Cochran's formula online: sample size calculator

Cochran's formula gives a study's sample size from the confidence level (z), the proportion of the population with the attribute (p and q) and the margin of error (d), and when the population size (N) is known it corrects the result for a finite population. For example, a population of 1000 at 95% confidence and a 0.05 margin of error needs a sample of 278.

Cochran's formula has four components, and each must be reported in your methodology chapter:

• z: the normal distribution value for the confidence level; 1.645 for 90%, 1.96 for 95% and 2.576 for 99%. Some textbooks call it t.
• p and q: p is the proportion of the population that has the attribute under study and q the proportion that doesn't, i.e. 1 minus p. If you don't know it, set p to 0.5 so that q is 0.5 too; this yields the largest sample size and is the most conservative choice.
• d: the margin of error, or estimate precision; 0.05 is the most common value. A smaller margin needs a larger sample: halving d roughly quadruples the sample size.
• N: the population size; when known, it is used for the finite population correction.

The calculator always rounds up, because the sample size must be a whole number and rounding down means slightly less precision than you asked for.

If the population is very large — say, every university student in the country — or you don't know its exact size, the first part of the formula is enough: leave the population box empty. With 95% confidence, p = 0.5 and d = 0.05 the result is 384.16, usually written as 384 in textbooks; because the calculator rounds up, it shows 385.

If the population is finite and known — say, a company's staff or a school's students — the finite population correction lowers the sample size. The smaller the population, the bigger the reduction; for populations in the hundreds of thousands the difference all but disappears and the number stays close to 384.

A common question: how large a sample does a population of 1000 need at 95% confidence and a 0.05 margin of error? Step by step:

So you need at least 278 complete questionnaires. The one-line form of the formula gives 277.54 for the same population, which also rounds up to 278, and the Morgan table suggests 278. The table below shows the same calculation with other inputs:

Both use 95% confidence, a 0.05 margin of error and p = 0.5; for other values, change 1.96, 0.05 and 0.5. If your Excel expects “/” as the decimal separator or “;” between arguments, use those instead of “.” and “,”.

If your population size is in this table, take the number directly. Every value uses p = 0.5 and is rounded up, and the “95% confidence, 0.05 margin” column is the usual thesis setting; for sizes between rows, use the calculator at the top of the page.

The Krejcie & Morgan (1970) table did the same calculation in advance for common population sizes, with 95% confidence (χ² = 3.841), a proportion of 0.5 and a margin of error of 0.05. The third column is Cochran's formula with the finite population correction, so you can see both methods side by side.

The full 90-row Krejcie and Morgan table, its formula and a calculator that gives the table value for any population size are on the page Morgan table.

• The Morgan table is Cochran's calculation done in advance for common population sizes at 95% confidence, a proportion of 0.5 and a 0.05 margin of error; its numbers differ from Cochran's by at most one person.
• If you need a different confidence level or margin of error (say 99% or 0.03), know p from a pilot study, or don't know the population size, use Cochran's formula.
• If your supervisor asked for the Morgan table and your population size is in it, take the number from the table and cite Krejcie and Morgan (1970).
• For a population between two rows of the table, don't guess: get the exact number from the Morgan table calculator or the calculator on this page.

For example, for the 1,200 teachers of one province, Cochran gives 292 and the Morgan table 291; both numbers are acceptable in a thesis — just cite the method you used and its source.

Sample size means complete, usable questionnaires — not the number of people who open the link. In an online questionnaire some people always drop out halfway, and some give careless or duplicate answers that have to be discarded.

To compensate, divide the sample size by the share of usable responses you expect. If you think 20% of responses will be discarded, collect at least 482 responses instead of 385 (385 divided by 0.8).

Building and publishing are free, and the free account gets 50 responses a month; more responses unlock with any monthly, quarterly or yearly plan, and no response is ever lost. The “suspicious responses” report separates very fast answers and straight-lined matrices, and the “incomplete responses analysis” and “drop-off point per question” reports show where respondents abandoned the questionnaire. For the questionnaire itself, see Build an online questionnaire for details.

1. 1

   Build the questionnaire

   Build your online questionnaire for free, or, if a validated instrument exists for your construct, start from the ready-made standard questionnaires; either way you get a link to send to your sample.

2. 2

   Check reliability

   With your pilot data, check reliability with the Cronbach's alpha calculator; 0.7 or higher is usually acceptable.

3. 3

   Report the method

   In the methods chapter, write: “Using Cochran's formula with a 95% confidence level and a 0.05 margin of error, the sample size for a population of 1000 was set at 278.” The calculator at the top of the page writes this sentence with your own numbers.