Cochran's Sample Size Formula: Why Everyone Keeps Getting 384

For estimating a single proportion in a large population, the familiar formula is:

Where:

• n₀ = initial sample size;
• Z = critical value for the selected confidence level;
• p = expected population proportion;
• 1 - p = the complementary proportion;
• e = desired absolute precision, commonly called the margin of error.

For 95% confidence:

Z ≈ 1.96

The formula is a planning tool for a specific estimation problem.

It is not a universal sample-size formula for every study.

For a binary proportion, p(1-p) is largest when p = 0.50.

That means using 50% gives the largest initial sample size for a fixed confidence level and precision.

This makes it a common conservative planning value when the expected proportion is genuinely uncertain.

But if credible prior information exists, you may justify another expected proportion.

Do not write:

“50% was used because Cochran recommends 50%.”

Explain the actual reason:

uncertainty about the expected proportion and the conservative variance assumption.

Suppose previous credible research suggests:

p = 0.20

Use:

• p = 0.20
• 1 - p = 0.80
• Z = 1.96
• e = 0.05

This gives approximately:

246 participants

Same confidence level.

Same precision.

Different assumed proportion.

Different sample size.

That alone should make one thing clear:

“Cochran = 384” is not the method.

Suppose you change your desired precision from ±5 percentage points to ±3 percentage points while keeping 95% confidence and p = 0.50.

Because precision appears in the denominator as $e^2$, tightening the margin of error substantially increases the required sample.

More precision costs sample size.

This is one reason sample-size planning is not simply about finding the smallest defensible number.

It is about deciding how much uncertainty you can tolerate for the study objective.

If the eligible population is finite and not very large relative to the initial sample, a finite population correction can reduce the sample requirement.

Use:

Where:

• n₀ = initial large-population estimate;
• N = finite population size.

If:

• n₀ ≈ 385;
• N = 1,000;

then:

n ≈ 278

Why?

Sampling 278 people from a population of only 1,000 captures a much larger fraction of the population than sampling 278 people from a population of several million.

Suppose the corrected sample requirement is:

278 completed participants

and you expect:

90% response

Calculate the number to approach as:

So you would plan to approach approximately:

309 people

to obtain about 278 completed observations if the response-rate assumption is correct.

This is more precise than casually saying:

“We added 10%.”

The required analysis sample and the number approached are different quantities.

The basic formula corresponds to a simple random-sampling framework.

But many real studies sample:

• villages;
• schools;
• health facilities;
• enumeration areas;
• households within clusters.

People within the same cluster can be more similar than independently selected people.

That can reduce statistical efficiency.

A planning adjustment called the design effect is therefore often used for clustered designs.

If:

• simple-random requirement = 385;
• justified design effect = 1.5;

then:

You would then consider non-response and other relevant adjustments.

But do not invent a design effect simply to complete the formula.

It should come from credible prior evidence or a defensible planning assumption.

And remember:

Adjusting the sample size for clustering does not remove the need to account for clustering in the analysis.

No.

This is where Day 10 differs from the broader Day 2 guide.

Cochran's single-proportion formula is useful when your primary objective is something like:

To estimate the proportion of women using modern contraception with specified absolute precision.

It is not automatically the correct approach if your objective is:

To compare contraceptive use between urban and rural participants.

Or:

To estimate an adjusted association between service access and contraceptive use.

Or:

To estimate a mean knowledge score.

Or:

To detect a specified difference between intervention and comparison groups.

Those require sample-size approaches aligned with those estimands and designs.

Because the raw calculation with the common assumptions is approximately:

384.16

Some texts refer to this as approximately 384.

For planning recruitment, rounding upward produces:

385

The difference is arithmetic, not a methodological controversy.

What matters much more is whether the assumptions fit your study.