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Cochran's formula calculator

The textbook survey sample size formula, with the finite population correction and every step written out so you can copy it into your methodology chapter.

Confidence level
0.5 = maximum variability
0.05 = ±5%
323respondents
Step 1 n₀ = z² · p · q / e² = 1.96² × 0.5 × 0.5 / 0.05² = 384.15 Step 2 n = n₀ / (1 + (n₀ − 1) / N) = 384.15 / (1 + 383.15 / 2,000) = 322.39 → 323

Using Cochran's (1977) formula with 95% confidence, p = 0.5 and e = 0.05, corrected for a population of 2,000, the required sample size is 323.

Cochran vs Slovin, side by side

Move p and the confidence level above. Cochran's answer changes because the formula knows about variability and confidence. Slovin's does not.

Same population, two formulas

Cochran (your settings)
323
Slovin, N / (1 + Ne²)
334

They match only at 95% confidence and p = 0.5 (up to rounding). Set p to 0.1 and Cochran drops sharply, because a rare trait is easier to pin down to ±5 points. Set 99% and Cochran rises. Slovin stays put either way, which is why methodologists such as Tejada & Punzalan (2012) advise against it.

What each symbol means

  • z: critical value of the standard normal distribution for your confidence level (1.645 for 90%, 1.960 for 95%, 2.576 for 99%).
  • p: the proportion you expect to have the attribute. q = 1 − p.
  • e: desired margin of error as a decimal.
  • N: population size, used only in the finite population correction.

Worked example

A university wants to estimate the share of its 4,500 postgraduate students who work part-time, to within ±5 points at 95% confidence, with no prior estimate. n₀ = 1.96² × 0.5 × 0.5 / 0.05² = 384.16. Corrected: 384.16 / (1 + 383.16 / 4,500) = 354.0, so survey 354 students. At a 40% response rate, invite 886.

Cochran's formula for continuous data

For estimating a mean, the equivalent is n₀ = z²σ²/e², where σ is the population standard deviation and e the margin in the original units. The sample size calculator handles both.

Frequently asked questions

What is Cochran's formula?

Cochran's formula, n₀ = z²·p·q / e², gives the sample size needed to estimate a population proportion with margin of error e at a confidence level whose critical value is z. It comes from William G. Cochran's Sampling Techniques (1977).

What is Cochran's formula for a finite population?

Apply the correction n = n₀ / (1 + (n₀ − 1)/N), where N is the population size. For n₀ = 384.16 and N = 2,000 this gives 322.

Why does Cochran's formula give 385?

With 95% confidence (z = 1.96), p = 0.5 and e = 0.05: 1.96² × 0.25 / 0.0025 = 384.16, which rounds up to 385. That is why 385 appears in so many theses.

Cochran or Slovin: which should I use?

Cochran, because it lets you state your confidence level and the expected variability. Slovin's formula silently assumes 95% confidence and p = 0.5 and is not found in standard sampling texts.