Standard Error Calculator
Find the standard error of the mean from your data, or from a standard deviation and a sample size. See how precise your sample mean is, with a clear step-by-step solution.
Standard Error Calculator
Choose how you want to calculate, enter your numbers and get your result instantly.
Step-by-Step Solution
What Is Standard Error?
The standard error (SE), more fully the standard error of the mean, tells you how much a sample mean is likely to vary from one sample to another. A small standard error means your sample mean is a precise estimate of the true mean of the whole group. A large standard error means it is less precise.
Here s is the standard deviation of your sample and n is the number of values in it.
To find the standard error, first calculate the standard deviation of your data. You can then use the standard error to build a confidence interval.
How to Calculate Standard Error
You can find the standard error by hand in three steps.
1. Find the standard deviation
Calculate the standard deviation of your data. Our Standard Deviation Calculator shows how, for both a sample and a population.
2. Count the values
Find n, the number of values in your sample, and take its square root.
3. Divide
Divide the standard deviation by the square root of n. The result is the standard error of the mean.
Standard Error Calculator Examples
Example 1: from a standard deviation and a sample size
A sample of 25 people has a standard deviation of 10.
SE = 10 ÷ √25 = 10 ÷ 5 = 2.
Example 2: a larger sample
Another sample has the same standard deviation of 10, but includes 100 people.
SE = 10 ÷ √100 = 10 ÷ 10 = 1. With four times as many values, the standard error is cut in half.
Example 3: from a data set
Find the standard error of the mean for this data, using the sample standard deviation:
The mean is 5, the sample standard deviation is about 2.1381 and n = 8. So SE = 2.1381 ÷ √8, which is about 0.7559.
Standard Error vs Standard Deviation
These two are often confused, but they describe different things.
- Standard deviation describes how spread out the individual values in your data are.
- Standard error describes how precisely your sample mean estimates the true mean.
The standard deviation does not systematically shrink as you collect more data. The standard error does, because it is divided by the square root of the sample size.
How Sample Size Affects Standard Error
Because n is in the denominator, a larger sample gives a smaller standard error. To cut the standard error in half, you need about four times as many values. This is why larger studies give more precise estimates of the mean.
The standard error is also used to build confidence intervals and to run tests on means, which makes it a basic building block in statistics.
More Standard Error Examples
These extra worked examples show how to find the standard error from a small data set, how it changes with sample size, how many values you need for a target precision, and how to compare two groups. Remember that SE = s ÷ √n.
Example 4: A small data set
Find the standard error of the mean for 6, 8, 10, 12, 14.
Answer: with the sample standard deviation, the standard error is about 1.4142.
The mean is (6 + 8 + 10 + 12 + 14) ÷ 5 = 10. The deviations are −4, −2, 0, 2 and 4, and the squared deviations are 16, 4, 0, 4 and 16, which add up to 40. The sample variance is 40 ÷ 4 = 10, so the sample standard deviation is √10 ≈ 3.1623. The standard error is 3.1623 ÷ √5 = √2 ≈ 1.4142.
If you used the population standard deviation instead, the variance would be 40 ÷ 5 = 8, the standard deviation would be √8 ≈ 2.8284, and the standard error would be about 1.2649. For the standard error of a sample mean, the sample standard deviation is the usual choice.
Example 5: Growing the sample
A measurement has a standard deviation of about 20. Find the standard error when n = 16, n = 64 and n = 400.
Answer: the standard error is 5, then 2.5, then 1.
For n = 16: SE = 20 ÷ √16 = 20 ÷ 4 = 5. For n = 64: SE = 20 ÷ √64 = 20 ÷ 8 = 2.5. For n = 400: SE = 20 ÷ √400 = 20 ÷ 20 = 1. Each time the sample size is multiplied by 4 (and then by about 6), the standard error drops, but the standard deviation of the data stays near 20. This shows the difference between the two: more data makes the mean more precise, but it does not make the individual values less spread out.
Example 6: How many values do you need?
The standard deviation is about 15. How large must the sample be for the standard error to be 1, and for it to be 0.5?
Answer: you need about 225 values for a standard error of 1, and about 900 values for 0.5.
Rearrange SE = s ÷ √n to n = (s ÷ SE)². For SE = 1: n = (15 ÷ 1)² = 225. For SE = 0.5: n = (15 ÷ 0.5)² = 30² = 900. Check: 15 ÷ √225 = 15 ÷ 15 = 1 and 15 ÷ √900 = 15 ÷ 30 = 0.5. Cutting the standard error in half needs four times as many values.
Example 7: Comparing the precision of two groups
Group A has a mean of 72, a standard deviation of 9 and n = 36. Group B has a mean of 70, a standard deviation of 6 and n = 9. Which group has the more precise mean?
Answer: Group A has the more precise mean, even though its values are more spread out.
Group A: SE = 9 ÷ √36 = 9 ÷ 6 = 1.5. Group B: SE = 6 ÷ √9 = 6 ÷ 3 = 2. Group A has the larger standard deviation, but its larger sample gives a smaller standard error. So the standard error, not the standard deviation, tells you how precisely each sample mean estimates the true mean.
Common mistakes when finding the standard error
A common mistake is using the standard deviation when you need the standard error, or the other way around. Another is dividing by n instead of √n, which makes the standard error far too small. Also be careful to take the square root of the sample size before dividing. Once you have the standard error, you can build an interval with our Confidence Interval Calculator.
Created by Sajid Ali, SmartStatCalc · Last updated: 7 November 2026
Found a mistake or have a suggestion? Contact us.
How to Use Our Standard Error Calculator
- Choose a calculation method: your own data set, or a standard deviation and sample size you already know.
- Enter your numbers in the boxes. For a data set, choose the type of standard deviation.
- Click the Calculate Standard Error button.
- Read the standard error, along with the mean, the standard deviation and n.
- Use the step-by-step solution to see how the answer was found.
Click Reset at any time to clear everything and start again.
Why Use SmartStatCalc?
SmartStatCalc is built to make everyday statistics quick and easy to understand. Instead of working out the standard deviation and the square root by hand, you can enter your data and see a clear result right away.
Who Can Use a Standard Error Calculator?
Students use the standard error in statistics and science classes. Researchers use it to report how precise their sample means are, and analysts use it when they compare averages from surveys, experiments or tests.
Frequently Asked Questions
What is the standard error in simple words?
It shows how far your sample mean is likely to be from the true mean of the whole group.
What is the formula for standard error?
The standard error of the mean is SE = s ÷ √n, where s is the standard deviation and n is the sample size.
What is the difference between standard error and standard deviation?
The standard deviation describes the spread of the values in your data. The standard error describes how precise the sample mean is.
Does a larger sample give a smaller standard error?
Yes. A larger sample size makes the square root of n bigger, so the standard error becomes smaller.
Should I use the sample or the population standard deviation?
When you want the standard error of a sample mean, the sample standard deviation is the usual choice. If you know the true population standard deviation, you can use that instead.
Can the standard error be zero?
It is zero only when the standard deviation is zero, which means all values in the data are the same.
What is a good standard error?
A smaller standard error means a more precise estimate. What counts as small depends on the size of your mean and on what you are measuring.
Is this Standard Error Calculator free?
Yes. SmartStatCalc's Standard Error Calculator is free to use online.
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SmartStatCalc. (2026). Standard Error Calculator. https://smartstatcalc.com/standard-error-calculator/