Standard Error Calculator

What the standard error measures

A sample mean is an estimate, and estimates wobble. Measure eight seedlings, take the mean, then take another eight and the second mean will not match the first. The standard error is the standard deviation of that wobble: if you could repeat the study many times, the means would scatter around the true value, and the standard error is the typical size of that scatter.

This is not the same job as a standard deviation. The standard deviation describes how far individual seedlings sit from their own group mean, and it does not shrink when you measure more seedlings. The standard error describes how far the group mean sits from the true mean, and it does shrink, because averaging more values cancels out more of the noise. A study that reports only the standard deviation tells you about the plants. A study that reports the standard error tells you about the estimate.

NIST's statistics handbook builds the same idea from the sampling distribution: the variance of a mean of n independent measurements is the population variance divided by n, so the standard error is sigma divided by the square root of n. Every formula on this page is that relationship, written for a different kind of estimate.

The practical payoff is a scale for surprises. A mean that sits 0.4 standard errors from a target is unremarkable. A mean that sits 4 standard errors away is worth investigating. The standard deviation calculator handles the first question, the spread of the data itself, when that is what you need.

The formula for a mean, worked through

With a sample standard deviation s and n observations, the standard error of the mean is s divided by the square root of n. The sample standard deviation uses n minus 1 in the denominator, because the deviations are measured from the sample mean rather than the unknown true mean.

Eight seedling heights in centimeters were 12, 15, 11, 14, 13, 16, 10 and 15. They add to 106, so the mean is 106 ÷ 8 = 13.25 cm. The deviations from 13.25 are -1.25, 1.75, -2.25, 0.75, -0.25, 2.75, -3.25 and 1.75, and their squares add to 31.5.

Divide 31.5 by 7 degrees of freedom and the sample variance is 4.5, so s = 2.1213 cm. The square root of 8 is 2.8284, and 2.1213 ÷ 2.8284 = 0.75 cm. That is the standard error: on repeated samples of eight, the mean would typically land about 0.75 cm from the true mean.

Notice the pull of that last step. Doubling the sample size divides the standard error by 1.4142, not by 2. Four times the sample size is needed to halve the error. That square root is why big surveys cost so much more than small ones for each extra digit of precision.

Turning the error into an interval

An interval for the mean is the estimate plus or minus a critical value times the standard error. When the population standard deviation is unknown and the sample is small, the critical value comes from the t distribution with n minus 1 degrees of freedom. For the seedlings, the 95 percent value at 7 degrees of freedom is 2.3646, so the interval is 13.25 ± 2.3646 × 0.75, which is 13.25 ± 1.7735, or 11.48 to 15.02 cm.

A larger sample uses a smaller t value and also a smaller standard error. A summary of 36 packages with mean 68.4 in and s 2.5 in gives 2.5 ÷ 6 = 0.4167, a t value of 2.0301 at 35 degrees of freedom, and a margin of 0.8459. The interval runs from 67.55 to 69.25 in.

Read the first row and you see why the t distribution matters. With 5 degrees of freedom the 95 percent value is 2.5706, about 31 percent wider than the normal value of 1.96. With 120 degrees of freedom the gap is 1 percent. The NIST/SEMATECH handbook section on confidence limits for the mean states the same formula and adds the caution that matters most: a 95 percent interval is not a 95 percent chance that this particular interval contains the true mean. The confidence belongs to the method, not to the finished interval.

Proportions, polls and margins of error

A share has its own standard error. If a poll of n people finds a share p, the standard error is the square root of p times 1 minus p divided by n. The formula looks different but the logic is identical: a proportion is an average of zeros and ones, and its spread depends on how close the share sits to 50 percent.

Take a poll of 600 adults where 52 percent answer yes. The standard deviation of a single answer is 0.4996, and 0.4996 ÷ 24.4949 = 0.0204, which is 2.04 percentage points. The 95 percent margin is 1.96 × 2.0396 = 3.9976 points, so the interval runs from 48.0 to 56.0 percent. News reports call that margin of error, and it is exactly a critical value times a standard error. The OpenStax introduction to statistics reaches the same formula from the central limit theorem.

Two cautions keep the number honest. The normal approximation needs at least about 10 expected successes and 10 expected failures, which is why this page flags small counts instead of printing a confident interval. And a poll of a volunteer panel is not a random sample of 600 people, so the real error is larger than the formula says. Design effects of 1.5 to 2 are common in phone and online panels, which multiplies the margin by 1.22 to 1.41.

Comparing two means

When two groups are compared, the estimate is the difference between the means and its standard error combines both groups: the square root of s1 squared over n1 plus s2 squared over n2. The errors add in quadrature because the two samples are independent.

A test group of 25 students averages 74.0 with s 3.2, and a control group of 20 averages 70.5 with s 4.0. The two variance terms are 0.4096 and 0.8, so the standard error of the difference is the square root of 1.2096, which is 1.0998 points. The difference itself is 3.5 points. Degrees of freedom from the Welch formula come to 35.97, the t value is 2.0281, and the margin is 2.2304, so the interval for the gap runs from 1.27 to 5.73 points.

Because the interval stays above zero, the data are consistent with a real gain in the test group. Had the interval spanned zero, the honest reading would be that the study cannot separate the two groups. One habit to drop: comparing two confidence intervals by eye and calling overlap a lack of significance. Two intervals can overlap while the difference between the means is still clearly positive, because the standard error of the difference is smaller than either individual error.

Finite population correction

All of the above assumes sampling from a population so large that removing a few items changes nothing. When the sample is a large slice of a small population, the standard error shrinks by the square root of N minus n over N minus 1, where N is the population size and n is the sample size.

A sample of 25 invoices out of 500 has s 3.2 and a plain standard error of 0.64. The correction factor is the square root of 475 ÷ 499, which is 0.9757, so the corrected standard error is 0.624. The effect is small here because the sample is 5 percent of the population. At a sample of 250 out of 500 the factor becomes 0.708 and the standard error drops by nearly 30 percent.

The usual rule of thumb is to apply the correction when the sample exceeds 5 percent of the population. Below that the factor is above 0.975 and rarely changes a decision.

Sample size, rounding and common mistakes

Because the standard error falls with the square root of n, cutting it in half costs four times the data. The seedling study has a standard error of 0.75 cm at 8 plants. At 32 plants the same s gives 2.1213 ÷ 5.6569 = 0.375 cm, and reaching 0.25 cm would take 72 plants, since 2.1213 ÷ 8.4853 = 0.25.

Round the standard error to two significant digits and keep the interval endpoints at the precision of the original data. A standard error of 0.1832 on a mean measured to one decimal belongs in print as 0.18, and the interval as 500.1 to 500.5 rather than 500.117 and 500.483.

Four mistakes account for most wrong answers. Reporting the standard deviation and calling it the standard error overstates precision by a factor of the square root of n. Using 1.96 with 8 observations understates the margin, since the correct t value is 2.3646. Dividing the sum of squared deviations by n instead of n minus 1 shrinks s, and the bias is worst in small samples. Treating the standard error as a range for individual values is the fourth: the interval around a mean says nothing about where the next seedling will land.

One more check is worth the ten seconds. If the standard error is more than a third of the standard deviation, the sample size is below 10, and a normal-based interval is a rough guide at best.