About the Standard Error Calculator
When you calculate the mean of a sample, you hope it is close to the true mean of the whole population — but a different sample would give a slightly different answer. The standard error measures how much sample means typically vary from one sample to the next. It tells you how precise an estimate is, and it is the building block of confidence intervals, margins of error and most statistical tests.
This standard error calculator finds the standard error of the mean from a data list or from a standard deviation and sample size, and the standard error of a proportion from a percentage and sample size. It also gives the approximate 95% margin and, where possible, a rough 95% interval.
How to Use the Standard Error Calculator
Choose From data, From SD and n or For a proportion.
For data, enter your numbers; the calculator uses the sample standard deviation.
For summary figures, enter the standard deviation and the sample size.
For a proportion, enter the percentage in the sample and the sample size.
The Formulas
standard error of the mean SE = s ÷ √n
standard error of a proportion SE = √( p(1 − p) ÷ n )
approximate 95% margin 1.96 × SEHere s is the sample standard deviation, n the sample size and p the sample proportion as a decimal.
Step-by-Step Example
A standard deviation of 12 from a sample of 36.
√36 = 6
SE = 12 ÷ 6 = 2
95% margin ≈ 1.96 × 2 = 3.92The sample mean is likely to be within about ±3.92 of the population mean.
The data 2, 4, 4, 4, 5, 5, 7, 9.
Mean 5, sample SD 2.13809, n = 8
SE = 2.13809 ÷ √8 = 2.13809 ÷ 2.82843 = 0.755929
Rough 95% interval: 5 ± 1.48 → 3.52 to 6.48A proportion of 40% in a sample of 400.
SE = √(0.4 × 0.6 ÷ 400) = √0.0006 = 0.0245 = 2.45 points
95% margin ≈ ±4.8 points → about 35.2% to 44.8%Standard Deviation Versus Standard Error
These are easily confused. The standard deviation describes the spread of the individual values: how different one person, part or measurement is from another. The standard error describes the precision of an average: how much the sample mean would wobble if you repeated the whole study. Collecting more data does not make the standard deviation smaller — people stay just as varied — but it does shrink the standard error, because an average of many values is steadier than an average of a few.
Research papers often report the mean ± SE, which makes results look more precise than ± SD. When reading a paper, check which one is being shown.
The Square Root Law
Because the standard error divides by √n, precision improves only with the square root of the sample size. Doubling the sample reduces the standard error by about 29 percent; quadrupling it halves the error; and a hundred times as much data is needed to cut it to a tenth. This is why surveys of about 1,000 people are so common: the margin of error for a proportion is then about ±3 percentage points, and going much lower becomes expensive quickly.
From Standard Error to Confidence Interval
For large samples, about 95 percent of sample means fall within 1.96 standard errors of the true mean, so the interval mean ± 1.96 × SE is an approximate 95 percent confidence interval. For small samples, the t distribution gives a slightly wider multiplier — about 2.36 for a sample of 8, for example — so the rough interval shown here is a quick guide rather than an exact result. Use a confidence interval calculator for the precise figure.
Standard Error of a Proportion
Opinion polls, election surveys and A/B tests report proportions: the percentage who agree, vote for a party or click a button. The standard error is largest when the proportion is near 50 percent and smaller near 0 or 100 percent. For a sample of 400 it is 2.5 points at 50 percent but only 1.1 points at 5 percent. The formula assumes a simple random sample; real surveys with weighting or clustering usually have larger errors.
Standard Error of Other Statistics
The idea extends beyond means and proportions. Every estimate calculated from a sample — a difference between two means, a regression slope, a correlation — has its own standard error describing how much it would vary between samples. For the difference between two independent means, the standard errors combine as the square root of the sum of their squares: errors of 3 and 4 give a combined error of 5.
Understanding Your Result
The headline is the standard error.
The margin at 95% line gives 1.96 × SE, the approximate margin of error.
The rough 95% interval line gives the estimate plus and minus that margin, when the mean or proportion is known.
The worth knowing line explains how the standard error changes with sample size.
When Should You Use This Calculator?
Use it to judge how precise a sample mean or survey percentage is.
Use it to add error bars to charts of averages.
Use it as a step toward confidence intervals, t-tests and z-tests.
Use it to see how much a larger sample would improve precision.
Common Mistakes
Reporting the SE as the spread of the data. The SE describes the mean's precision.
Using the population SD formula. The standard error uses the sample SD.
Expecting precision to grow in step with sample size. It grows with √n.
Using 1.96 for very small samples. Use a t value instead.
Ignoring survey design. Weighted or clustered samples have larger errors than the simple formula suggests.