Cochran's Formula Calculator

How to use this calculator

1. Choose a confidence level you can justify95% is the convention in most academic research and is rarely questioned. 99% demands a substantially larger sample; 90% is defensible for exploratory work if you say so explicitly.
2. Set the expected proportion honestlyIf you have no prior estimate, use 50%. That maximises p(1−p) and therefore the sample size, which is the conservative and safe choice. Using a lower figure without evidence understates the sample you need.
3. Enter the population only if you know itCochran's formula works for unknown or effectively infinite populations. Supplying a population applies the finite correction, which reduces the sample — often substantially for small populations.
4. Check the sensitivity rowsThe breakdown shows the sample at other margins of error. Panels ask what happens if you loosen the margin, and having the figure ready is worth more than the headline number.

How the formula works

Cochran's formula answers a more general question than Slovin's: how many responses do I need to estimate a proportion to within a stated margin, at a stated confidence, when I may not know the population size at all?

The numerator has two parts. Z² reflects the confidence level — how many standard errors wide the interval must be. The p(1−p) term is the variance of a proportion, and it peaks at p = 0.5. That is why 50% is the safe default: it is the point of maximum uncertainty, so a sample sized for it is adequate whatever the true proportion turns out to be.

The denominator, e², is why precision is so expensive. Halving the margin of error from 5% to 2.5% quadruples the sample, because e is squared. That single relationship explains most of the cost difference between a serviceable survey and a precise one.

Notice what is absent: the population size. Cochran's base formula does not use it, because for a large population the sample needed barely changes with N. The finite population correction is applied afterwards, and it only makes a meaningful difference when the population is small relative to n₀.

Worked example: a barangay survey

A study at 95% confidence, a 5% margin of error, and no prior estimate of the proportion.

Z is 1.96, p is 0.5 so p(1−p) is 0.25, and e² is 0.0025. n₀ = 1.96² × 0.25 ÷ 0.0025 = 3.8416 × 0.25 ÷ 0.0025 = 384.16, so 385 respondents.

That 385 figure is worth memorising. It is the sample size for a proportion at 95% confidence and a 5% margin for any population large enough that the correction does not bite — which is why national surveys of very different countries so often use samples of around 400.

Now suppose the population is a barangay of 2,000 residents. The correction gives 384.16 ÷ (1 + 383.16 ÷ 2000) = 384.16 ÷ 1.1916 = 322.4, so 323 respondents. The correction saved 62 interviews.

With a population of 500 the correction bites much harder: 384.16 ÷ (1 + 383.16 ÷ 500) = 217.5, so 218 — nearly half the uncorrected figure.

Cochran's against Slovin's

Slovin's is simpler and is widely accepted in Philippine undergraduate work. Cochran's is more defensible because every assumption is visible and adjustable, and it is the right choice if your methodology states a confidence level other than 95%, if you have a prior estimate of the proportion, or if the population is unknown.

Choosing p, and why 0.5 is the safe answer

The p(1−p) term is the variance of a binomial proportion, and it is not symmetric in its effect. At p = 0.5 it equals 0.25, its maximum. At p = 0.2 or 0.8 it is 0.16, and at p = 0.05 or 0.95 it is just 0.0475.

That means an expected proportion far from 50% dramatically reduces the required sample. At 95% confidence and a 5% margin, p = 0.5 needs 385 respondents while p = 0.1 needs only 139.

The temptation is obvious and the risk is real. If you assume p = 0.1 and the true proportion turns out to be 0.4, your sample is too small and the margin of error you claimed is wrong. Using a prior estimate is legitimate when it comes from a pilot study or comparable published research — not when it comes from expectation.

For thesis work, using 0.5 and stating that you did so because no prior estimate was available is both defensible and easy to defend. Using a lower figure requires you to cite where it came from, and a panel will ask.

Assumptions and limitations

• Sizes a sample for estimating a proportion, not a mean or a difference between groups
• Assumes simple random sampling — it does not validate a convenience or purposive sample
• Does not adjust for anticipated non-response, which should be added on top
• For stratified designs, use it for the overall total then allocate proportionally across strata
• The finite population correction assumes the population size is known and accurate
• Gives a minimum for the stated precision — it does not guarantee the sample is representative

Questions people ask about this in Philippines