Standard Error Calculator
Work out the standard error of the mean from a raw list of numbers, or from a sample standard deviation and sample size.
How to use this calculator
Pick Raw data if you have the individual values from your sample data. Pick Summary data if you already know the sample standard deviation and the sample size.
For raw data, type your numbers separated by commas and press Calculate. The tool shows the mean, the sum of squares, the sample standard deviation and the standard error. For summary data, enter the standard deviation and the sample size. Press Reset to clear the fields and start again.
It works like other statistics calculators: enter your numbers, press one button and read the result. The steps below show what happens behind the scenes, so you can check the answer by hand.
What is standard error?
The standard error (SE) tells you how far a sample mean is likely to sit from the true mean of the population. It measures the precision of your estimate.
A small standard error means your sample mean is a reliable estimate. A large one means there is more variability, so the sample mean could be further off.
Standard error is often called the standard error of the mean, or SEM. It is closely related to standard deviation, but the two answer different questions. Standard deviation describes the spread of the values inside one data set. Standard error describes how much the sample mean would change if you drew many samples from the same population.
Standard error formula
s is the sample standard deviation and n is the sample size.
SE = √( Σ(x − x̄)² / (n − 1) ) / √n
The raw data mode uses the sample standard deviation, which divides by n − 1. If you know the population standard deviation (σ) instead, use Summary data and enter σ as the standard deviation. The formula stays the same: SE = σ / √n.
How to calculate standard error step by step
2. Add the values and divide by n to get the mean.
3. Subtract the mean from each value and square each difference.
4. Add the squared differences to get the sum of squares.
5. Divide the sum of squares by n − 1, then take the square root. This is the sample standard deviation.
6. Divide the standard deviation by the square root of n. This is the standard error.
Worked examples
n = 6, so the mean is 72 / 6 = 12.
The squared differences are 64, 16, 49, 0, 64 and 121. Their sum is 314.
The sample standard deviation is √(314 / 5) ≈ 7.925.
The standard error is 7.925 / √6 ≈ 3.235.
The standard error is 36.78 / √49 = 36.78 / 7 ≈ 5.254.
How sample size affects standard error
The standard error shrinks as the sample grows, but only with the square root of n. To cut the standard error in half, you need four times as many data points. The table below uses a fixed standard deviation of 10.
| Sample size (n) | Standard error (s = 10) |
|---|---|
| 4 | 5.00 |
| 25 | 2.00 |
| 100 | 1.00 |
| 400 | 0.50 |
Using standard error to build a confidence interval
The standard error is the starting point for a confidence interval. A confidence interval gives a range that is likely to contain the true population mean. The general form is:
This calculator gives you the SE. You then multiply it by a critical value. For a large sample and 95% confidence, the critical value is about 1.96. For a small sample, use the t critical value for your degrees of freedom (n − 1).
Common mistakes
Forgetting the square root. The standard error is s divided by √n, not s divided by n.
Using the wrong standard deviation. Dividing by n instead of n − 1 gives the population formula. For sample data, use n − 1.
Mixing up SE and SD. Use the standard deviation to describe how spread out your data is. Use the standard error to describe how precise your mean is.
Entering variance instead of standard deviation. In Summary data mode, the field expects the standard deviation. If you have the variance, take its square root first.
Frequently asked questions
What is the difference between raw and summary data?
Raw data is the list of values you measured. Summary data skips the list and gives the sample standard deviation and the sample size directly.
Why does a bigger sample reduce the standard error?
A larger sample averages out random variation. Because the standard error divides by √n, more data points always produce a smaller standard error, as long as the standard deviation stays about the same.
Is standard error the same as standard deviation?
No. Standard deviation measures the spread within one sample. Standard error measures how much the sample mean would vary between samples. For any sample larger than one, the standard error is smaller than the standard deviation.
Can I use this for a population standard deviation?
Yes. Switch to Summary data and enter the population standard deviation (σ) and n. The result is σ / √n.
How many data points do I need?
The raw data mode needs at least two values, because the sample standard deviation divides by n − 1. In practice, more data points give a more reliable estimate.
What counts as a good standard error?
There is no universal cutoff. It depends on the scale of your data and how precise you need to be. A smaller SE relative to the mean means a more precise estimate.