Standard Error Calculator
How Much Would the Sample Mean Wobble If You Repeated It?
Standard deviation describes spread within a single data set; standard error describes something different — how much the sample mean itself would vary if you drew a new sample of the same size over and over. It's the bridge between "here's what I measured" and "here's how much I'd trust that measurement," and it's the quantity behind every confidence interval and margin of error you'll see in a poll or a clinical study.
The Formula
s is the sample standard deviation and n is the sample size. Standard error shrinks as the sample grows — more data gives a more stable estimate of the true mean, but the relationship isn't linear.
Diminishing Returns From Larger Samples
Because the sample size sits under a square root, doubling your sample doesn't halve the standard error — you need four times the data to cut it in half:
| Sample size (n) | Standard error |
|---|---|
| 10 | 3.16 |
| 30 | 1.83 |
| 100 | 1.00 |
| 500 | 0.45 |
| 1,000 | 0.32 |
Going from 100 to 1,000 observations only cuts the standard error by about two-thirds, not by a factor of ten — a useful reality check before committing to a much larger, more expensive study.
Where This Matters
- Polling and surveys — the "margin of error" reported alongside a poll result is built directly from the standard error.
- Clinical trials — determines how precisely a treatment effect has been estimated.
- Quality assurance — judging whether a shift in a measured average reflects a real process change or just sampling noise.
- A/B testing — comparing two standard errors is a step toward deciding whether an observed difference is statistically meaningful.
How to Use This Calculator
- Choose From Sample Data if you have the raw values, or From Std Dev & Sample Size if you already know those two numbers.
- Enter your comma-separated values, or enter the standard deviation and sample size directly.
- Select Calculate to get the standard error.
Related Calculations
Standard error feeds directly into the Confidence Interval Calculator. To compute the standard deviation it's based on, use the Standard Deviation Calculator.