Sampling Error Calculator
Calculate sampling error (standard error and margin of error) for proportions and means. Includes finite population correction, error decomposition, and sample size comparison.
Standard Error Calculator
Calculate standard error for means, proportions, differences, or raw data. Includes margin of error, finite population correction, and SE vs sample size comparison.
Standard Error Calculator
Standard Error⧉
2.190890
SE of the estimator
Margin of Error⧉
±4.2950
At 95% confidence (z* = 1.960)
SE (no FPC)⧉
2.190890
Standard error before FPC
FPC Factor⧉
Not applied
Infinite population
Std Deviation⧉
12.0000
Sample standard deviation
SE by Sample Size
| n | SE | MOE (95%) |
|---|---|---|
| 5 | 5.366563 | ±10.5206 |
| 10 | 3.794733 | ±7.4392 |
| 25 | 2.400000 | ±4.7049 |
| 50 | 1.697056 | ±3.3269 |
| 100 | 1.200000 | ±2.3525 |
| 250 | 0.758947 | ±1.4878 |
| 500 | 0.536656 | ±1.0521 |
| 1,000 | 0.379473 | ±0.7439 |
| 5,000 | 0.169706 | ±0.3327 |
Visual: SE Decreases with √n
n=5
5.36656
n=10
3.79473
n=25
2.40000
n=50
1.69706
n=100
1.20000
n=250
0.75895
n=500
0.53666
n=1000
0.37947
n=5000
0.16971
SE ∝ 1/√n — quadrupling n halves the SE
Planning notes, formulas, and examples
About the Standard Error Calculator
The standard error (SE) measures the precision of a sample statistic as an estimate of a population value. It describes how much a mean, proportion, or difference would vary from sample to sample if you repeated the same study design many times.
This calculator covers several common cases: the SE of a mean, a proportion, a difference between means, a difference between proportions, and SE derived directly from raw data. It also reports the corresponding margin of error and can apply a finite population correction when the sample is a large share of the full population.
That makes it useful for confidence intervals, hypothesis tests, survey interpretation, and planning how much sample size you need to reach a given level of precision.
When This Page Helps
Standard error sits underneath most introductory inference, but the formula changes with the estimator and the study design. It is easy to mix up standard deviation, standard error, and margin of error if you are moving quickly between notes, software, and hand calculations.
Keeping the estimator choice, confidence level, and finite-population option together helps make the result easier to interpret and easier to explain in a report or study plan.
How to Use the Inputs
- Select the type of standard error to compute.
- Enter the required statistics (SD and n for means, p̂ and n for proportions, etc.).
- Or enter raw data directly for automatic computation.
- Set the confidence level for the margin of error.
- Optionally enter population size for finite population correction.
- Review the SE, margin of error, and sample size comparison.
- Use the visual chart to understand the √n relationship.
Formula used
SE of Mean: SE = s / √n
SE of Proportion: SE = √(p̂(1−p̂)/n)
SE of Difference of Means: SE = √(s₁²/n₁ + s₂²/n₂)
SE of Difference of Proportions: SE = √(p̂₁(1−p̂₁)/n₁ + p̂₂(1−p̂₂)/n₂)
Margin of Error: MOE = z* × SE
With FPC: SE_adj = SE × √((N−n)/(N−1))
Key relationship: SE ∝ 1/√nExample Calculation
Result: SE = 2.1909, MOE = ±4.29
With s = 12 and n = 30, the standard error of the mean is 12/√30 = 2.19. At 95% confidence (z* = 1.96), the margin of error is ±4.29. This means the sample mean is expected to be within about 4.3 units of the true population mean.
Tips & Best Practices
- Standard error ≠ standard deviation. SD measures data spread; SE measures precision of the estimator. SE is always smaller than SD by a factor of √n.
- SE decreases with √n: to cut SE in half, you need 4× the sample size.
- For proportions near 0.5, SE is maximized. For proportions near 0 or 1, SE is small (but the normal approximation may fail).
- The SE of a difference is always larger than the SE of either individual estimate due to combined uncertainties.
- Finite population correction reduces SE when you sample > 5% of the population.
- When reporting results, always include either the SE or the confidence interval — a point estimate alone is incomplete.
Standard Error as the Foundation of Inference
Nearly every inferential procedure in statistics uses the standard error. Confidence intervals are point estimate ± z* × SE. Test statistics are (estimate − null) / SE. Power analysis uses SE to determine the sample size needed to detect an effect. Understanding SE is understanding the precision of your data.
The 1/√n Law and Its Implications
The inverse square root relationship between SE and n has profound practical implications. Getting from ±10% precision to ±5% requires 4× the data. Getting to ±1% requires 100× the starting sample. This diminishing returns curve is why most studies settle for "good enough" precision rather than pursuing perfection — the cost grows quadratically with precision.
SE for Complex Estimators
Beyond simple means and proportions, standard errors exist for regression coefficients, correlation coefficients, percentiles, and virtually any sample statistic. When analytical formulas aren't available, bootstrap methods estimate SE by resampling from the data. The concept extends to any statistic that varies across samples.
Sources & Methodology
Last updated:
Frequently Asked Questions
Standard deviation (SD) measures variability of individual observations within a sample. Standard error (SE) measures variability of the sample statistic (like the mean) across different samples. SE = SD/√n, so it's always smaller than SD for n > 1.
With more observations, extreme values average out and the sample statistic becomes more stable. The Central Limit Theorem guarantees this: the variance of the sample mean is σ²/n, giving SE = σ/√n.
Confidence intervals are generally preferred for scientific communication because they're more intuitive (the range of plausible values). SE is useful for technical audiences and as an input to meta-analyses. Some journals require one or the other.
When comparing two independent estimates, the SE of their difference combines both uncertainties: SE_diff = √(SE₁² + SE₂²). This applies to both mean differences and proportion differences.
Compute the sample standard deviation (s), then divide by √n. The raw data mode in this calculator does this automatically, computing s from your data and then SE = s/√n.
Meta-analyses weight studies by the inverse of their squared SE. Studies with smaller SE (larger samples, less variability) get more weight because they provide more precise estimates of the effect.
More in this topic
Margin of Error Calculator
Calculate margin of error from sample data or find required sample size for a desired MOE. Supports proportions and means with finite population correction.
Sampling Distribution of Sample Proportion Calculator
Explore the sampling distribution of p̂. Calculate mean, standard error, probabilities, and quantiles. Visualize the bell curve with normal approximation conditions.