% |
|---|---|---|
| 5 | 4 | 2.776 |
| 10 | 9 | 2.262 |
| 20 | 19 | 2.093 |
| 30 | 29 | 2.045 |
| 60 | 59 | 2.001 |
| Very large | - | 1.960 |
Using 1.96 with a sample of five gives an interval about 30% too narrow, which overstates how much the result is worth.
What a 95% interval actually claims
It does not say there is a 95% probability the true mean lies inside this particular interval. The true mean is a fixed number; it either does or it does not.
The claim is about the procedure. If the study were repeated many times and an interval computed each time, 95% of those intervals would contain the true mean. This one is one draw from that process. The distinction sounds pedantic and matters as soon as anyone starts treating the interval as a probability distribution over the answer.
The square root, and what precision costs
Improving precision gets expensive quickly, because of that square root.
| To achieve | Sample needed |
|---|---|
| Current standard error | n |
| Half the standard error | 4n |
| A quarter of it | 16n |
| A tenth of it | 100n |
A survey of 1,000 people has a margin of error near 3 percentage points. Reaching 1 point needs about 9,000, and reaching 0.3 needs around 100,000. This is why national polls cluster around a thousand respondents: it is roughly where the cost of another interview stops buying useful precision.
Standard error of other statistics
The mean is the common case, and the idea generalises. A proportion has standard error √(p(1-p) ÷ n), which is largest at p = 0.5 and shrinks toward the extremes. A regression slope, a difference between two means and a correlation each have their own formula.
All of them answer the same question: how much would this statistic bounce around if the study were run again with a fresh sample from the same population.
What the standard error does not cover
It measures sampling variability alone. It says nothing about bias, and a biased sample produces a tight confidence interval around the wrong number.
A poll that only reaches people who answer landlines can have a standard error of half a percentage point and be several points off, because the people it missed differ from the people it reached. No sample size fixes that, and a small standard error can lend false confidence to a badly drawn sample.
The standard error of a proportion
Poll results use a different formula for the same idea:
SE = √(p(1 - p) ÷ n)
At p = 0.5 with 1,000 respondents that is 1.58 percentage points, and the 95% margin of error is 1.96 times it, or 3.1 points. That is the figure quoted in the small print of every published poll.
The formula peaks at p = 0.5 and shrinks toward the extremes: a result of 10% on the same sample has a standard error of 0.95 points rather than 1.58. This is why a party polling at 5% has a tighter interval than one polling at 50%, on identical sample sizes.
Pushing the margin from 3.1 points to 1.0 needs 9,604 respondents, and reaching 0.3 needs 106,711. The cost curve is what fixes most national polls near a thousand.
Comparing two means
Comparing two groups needs the standard error of the difference, which combines both:
SE of difference = √(SE1² + SE2²)
Two groups each with a standard error of 0.5 give a difference with a standard error of √0.5 = 0.707, larger than either on its own. Uncertainty adds when quantities are compared, which is why detecting a difference between two groups needs more data than estimating one group well.
A common shortcut, checking whether two confidence intervals overlap, is conservative and often wrong. Intervals can overlap while the difference is still statistically significant, so testing the difference directly is the correct approach.
What error bars should say
An error bar is meaningless without a label, since three different things get plotted with identical marks: the standard deviation, the standard error, and a confidence interval. Each is a different size on the same data, with the standard deviation the largest and the standard error the smallest.
The convention worth following is to state in the caption which one is shown and, for a confidence interval, at what level. Readers cannot infer it from the picture, and a figure showing standard errors looks roughly three times tighter than the same data showing standard deviations at n = 10.
The central limit theorem underneath it
The formula works because of a result that holds regardless of the shape of the underlying data. Sample means drawn from any distribution with a finite variance tend toward a normal distribution as the sample size grows, centred on the true mean with a spread of s divided by the square root of n.
That is why a confidence interval built this way is reasonable even when the individual measurements are visibly skewed. The individual values need not be normal; the average of enough of them is. In practice n above about 30 is treated as sufficient for moderately skewed data, and heavily skewed or long-tailed data needs more, or a method that does not rely on the approximation.
Common mistakes
Reporting standard error as spread. Error bars labelled SEM look far tighter than the data really is; say which one is plotted.
Using 1.96 for small samples. Use the t value for n-1 degrees of freedom.
Using the population standard deviation formula. Sample standard deviation divides by n-1, not n.
Treating a tight interval as accuracy. It measures precision, and precision around a biased estimate is still wrong.
Common questions
Frequently asked questions
Standard deviation describes how spread out the individual values are and settles at the population value as the sample grows. Standard error describes how precisely the mean is known and shrinks toward zero as the sample grows. They answer different questions.
Increase the sample size, or reduce the variability in the measurements themselves. Because of the square root, halving the standard error requires four times the sample, so improving measurement precision is often the cheaper route.
Because the sample mean is estimated from the same data, which makes the deviations slightly too small on average. Dividing by n-1 rather than n corrects that bias. The difference matters at small sample sizes and fades as n grows.
Use the t value whenever the standard deviation is estimated from the sample, which is almost always. With 4 degrees of freedom it is 2.776 rather than 1.96, so using the normal value would give an interval roughly 30% too narrow.
It means the procedure produces intervals that contain the true value 95% of the time across repeated studies. It does not mean there is a 95% probability that this specific interval contains the true mean, since the true mean is fixed rather than random.
Only when the sample size is 1, where they are equal and neither is meaningful. For any n above 1 the square root of n exceeds 1, so the standard error is always the smaller figure.
Rearrange the formula: n equals the standard deviation divided by the target standard error, all squared. Halving the current standard error needs four times the sample, and reducing it tenfold needs a hundred times.
No, it means the result is precise. Precision and accuracy differ: a biased sampling method produces a tight interval around the wrong value. Standard error measures sampling variability only, and no sample size corrects a bias.