Standard Error Calculator

Calculate the standard error (SE) of the mean or proportion and get a confidence interval automatically. Use it for sample size planning, research design, and survey analysis.

Calculation Mode
Confidence Level

Use sample SD if population SD is unknown.

Standard Error

±1.5

Standard error of the mean (SEM).

Z Value

1.96

Margin (95%)

±2.94

Lower Bound

97.06

Upper Bound

102.94

Confidence interval: (97.06, 102.94) at the 95% confidence level.

What This Calculator Does

This standard error calculator computes SE for both the sample mean and the sample proportion, then constructs a confidence interval at the chosen confidence level. Standard error quantifies sampling variability, so smaller SE means a more reliable estimate. The calculator is designed for researchers, students, analysts, and survey designers who need a quick, accurate SE without opening a statistics package.

Standard Error Formulas

For the Sample Mean

SE = σ / √n

  • σ: standard deviation (population or sample estimate)
  • n: sample size

For the Sample Proportion

SE = √(p(1 − p) / n)

  • p: sample proportion (decimal form, e.g. 0.5 not 50)
  • n: sample size

Confidence Interval

A confidence interval is estimate ± z × SE. The z critical value depends on the chosen confidence level: 1.645 for 90%, 1.96 for 95%, 2.576 for 99%. For small samples (n < 30) with unknown σ, use the t-distribution instead.

Worked Example (Mean)

Suppose a sample of n = 64 produces a mean of 100 with standard deviation 15. Then:

  • SE = 15 / √64 = 15 / 8 = 1.875
  • 95% margin = 1.96 × 1.875 ≈ 3.675
  • 95% CI = 100 ± 3.675 = (96.325, 103.675)

Worked Example (Proportion)

A poll of n = 1,000 voters finds 52% support for a candidate. Then:

  • SE = √(0.52 × 0.48 / 1000) = √0.0002496 ≈ 0.0158
  • 95% margin = 1.96 × 0.0158 ≈ 0.0310 (3.1 percentage points)
  • 95% CI for support = 52% ± 3.1% = (48.9%, 55.1%)

How to Interpret Standard Error

  • Precision: smaller SE = more precise estimate.
  • Reliability: a 95% CI has roughly 95% coverage probability in repeated sampling.
  • Comparisons: when comparing two groups, smaller SEs make differences easier to detect.
  • Sample size planning: solve n = (z × σ / target SE)² to find the required n.

Limitations

SE assumes simple random sampling. For clustered, stratified, or weighted samples, design effects (DEFF) inflate the variance and SE. The formulas here also assume approximate normality, which is reasonable for means when n ≥ 30 and for proportions when np and n(1−p) are at least 5. For non-normal or skewed data, use bootstrap or non-parametric methods.

Sources and References

  • Cochran, W. G. Sampling Techniques.
  • NIST/SEMATECH e-Handbook of Statistical Methods.
  • Lohr, S. Sampling: Design and Analysis.
  • Casella, G. & Berger, R. L. Statistical Inference.

Frequently Asked Questions

What is standard error?
Standard error (SE) measures how precisely a sample statistic estimates the true population parameter. It is the standard deviation of the sampling distribution of a statistic, most commonly the mean. A smaller standard error means the sample mean is more likely to be close to the true population mean, while a larger SE implies more sampling variability.
How is standard error of the mean calculated?
The standard error of the mean (SEM) equals the population standard deviation divided by the square root of the sample size: SE = σ / √n. When the population standard deviation is unknown, the sample standard deviation (s) is used as a substitute. SEM shrinks as sample size grows, but only with the square root of n — to halve SE you need to quadruple the sample size.
What is the standard error of a proportion?
For a sample proportion p, the standard error is SE = √(p(1 − p) / n). The variance of a Bernoulli variable is p(1 − p), and dividing by n gives the variance of the sample proportion. SE is largest when p = 0.5 and smallest when p approaches 0 or 1, which is why rare-event surveys need larger samples for the same precision.
What is the difference between standard error and standard deviation?
Standard deviation (SD) describes the variability of individual data points around the mean within a single sample. Standard error (SE) describes the variability of the sample mean across repeated samples. SE is always smaller than SD because averaging reduces variability. As sample size grows, SE shrinks toward zero, but SD stays roughly the same.
How does sample size affect standard error?
Standard error decreases as the square root of sample size grows. Doubling the sample size cuts SE by about 29% (1/√2). Quadrupling the sample size cuts SE in half. This non-linear relationship is why high-precision research needs very large samples, and why pilot studies are used to estimate SD before planning the full study.
Can I get a confidence interval from the standard error?
Yes. A confidence interval for the mean is x̄ ± z × SE, where z depends on the chosen confidence level (1.96 for 95%, 2.576 for 99%). For example, if the mean is 50 and SE is 2, a 95% confidence interval is roughly 46.08 to 53.92. This calculator returns both the SE and the corresponding confidence interval automatically.
What sample size do I need for a target SE?
Rearrange the formula: n = (z × σ / target SE)². For example, to achieve SE = 1 with σ = 10 at 95% confidence: n = (1.96 × 10 / 1)² ≈ 384. This is a common rule of thumb for surveys aiming for a ±3% margin at 95% confidence. Use the Margin of Error or Sample Size calculators for the full inverse calculation.
When should I use t-distribution instead of z?
Use the t-distribution when the population standard deviation is unknown and the sample size is small (typically n < 30). For larger samples the t-distribution converges to the z-distribution, so the z approximation is acceptable. This calculator uses z for simplicity; for small samples, replace 1.96 with the t critical value at n − 1 degrees of freedom.

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