Measure and test
Standard error calculator
Get the standard error of a mean from the standard deviation and sample size or from raw data, of a proportion such as a conversion rate, and of the difference between two rates. Each result shows the steps and the confidence interval it implies.
Four ways in
From SD and n, from raw data, for a proportion, or for the difference in an A/B test.
Steps shown
Every result shows the formula with your numbers filled in.
Interval included
The confidence interval each standard error implies, with its t or z value.
Standard error of the mean from SD and n
Have a standard deviation and a sample size? This is the quickest route. Add the mean for the confidence interval. Free, no sign-up, and nothing you type leaves your browser.
Standard error of the mean
SE = 1.225
SE = s ÷ √n = 24.5 ÷ √400 = 24.5 ÷ 20 = 1.225.
95% confidence interval for the mean: 68.200 ± 2.408, from 65.792 to 70.608.
That uses t = 1.9659 with 399 degrees of freedom: margin of error = t × SE.
With 4 times the sample (1,600), the standard error would halve to 0.6125.
Formula
SE = s / sqrt(n)
interval = mean +/- t * SE t with n - 1 degrees of freedomShort answer
A standard error calculator measures how precisely a sample estimates the true value. For a mean, divide the sample standard deviation by the square root of the sample size: SE = s ÷ √n. For a proportion, SE = √(p(1 - p) ÷ n). Multiply the standard error by about 1.96 for a 95% margin of error.
- Standard error of the mean: SE = s ÷ √n, with s the sample standard deviation calculated with n - 1.
- Standard error of a proportion, such as a conversion rate: SE = √(p(1 - p) ÷ n).
- The standard deviation describes the spread of your data; the standard error describes how much the mean would move from sample to sample.
- Quadrupling the sample size halves the standard error, because it shrinks with the square root of n.
- A 95% confidence interval is the estimate ± about 2 standard errors: t × SE for a mean, 1.96 × SE for a proportion.
Standard error from raw data
Paste your values. The calculator finds the mean and the sample standard deviation, which divides by n - 1, then the standard error.
Standard error of the mean
SE = 2.302
n = 10 values, mean = 49.9.
Sample standard deviation s = 7.27935 (sum of squared deviations 476.9 ÷ (n - 1) = 52.9889, then the square root).
SE = s ÷ √n = 7.27935 ÷ √10 = 2.302.
95% confidence interval for the mean: 49.900 ± 5.207, from 44.693 to 55.107.
That uses t = 2.2622 with 9 degrees of freedom: margin of error = t × SE.
Formula
mean = sum of x / n
s = sqrt( sum of (x - mean)^2 / (n - 1) )
SE = s / sqrt(n)Standard error of a proportion
For a rate or a share: conversions out of visitors, yes answers out of responses.
Standard error of a proportion
SE = 0.003082
p = 250 ÷ 5,000 = 0.05 (5%).
SE = √(p(1 - p) ÷ n) = √(0.05 × 0.95 ÷ 5,000) = 0.003082, which is 0.308 percentage points.
95% confidence interval: 5% ± 0.604 points (z = 1.9600), from 4.396% to 5.604%.
Formula
p = successes / n
SE = sqrt( p * (1 - p) / n )
interval = p +/- z * SEStandard error of the difference between two rates
For an A/B test: the standard error of variant B's conversion rate minus control A's.
Standard error of the difference
SE = 0.003182
Rates: A 5%, B 5.7%. Difference B - A = +0.700 percentage points.
SE = √(SE_A² + SE_B²) = √(0.002179² + 0.002318²) = 0.003182, which is 0.318 percentage points.
95% confidence interval for the difference: +0.700 ± 0.624 points, from +0.076 to +1.324 points.
For a significance test, the pooled SE is 0.0031824 and z = difference ÷ pooled SE = 2.1996. The statistical significance calculator turns that into a p-value.
Formula
SE = sqrt( pA * (1 - pA) / nA + pB * (1 - pB) / nB )
interval = (pB - pA) +/- z * SE
pooled SE = sqrt( p * (1 - p) * (1/nA + 1/nB) ) p = all conversions / all visitorsWhat is the standard error formula?
The standard error formula for a mean is SE = s ÷ √n: the sample standard deviation divided by the square root of the sample size. A proportion uses SE = √(p(1 - p) ÷ n), because the spread of a yes or no outcome is set by the rate itself. The standard error of a difference adds the two squared standard errors under one square root.
| Estimate | Standard error | Where |
|---|---|---|
| A mean | s / √n | s is the sample standard deviation, n the sample size |
| A proportion | √(p(1 - p) / n) | p is successes ÷ n, as a decimal |
| Difference of two proportions | √(p₁(1 - p₁)/n₁ + p₂(1 - p₂)/n₂) | Two independent groups, such as A and B in a split test |
| Difference of two means | √(s₁²/n₁ + s₂²/n₂) | Two independent groups (the unpooled, Welch form) |
How to calculate standard error, step by step
- Count the observations to get n.
- Find the mean: add the values and divide by n.
- Subtract the mean from each value and square the result.
- Add the squares and divide by n - 1 to get the sample variance.
- Take the square root to get the sample standard deviation, s.
- Divide s by √n. That is the standard error of the mean.
If you already have the standard deviation, start at step 6. For a proportion, skip the deviations: put p and n into √(p(1 - p) ÷ n).
Standard error of the mean: a worked example
Ten sessions on a landing page lasted 42, 55, 38, 61, 47, 52, 44, 58, 49, 53 seconds. The numbers are made up to show the arithmetic, and they are the raw data calculator's default values.
| Value (x) | x - mean | (x - mean)² |
|---|---|---|
| 42 | -7.9 | 62.41 |
| 55 | 5.1 | 26.01 |
| 38 | -11.9 | 141.61 |
| 61 | 11.1 | 123.21 |
| 47 | -2.9 | 8.41 |
| 52 | 2.1 | 4.41 |
| 44 | -5.9 | 34.81 |
| 58 | 8.1 | 65.61 |
| 49 | -0.9 | 0.81 |
| 53 | 3.1 | 9.61 |
| Total | 0 | 476.9 |
- Mean: 499 ÷ 10 = 49.9.
- Sample variance: 476.9 ÷ (10 - 1) = 52.9889.
- Sample standard deviation: √52.9889 = 7.2793.
- Standard error: 7.2793 ÷ √10 = 7.2793 ÷ 3.1623 = 2.3019 seconds.
- 95% confidence interval: 49.9 ± 2.262 × 2.3019 = 49.9 ± 5.21, from 44.69 to 55.11 seconds. With 9 degrees of freedom, t is 2.262 rather than 1.96.
Dividing by n instead of n - 1 would give the population standard deviation, 6.9058. That is right only when the ten values are every session there is. For a sample that stands in for more sessions, use n - 1, as Excel's STDEV.S does.
How do you convert standard deviation to standard error?
To convert a standard deviation to a standard error, divide it by the square root of the sample size. An average order value of 68.20 from 400 orders with a standard deviation of 24.5 has SE = 24.5 ÷ √400 = 24.5 ÷ 20 = 1.225, in the same currency as the orders.
The 95% interval for that average is 68.20 ± 1.966 × 1.225 = 68.20 ± 2.41, from 65.79 to 70.61. The numbers are illustrations, the defaults of the first calculator, not a typical order value.
Why a bigger sample shrinks the standard error
The standard error falls with the square root of the sample size, so four times the data halves it. Keep the standard deviation at 24.5 and change only n:
| Sample size (n) | √n | Standard error |
|---|---|---|
| 25 | 5 | 4.900 |
| 100 | 10 | 2.450 |
| 400 | 20 | 1.225 |
| 1,600 | 40 | 0.6125 |
| 6,400 | 80 | 0.3063 |
Precision gets expensive: going from 400 to 6,400 observations, sixteen times the data, only cuts the standard error to a quarter. The standard deviation itself does not shrink with more data; it settles near the true spread.
Standard deviation vs standard error: what is the difference?
The standard deviation describes how far individual values sit from their mean. The standard error describes how far the sample mean is likely to sit from the true mean. Report the standard deviation to show how varied your data are, and the standard error, or better a confidence interval, to show how precisely you have measured the average.
| Standard deviation (SD) | Standard error (SE) | |
|---|---|---|
| Measures | Spread of the individual values | Precision of the sample mean |
| Formula | √(sum of (x - mean)² ÷ (n - 1)) | SD ÷ √n |
| As the sample grows | Settles near the true spread | Keeps shrinking toward 0 |
| Use it to | Describe your data | Build a confidence interval or a test |
| Example on this page | 7.2793 seconds | 2.3019 seconds |
A common mistake is to draw error bars with the standard error and read them as the spread of the data. Say which one your bars show.
How do you find the standard error of a proportion?
To find the standard error of a proportion, multiply the proportion by one minus itself, divide by the sample size and take the square root: SE = √(p(1 - p) ÷ n). A conversion rate is a proportion, so this is the standard error of a conversion rate.
- A page converts 250 of 5,000 visitors: p = 0.05 (5%).
- p(1 - p) ÷ n = 0.0000095.
- SE = √0.0000095 = 0.00308, or 0.31 percentage points.
- 95% interval: 5% ± 1.96 × 0.31 = 5% ± 0.60 points, from 4.40% to 5.60%.
The counts are illustrations, not a benchmark conversion rate. This formula, the Wald interval, gets unreliable with few conversions and fails at 0% or 100%, where it gives a standard error of 0. For those cases the confidence interval calculator gives a Wilson interval.
Standard error of the difference in an A/B test
The standard error of the difference between two conversion rates is the square root of the sum of their squared standard errors. Independent errors add as squares, not straight, so the difference is less precise than either rate on its own but less uncertain than the two standard errors added together.
- Control A converts 500 of 10,000 visitors (5%); variant B converts 570 of 10,000 (5.7%). The difference is 0.70 percentage points.
- SE of A = 0.00218; SE of B = 0.00232.
- SE of the difference = √(0.00218² + 0.00232²) = 0.00318, or 0.32 points.
- 95% interval for the difference: 0.70 ± 0.62 points, from 0.08 to 1.32 points.
A significance test uses a slightly different standard error: it pools both groups (5.35% overall) because it assumes, for the test, that the two rates are equal. Here the pooled SE is 0.00318 and z = difference ÷ pooled SE = 2.20. The statistical significance calculator runs that test and gives the p-value; the A/B test sample size calculator tells you how many visitors make the standard error small enough before you start. The numbers are made up to show the arithmetic.
The standard error of a difference between two means
For averages, such as order value in A and B, the same rule applies with SE = √(s₁²/n₁ + s₂²/n₂). With means of 68.20 and 71.40, standard deviations of 24.5 and 27.1, and 400 and 380 orders: SE = √(1.225² + 1.390²) = 1.853. The 95% interval for the difference of 3.20 is ± 1.963 × 1.853 = ± 3.64, from -0.44 to 6.84, with t on about 760 degrees of freedom (Welch's approximation). An interval that includes 0 means the data do not rule out no difference at all.
How do you get a confidence interval from the standard error?
To get a confidence interval from the standard error, multiply the standard error by the critical value for your confidence level and add the result to, and subtract it from, the estimate. That product is the margin of error. A proportion uses z from the normal distribution; a mean uses t with n - 1 degrees of freedom, which is larger for small samples.
| Confidence level | z | t, n = 10 | t, n = 30 |
|---|---|---|---|
| 90% | 1.645 | 1.833 | 1.699 |
| 95% | 1.960 | 2.262 | 2.045 |
| 99% | 2.576 | 3.250 | 2.756 |
Every result above shows its interval. For the interval on its own, with a Wilson interval for small counts, use the confidence interval calculator. For a survey, where the question is usually how many responses you need, the margin of error calculator works backward from the margin you want to a sample size.
How to calculate standard error in Excel or Google Sheets
To calculate standard error in Excel or Google Sheets, divide the sample standard deviation by the square root of the count: =STDEV.S(A2:A11)/SQRT(COUNT(A2:A11)). Neither program has a single standard error function. With the ten example values in A2:A11 the formula returns 2.3019.
| To get | Formula | Result |
|---|---|---|
| Standard error of the mean | =STDEV.S(A2:A11)/SQRT(COUNT(A2:A11)) | 2.3019 |
| Sample standard deviation | =STDEV.S(A2:A11) | 7.2793 |
| Margin of error for the mean, 95% | =CONFIDENCE.T(0.05,STDEV.S(A2:A11),COUNT(A2:A11)) | 5.21 |
| Standard error of a proportion | =SQRT(B1*(1-B1)/B2), rate in B1, sample size in B2 | 0.00308 for 5% of 5,000 |
Use STDEV.S, not STDEV.P: STDEV.P divides by n and is for a complete population. In Excel, the Descriptive Statistics tool in the Analysis ToolPak also reports a "Standard Error" line, which is the same standard error of the mean.
What is a good standard error?
A good standard error is one small enough that the confidence interval around your estimate is narrower than the smallest difference that would change your decision. There is no fixed threshold: a standard error of 0.3 percentage points is plenty for telling a 5% rate from a 10% one, and too wide for telling 5.0% from 5.3%.
- Compare it with the effect you care about. If the 95% margin, about 2 standard errors, is wider than that effect, the data cannot tell you the answer yet.
- Compare it with the estimate. SE ÷ mean shows the relative precision: an SE of 2 on a mean of 50 is 4% of the estimate.
- Grow the sample to shrink it. Four times the observations halves the standard error.
On a website, a precise number tells you how a page performs, not why. Convertica's free CRO audit scores the page itself on eight checks and gives you three fixes to test.
Related calculators
The standard error feeds intervals and tests. These tools take the next step.
See all free tools and how they fit together.
Standard error calculator FAQ
How does a standard error calculator work?
A standard error calculator divides the spread of your data by the square root of the sample size. From raw data it first finds the mean and the sample standard deviation (with n - 1), then SE = s ÷ √n. For the ten values in the example on this page, s = 7.2793 and SE = 7.2793 ÷ √10 = 2.3019.
What is the formula for standard error?
The standard error of the mean is SE = s ÷ √n, where s is the sample standard deviation and n the sample size. The standard error of a proportion is SE = √(p(1 - p) ÷ n). For the difference between two independent estimates, add the squared standard errors and take the square root: SE = √(SE1² + SE2²).
What is the difference between standard deviation and standard error?
The standard deviation measures how spread out the individual values are. The standard error measures how much the sample mean would vary if you repeated the sampling. The standard error equals the standard deviation divided by √n, so it is always smaller for samples of 2 or more and shrinks as the sample grows, while the standard deviation does not.
How do you convert standard deviation to standard error?
Divide the standard deviation by the square root of the sample size: SE = SD ÷ √n. A standard deviation of 24.5 from 400 observations gives SE = 24.5 ÷ 20 = 1.225. To go back, multiply: SD = SE × √n. Use the sample size of the group the mean came from.
Why does the sample standard deviation divide by n - 1?
Dividing by n - 1 instead of n corrects for measuring the spread around the sample mean rather than the unknown true mean, which makes the squared deviations come out slightly too small. This is Bessel's correction. It is what Excel's STDEV.S and this calculator use; STDEV.P divides by n and is for a complete population.
What is a good standard error?
There is no universal good value: a standard error is small or large only compared with the difference you need to detect. If a 95% interval (about 2 standard errors each side) is narrower than the smallest change that would alter your decision, the estimate is precise enough. If not, collect more data: four times the sample halves the standard error.
How do you calculate standard error in Excel?
Excel has no standard error function, so combine two: =STDEV.S(A2:A11)/SQRT(COUNT(A2:A11)) gives the standard error of the mean for the values in A2:A11. Google Sheets uses the same formula. For a proportion in B1 with the sample size in B2, use =SQRT(B1*(1-B1)/B2).
Is standard error the same as margin of error?
No. The margin of error is the standard error multiplied by a critical value for your confidence level. At 95% that is about 1.96 for a proportion, or t for a mean, so the margin of error is roughly twice the standard error. In the conversion rate example, SE = 0.31 points and the 95% margin of error is 0.60 points.
