Are Slovin's formula and Taro Yamane's formula the same?

Yes. Both use n = N ÷ (1 + N·e²). Yamane (1967) popularised the expression; many theses cite it under either name. Cite the source your adviser or department expects.

When should I use Cochran's formula instead of Slovin's?

Cochran's formula lets you set the confidence level and the expected proportion p, so it is more defensible when a reviewer asks how confidence was chosen. Slovin's formula assumes p = 0.5 and roughly 95% confidence implicitly.

What margin of error should I use?

5% (0.05) is the most common choice in survey research. Smaller margins such as 3% need much larger samples; larger margins such as 10% are sometimes accepted for pilot or exploratory studies.

Should I round the sample size up or down?

Always round up to the next whole respondent, because rounding down produces a sample slightly smaller than the formula requires.

SAMPLE SIZE CALCULATOR

Sample size calculator with Slovin, Yamane, and Cochran formulas.

Find the minimum number of respondents for a survey or thesis. Every step is shown so you can copy the working into your methodology chapter.

INPUTS

Slovin's formula

STEP-BY-STEP

Use when the population size is known and you only need to set the margin of error.

POPULATION SIZE (N)MARGIN OF ERROR (%)EXPECTED RESPONSE RATE (%)

How to calculate sample size for a thesis or survey

Slovin's formula (Taro Yamane)

n = N ÷ (1 + N·e²)

N is the population size and e is the margin of error as a decimal. For N = 1,200 and e = 0.05: n = 1,200 ÷ (1 + 1,200 × 0.0025) = 1,200 ÷ 4 = 300 respondents.

The formula is short and easy to defend in a methods chapter, which is why it is common in undergraduate and master's research. It quietly assumes maximum variability (p = 0.5).

Cochran's formula with finite population correction

n₀ = z²·p(1 − p) ÷ e² → n = n₀ ÷ (1 + (n₀ − 1) ÷ N)

At 95% confidence (z = 1.96), p = 0.5, and e = 0.05, n₀ = 384.16. For a population of 1,000 the correction gives 277.7, so you need 278 respondents.

Use Cochran when a reviewer may ask about the confidence level, or when a pilot study suggests p is far from 0.5.

Choosing the margin of error

5% — the default for most survey and thesis research
3% — national or high-stakes surveys; needs a much larger sample
10% — sometimes accepted for pilot or exploratory studies

How to write it in your methodology

State the population, the formula, the margin of error (and confidence level for Cochran), and the final rounded sample. The calculator generates this sentence for you; cite Yamane (1967) or Cochran (1977) as your department prefers.

Always round up. If you expect non-response, divide by the expected response rate to decide how many people to invite.

References

Yamane, T. (1967). Statistics: An Introductory Analysis (2nd ed.). Harper & Row.
Cochran, W. G. (1977). Sampling Techniques (3rd ed.). John Wiley & Sons.
Israel, G. D. (1992). Determining Sample Size. University of Florida IFAS Extension, PEOD6.

Sample Size Calculator

Find the minimum sample size for a survey or thesis using Slovin/Yamane or Cochran's formula, with confidence level, margin of error, population size, response-rate adjustment, and a copy-ready methods sentence.

Compute minimum respondents for a thesis survey from a known population

Compare Slovin/Yamane with Cochran's formula and finite population correction

Inflate the sample for expected non-response

KEY FUNCTIONS

Slovin / Taro Yamane formulaCochran formula with finite population correction90%, 95%, and 99% confidence levelsResponse-rate adjustmentStep-by-step working and methods sentence

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