How to Determine Sample Size: Cochran's Formula, the Morgan Table or Power Analysis
Cochran and Morgan were built to estimate a proportion in a population, not to test a relationship. With worked examples, this guide shows when to use which method, how to apply the finite population correction and allow for attrition, and what power analysis gives for correlation, regression and group comparisons.
"The sample size was set at 384 using Cochran's formula" may be the most common sentence in thesis methods chapters, and in many of them it isn't the right choice. Not because Cochran is wrong, but because it was built for a different question. This guide puts the three main methods side by side with fully worked examples and ends with ready-to-adapt text for your methods chapter.
First ask: what is the analysis for?
| If your main question is… | Suitable method | Example |
|---|---|---|
| What percentage of the population has a characteristic? | Cochran's formula or the Morgan table | The share of students satisfied with online learning |
| What is the population mean of a variable? | Cochran for a mean (with a standard deviation) | Teachers' mean job-satisfaction score |
| Is there a relationship between two variables? | Power analysis for correlation | Leadership style and organisational commitment |
| Do two or more groups differ? | Power analysis for t or ANOVA | Burnout across three shifts |
| How well do several variables predict an outcome? | Power analysis for regression | Predicting purchase intention from five variables |
| Structural equation modelling or factor analysis | Method-specific rules | A technology acceptance model |
Most theses have both description and hypothesis tests. The simple rule: size the sample for the most demanding analysis you will run, and report both numbers.
Cochran's formula, part by part
n₀ = z² × p × (1 − p) / d²
n = n₀ / (1 + (n₀ − 1) / N)The first line is for a large or unknown population; the second is its correction for a finite population:
- z: 1.96 for 95% confidence (1.645 for 90%, 2.576 for 99%).
- p: the proportion with the characteristic. If you have no figure from earlier research or a pilot, use 0.5; p(1−p) peaks at 0.5, giving the most conservative (largest) sample.
- d: the margin of error, usually 0.05, meaning "the percentage I report may be off by up to 5 percentage points either way."
- N: the population size, if known.
Example 1: unknown or very large population
n₀ = 1.96² × 0.5 × 0.5 / 0.05² = 3.8416 × 0.25 / 0.0025 = 384.16 → 385Sources usually write 384.16 as 384, but sample sizes are rounded up, so 385. The figure is almost the same for a population of one million or ten million; once a population is very large, its size no longer matters.
Example 2: finite population of 1,200 teachers
n = 384.16 / (1 + 383.16 / 1200) = 384.16 / 1.3193 = 291.18 → 292The finite population correction matters when the sample is a sizeable share of the population. For a population of 300 the result is 169; for 50,000 it is 382, nearly the same 385.
Example 3: when you know p
If earlier research found that about 20% of students work part-time, p = 0.2:
n₀ = 3.8416 × 0.2 × 0.8 / 0.0025 = 245.86 → 246With a population of 1,200, 205 people. Always cite the source of p; an examiner will push an unsourced p back to 0.5.
Example 4: Cochran for a mean
If the goal is to estimate a mean (say, the mean satisfaction score on a 1–5 scale), the variance replaces p(1−p) and d is in the variable's own units:
n₀ = z² × s² / d² s = 0.8 , d = 0.1 → 3.8416 × 0.64 / 0.01 = 245.86 → 246Take the standard deviation (s) from a pilot. d = 0.1 on a five-point scale means you want the mean to within 0.1 points; with d = 0.15 the same calculation gives 110. Don't pick d just to land on a convenient number; justify it.
How much do confidence and margin of error change the result?
| Confidence | Margin of error | Unknown population | Population of 1,000 |
|---|---|---|---|
| 90% | 0.05 | 271 | 214 |
| 95% | 0.05 | 385 | 278 |
| 95% | 0.03 | 1,068 | 517 |
| 99% | 0.05 | 664 | 400 |
Run examples 1–3, or any other combination, in the Cochran sample size calculator; it also shows the calculation steps for your thesis.
The Morgan table: Cochran, precomputed
The Krejcie and Morgan table (1970) performs the same calculation for common population sizes under three fixed assumptions: 95% confidence (χ² = 3.841), a proportion of 0.5 and a margin of error of 0.05. That is why its numbers differ from the corrected Cochran figure by at most one person, and that one comes from rounding.
| Population | Morgan table | Cochran (rounded up) |
|---|---|---|
| 100 | 80 | 80 |
| 200 | 132 | 132 |
| 300 | 169 | 169 |
| 500 | 217 | 218 |
| 1,000 | 278 | 278 |
| 2,000 | 322 | 323 |
| 5,000 | 357 | 357 |
| 10,000 | 370 | 370 |
| 50,000 | 381 | 382 |
| 1,000,000 | 384 | 385 |
See the full table for every population size in the Morgan table. The table's limit: it is valid only for those three assumptions. If you want a 3% margin or 99% confidence, or you already know p, use Cochran directly. And like Cochran itself, the Morgan table is for estimating a proportion, not testing a relationship.
Power analysis: when your hypothesis is about a relationship or a difference
Cochran asks "how precise is my estimate?" Power analysis asks "if an effect really exists, what are the chances my test detects it?" Four numbers are tied together, and any three give you the fourth:
- Significance level (α): usually 0.05.
- Power: usually 0.80, an 80% chance of detecting a real effect.
- Effect size: how large the effect you're looking for is; ideally from similar studies.
- Sample size: what you're after.
Without similar studies, Cohen's conventions (1988) are standard:
| Analysis | Effect index | Small | Medium | Large |
|---|---|---|---|---|
| Comparing two means | d | 0.2 | 0.5 | 0.8 |
| Correlation | r | 0.1 | 0.3 | 0.5 |
| ANOVA | f | 0.10 | 0.25 | 0.40 |
| Multiple regression | f² | 0.02 | 0.15 | 0.35 |
Sample size for a medium effect (α = 0.05, power 0.80, two-tailed)
| Design | Sample size |
|---|---|
| Two independent groups, d = 0.5 | 64 per group (128 in total) |
| Correlation of two variables, r = 0.3 | About 85 |
| Three-group ANOVA, f = 0.25 | 53 per group (159 in total) |
| Regression with 3 predictors, f² = 0.15 | 77 |
| Regression with 5 predictors, f² = 0.15 | 92 |
| Regression with 10 predictors, f² = 0.15 | 118 |
Two things stand out. First, testing a relationship with a medium effect often needs far fewer than 384 people. Second, with a small effect the picture flips: a correlation of r = 0.1 needs about 780 people, and comparing two groups at d = 0.2 needs about 394 in each group. So 384 is neither always too many nor always enough.
Run these calculations in the free G*Power software and put a screenshot of the output in your appendix, so an examiner can reproduce the number.
Structural equation modelling and factor analysis
There is no single formula for these methods, and rules of thumb are the norm: 5 to 10 responses per item for factor analysis, and at least 200 people for covariance-based structural equation modelling. In PLS the "ten times the largest number of paths pointing at one construct" rule is common, but it is a floor, not a target; for a demanding examiner, add a regression power analysis for that construct.
Attrition: the final number is not the number you collect
Sample size means the number of complete, usable questionnaires. Online, some responses stay unfinished and some are excluded for carelessness (finished in seconds, the same column for every item). If you expect 20% attrition:
n_collect = n / (1 − 0.20) = 385 / 0.8 = 481.25 → 482A pilot is the best source for estimating attrition. In an online questionnaire, the "drop-off point per question" and "suspicious responses" reports give you this number from real data.
Describe your sampling method honestly
The formula only tells you "how many," not "who." A sample of 400 gathered by posting a link in two messaging groups is a convenience sample, not a random one, and Cochran's formula assumes random sampling. If you have a list of the population (an organisation's staff, a faculty's students), draw a random sample and send those people personal invitations. If you don't, call the method "convenience" or "snowball" and mention it under the study's limitations; honesty about method is easier to defend than a big claim.
Sample text for your methods chapter
The study population comprises [1,200] [upper-secondary teachers in …] in the [2025–2026] school year. The sample size was set at 292 using Cochran's formula for a finite population, with 95% confidence (z = 1.96), a proportion of 0.5 and a margin of error of 0.05; allowing for 20% attrition, the questionnaire was sent to 365 people. For the hypothesis tested by [a regression with five predictors], a power analysis in G*Power (medium effect size f² = 0.15, α = 0.05, power 0.80) indicated a minimum of 92 participants, which the chosen sample exceeds. Sampling was [stratified random by education district].
Common mistakes
- Cochran for a study whose only hypotheses are correlational, with no mention of power.
- Rounding 291.18 down to 291; sample sizes are always rounded up.
- Using the Morgan table with a 3% margin of error; the table is for 5% only.
- Sizing the sample for the whole population, then comparing subgroups of 15.
- Claiming "simple random sampling" for a link shared on social media.
- Collecting exactly the target number, then dropping incomplete responses and ending up below the minimum.
Frequently asked questions
My population is 80 people. Should I sample?
If you can, ask everyone (a census). Cochran gives 67 for a population of 80; trying to reach all 80 is simpler and removes the sampling-method debate entirely.
Cochran or Morgan: which is more valid?
Neither; they are the same calculation. Cochran is flexible (you change confidence, margin and p), and Morgan is a ready-made table for the common case. Whichever you use, cite it.
Why does everyone write 384 when the calculator gives 385?
The formula gives 384.16. Tables round it to the nearest whole number; the stricter rule is to round up so the error never exceeds the target. Both are accepted in theses.
How many people for a pilot?
About 30 from the population who won't be in the main sample; enough to see Cronbach's alpha, completion time and ambiguous items. Details are in questionnaire validity and reliability.
Can an online questionnaire reach the target sample size?
Yes, if you plan for attrition from day one, block duplicate responses and check progress daily. The practical guide: running your thesis questionnaire online.
Got your number? Build the questionnaire in Porsino's online questionnaire builder, set the response cap to your sample size plus attrition and watch progress every day.