Standard Error Calculator
Get the standard error of a mean — from the standard deviation and n, or straight from raw data — of a proportion, or of the difference between two independent means or two proportions. Every result shows the formula with your numbers substituted, and the page explains exactly how the standard error differs from the standard deviation.
Fill in every field above — sample sizes must be whole numbers of 2 or more, and proportions between 0 and 1.
How to Use the Standard Error Calculator
- 1. Choose the estimate. A mean, a proportion, or the difference between two independent groups.
- 2. Enter your numbers. For a mean, give the standard deviation and n, or paste the raw data and let the calculator find the sample SD. Proportions can be typed as 0.45, 45% or 54/120.
- 3. Read the standard error. For two groups you get both the unpooled and the pooled version, labelled with where each one is used.
- 4. Check the working. Each formula appears with your values substituted, ready to copy into a methods section or a homework answer.
Standard Deviation vs Standard Error
The two are easy to mix up because one is built from the other. The standard deviation describes your data: how far individual values typically sit from the mean. The standard error describes your estimate: how far the sample mean (or proportion) would typically land from the true population value if you repeated the study many times. For a mean, SE = s ÷ √n.
| Standard deviation (s) | Standard error (SE) | |
|---|---|---|
| Describes | The spread of individual values | The precision of an estimate such as the mean |
| As n grows | Settles near the population SD | Shrinks in proportion to 1 ÷ √n |
| Report it to | Describe your sample | Show uncertainty: error bars, confidence intervals, test statistics |
Standard Error Formulas
- Mean: SE = s ÷ √n, where s is the sample standard deviation (n − 1 in its denominator).
- Proportion: SE = √(p̂(1 − p̂) ÷ n), where p̂ is the sample proportion.
- Difference between two independent means: SE = √(s₁² ÷ n₁ + s₂² ÷ n₂) without assuming equal variances (Welch). If you do assume them equal, pool the SDs first — sₚ = √(((n₁ − 1)s₁² + (n₂ − 1)s₂²) ÷ (n₁ + n₂ − 2)) — and use SE = sₚ√(1 ÷ n₁ + 1 ÷ n₂), as Student's t-test does.
- Difference between two proportions: SE = √(p̂₁(1 − p̂₁) ÷ n₁ + p̂₂(1 − p̂₂) ÷ n₂) for a confidence interval. The z-test of equal proportions uses the pooled p̄ = (x₁ + x₂) ÷ (n₁ + n₂) instead: SE = √(p̄(1 − p̄)(1 ÷ n₁ + 1 ÷ n₂)).
Worked Examples
A sample of 36 students has a standard deviation of 12.4 points, so SE = 12.4 ÷ √36 = 12.4 ÷ 6 ≈ 2.07. Quadruple the sample to 144 and the SE halves to about 1.03 — the SD of the scores would stay roughly where it was. For a proportion, 54 "yes" answers out of 120 give p̂ = 0.45 and SE = √(0.45 × 0.55 ÷ 120) ≈ 0.0454, or about 4.5 percentage points.
Comparing two classes — a mean of 78.2 (SD 10.5, n = 40) against 72.6 (SD 12.1, n = 35) — the difference is 5.6 points with an unpooled SE of √(10.5² ÷ 40 + 12.1² ÷ 35) ≈ 2.63. The pooled SD is about 11.27 and the pooled SE about 2.61; they are close here because the two SDs and sample sizes are similar.
Common Mistakes
- Reporting the SE where the SD belongs. The SE is always smaller, which makes data look less variable than it is. Describe a sample with the SD; use the SE for the precision of an estimate.
- Unlabelled error bars. Bars of ±1 SE, ±1 SD and a 95% confidence interval look identical on a chart. APA style asks you to say in the figure note what the bars represent.
- Treating the SE as the margin of error. The margin of error is a critical value times the SE — about 1.96 × SE for a 95% interval in large samples.
- Trusting √(p̂(1 − p̂) ÷ n) at the extremes. With a small n or p̂ near 0 or 1 the formula understates the uncertainty, and an interval built from it can spill past 0% or 100%.
Frequently Asked Questions
- How do you calculate the standard error from the standard deviation?
- Divide the standard deviation by the square root of the sample size: SE = s ÷ √n. With s = 12.4 and n = 36, SE = 12.4 ÷ 6 ≈ 2.07. Use the sample standard deviation, the one with n − 1 in its denominator.
- Should I report the standard deviation or the standard error?
- Report the SD when you describe your sample ("M = 12.44, SD = 1.73") and the SE — or better, a confidence interval — when you show how precisely a mean or difference has been estimated. Many journals prefer confidence intervals to bare SEs, and whichever you use, say which it is.
- What is a good standard error?
- There is no universal cut-off, because the SE is in the units of the data. Judge it against the size of the effect you care about: an SE of 2 points is small when you are looking for a 15-point difference and large when you are looking for a 3-point one. A smaller SE always means a more precise estimate.
- Why does the standard error get smaller with a larger sample?
- Averages of many values vary less than single values, because random highs and lows cancel out. The SE falls with the square root of n, so precision is expensive: halving the SE takes four times the data.
- What is the difference between the standard error and the margin of error?
- The margin of error is the standard error multiplied by a critical value for your confidence level. For a 95% interval with a large sample, that is about 1.96 × SE; with a small sample, the t critical value for n − 1 degrees of freedom is a little larger.
- When should I use the pooled standard error?
- Use it when you assume the two populations have equal variances — Student's t-test does — or, for proportions, when testing the null hypothesis that the two proportions are equal. For confidence intervals, and whenever the variances may differ, the unpooled SE is the safer choice.