Sampling and estimation calculators
Tools for working from a sample to the population it came from: how many people to survey, how precise an estimate is, and how a sample mean or proportion behaves from one sample to the next. The last group covers process control limits and data-quality checks.
Which calculator do I need?
| You have or want | Use |
|---|---|
| How many responses you need for a target margin of error | Sample size calculator |
| The ± figure for a poll result or a sample mean you already have | Margin of error calculator |
| A range that likely contains the population mean | Confidence interval calculator |
| A range for the population standard deviation or variance | Confidence interval for SD |
| The standard error of the mean from raw data, or the SD behind a published SE | Standard error calculator |
| The chance that a sample mean lands in a range, given μ, σ and n | Central limit theorem calculator |
| A binomial probability approximated with the normal curve | Normal approximation calculator |
| Control chart limits for a production process | Upper control limit calculator |
Plan a sample
Decide the sample size, or find the precision a given sample size buys you.
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Sample size calculator
Sample size for a survey proportion or a mean from the margin of error and confidence level, with finite population correction.
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Margin of error calculator
Margin of error for a mean or a survey proportion at 90%, 95% or 99% confidence, with the steps.
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Sampling error calculator
Margin of sampling error for a survey proportion or a mean at any confidence level.
Estimate a population value
Point estimates, standard errors and confidence intervals for a mean, a proportion or a standard deviation.
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Point estimate calculator
MLE, Laplace, Jeffrey and Wilson estimates of a proportion, the best one picked, or the estimate from a CI.
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Confidence interval calculator
Confidence interval for a mean from x̄, s and n, using t*, with the margin of error and the working.
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Confidence interval for standard deviation
Confidence interval for a population SD and variance from a sample SD, using the chi-square distribution.
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Standard error calculator
Standard error of the mean from raw data, or the standard deviation recovered from a published SE.
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Mean and standard error calculator
The mean and its standard error together, ready to report as mean ± SEM or plot as error bars.
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Standard deviation of the sample mean calculator
σ/√n from a standard deviation or raw data, with the finite population correction.
Sampling distributions
How x̄ and p̂ vary from sample to sample, and when the normal curve can stand in for the binomial.
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Central limit theorem calculator
Mean and standard error of x̄ from μ, σ and n, and the probability that the sample mean falls in a range.
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Normal probability calculator for sampling distributions
Probabilities for a sample mean or sample proportion, with the finite population correction.
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Sampling distribution of the sample proportion calculator
Mean and standard error of p̂, the np ≥ 10 check, and probabilities for p̂.
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Normal approximation calculator
Normal approximation to the binomial or Poisson, with and without the continuity correction, against the exact value.
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Continuity correction calculator
Turn P(X ≤ k), P(X = k) and the rest into the corrected normal bound, and compare with the exact answer.
Process control, forecasting and data checks
Keep a process within limits, project a growth rate forward, and test whether reported figures look natural.
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Upper control limit calculator
UCL and LCL for X̄-R, X̄-s, I-MR, p and c charts from data or summary values, with the chart.
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Cpk & process capability
Cp, Cpk, Cpu and Cpl from a process mean, SD and spec limits, with expected ppm out of spec.
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Exponential growth prediction calculator
Predict x(t) = x₀(1 + r)^t or solve for the start value, rate or time, with the doubling time.
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Benford's law calculator
First-digit distribution against Benford, with the chi-square test and Nigrini’s MAD conformity.
Standard deviation, standard error and margin of error
These three are easy to confuse because each is built from the one before. The standard deviation describes how spread out individual values are. The standard error describes how much the sample mean would vary if you repeated the study: SE = s / √n, so it shrinks as the sample grows while the SD does not. The margin of error is the standard error multiplied by a critical value (1.96 for 95% with a large sample), and a confidence interval is the estimate plus or minus that margin.
A worked comparison
A poll finds 50% support. With 1,000 respondents the 95% margin of error is 1.96 × √(0.5 × 0.5 / 1000) = 0.031, or ±3.1 percentage points. With 400 respondents it is 1.96 × √(0.25 / 400) = ±4.9 points. Going the other way, the sample size calculator shows that ±3 points needs 1.96² × 0.25 / 0.03² = 1,067.1, rounded up to 1,068 people. Precision improves with the square root of n: halving the margin of error takes four times the sample.
Where the normal curve comes from
The central limit theorem says the mean of a large enough sample is close to normally distributed, whatever the shape of the population, with standard deviation σ / √n. That is why confidence intervals and z-tests work on skewed data. The same idea lets you treat a binomial count as normal once np and n(1 − p) are both at least 10; the normal approximation and continuity correction calculators show how close the approximation gets to the exact answer.
Common mix-ups
- Reporting SE as if it were SD. Error bars labelled ± SE look much tighter than ± SD. To recover the SD, multiply by √n.
- Using the 1.96 critical value for small samples. For a mean with n below about 30, the t critical value is larger; the confidence interval calculator uses t for this reason.
- Forgetting the finite population correction. When the sample is more than about 5% of the population, the correction noticeably shrinks the standard error and the required sample size.
Guides to read alongside
Common questions
Does a larger population need a larger sample?
Barely, once the population is large. A survey of 1,068 people gives about ±3 points at 95% confidence whether the population is 100,000 or 100 million. Population size only matters when the sample is a sizeable share of it, which the finite population correction handles.
What does a 95% confidence interval mean?
If you repeated the sampling many times and built an interval each time, about 95% of those intervals would contain the true population value. It does not mean there is a 95% probability that this particular interval contains it; the interval either does or does not.
Why does the sample proportion need np ≥ 10?
The normal approximation to the sampling distribution of p̂ breaks down when successes or failures are rare, because the true distribution is then lopsided. Requiring at least 10 expected successes and 10 expected failures keeps it close enough to symmetric for normal-based intervals and tests.