Labeeb reference · every formula traced to its source · Last checked 28 September 2026
Sample Size Calculator for Theses: Cochran, Krejcie & Morgan, Yamane and Thompson (2026)
Work out a survey sample size with the four formulas you will meet in GCC theses, see the Krejcie and Morgan table, add a response-rate allowance, and learn when you need a G*Power analysis instead. The calculator runs in your browser and stores nothing.
Read this first: which sample size question are you answering?
There are two different questions, and each has its own method. Examiners check that you used the method that matches your research question.
- Describing a population (a prevalence, a percentage who agree, an average attitude score): you need enough responses to estimate a proportion within a margin of error. The Cochran, Krejcie and Morgan, Thompson and Yamane formulas answer this question, and so does Raosoft.
- Testing a hypothesis (comparing groups, testing a correlation, a regression or a path model): you need enough responses to detect an effect of a given size. That is a power analysis, usually done in G*Power. A survey formula is not a power analysis.
- All four survey formulas assume a simple random sample from a defined list of the population (the sampling frame). If you use convenience, snowball or cluster sampling, the number from a formula does not make the sample representative. Say so in your limitations.
- The number you calculate is a planning figure. Your supervisor and your ethics or research committee decide what your study needs, and your university or department may set its own rule.
Free calculator · runs in your browser, nothing is stored
Sample size calculator: survey and questionnaire studies
Enter your population size if you know it, choose a confidence level, and set the margin of error and expected proportion. Leave the expected proportion at 50% unless a previous study gives you a better figure. Add an expected response rate to see how many people to invite. All four formulas are shown side by side so you can report the one your department expects.
The results apply the published formulas to the numbers you enter. They are not a judgement of your study design. The first row reproduces Raosoft’s method (same formula, rounded up). The Krejcie and Morgan row is rounded to the nearest whole number, which is how the 1970 table was printed.
The formulas, and where each one comes from
Symbols: N is the population size, n the sample size, z the critical value for your confidence level (1.645 for 90%, 1.96 for 95%, 2.576 for 99%), p or P the expected proportion, and e or d the margin of error, written as a proportion (5% = 0.05). Full references are in the sources.
Cochran, with finite population correction
n₀ = z²p(1 − p) / e², then n = n₀ / (1 + (n₀ − 1) / N).
From Cochran (1977). At 95%, ±5% and p = 50%, n₀ = 384.16, so 385 for a very large population. Raosoft uses the same calculation, written as n = Nx / ((N − 1)E² + x), and rounds up (Rao & Rao).
Krejcie and Morgan (1970)
s = X²NP(1 − P) / (d²(N − 1) + X²P(1 − P))
X² is the chi-square value for 1 degree of freedom at the chosen confidence level (3.841 at 95%). The published table fixes P = .50 and d = .05 (Krejcie & Morgan, 1970). Arab methods texts often call it جدول مورجان. Because X² for 1 degree of freedom equals z², it is the same calculation as the Cochran card.
Steven K. Thompson
n = N·p(1 − p) / ((N − 1)(d² / z²) + p(1 − p))
The sample size for estimating a proportion in Thompson’s Sampling (Thompson, 2012). Arab texts often spell the name “Stephen” Thompson (معادلة ستيفن ثامبسون), and it is usually applied with z = 1.96, d = 0.05 and p = 0.50. Multiply the top and bottom by z² and it becomes the Krejcie and Morgan formula, so it gives the same answer.
Yamane (1967)
n = N / (1 + Ne²)
From Yamane (1967). It has no z and no p. It is what the formula above gives when z²p(1 − p) = 1, that is z = 2 and p = .50 (about 95.4% confidence), with N − 1 taken as N. Expect an examiner to question it if a thesis claims 99% confidence, uses a proportion other than 50%, or uses it for a hypothesis test.
Response-rate allowance
Number to invite = n / expected response rate, rounded up.
If you need 278 completed questionnaires and expect 60% to respond, invite at least 278 / 0.60 = 464. Base the expected rate on a pilot or a similar study, and report the achieved rate and the number of usable responses after data cleaning.
Krejcie and Morgan (1970) table: typical population sizes
The published values (95% confidence, ±5%, P = .50) next to what each formula gives for the same population. The K&M column is copied from the 1970 table. The other columns were calculated with the calculator above: Cochran, Raosoft and Thompson rounded up, and Yamane at a 5% margin.
| Population (N) | K&M table (s) | Cochran, Raosoft, Thompson | Yamane |
|---|---|---|---|
| 50 | 44 | 45 | 45 |
| 100 | 80 | 80 | 80 |
| 200 | 132 | 132 | 134 |
| 300 | 169 | 169 | 172 |
| 500 | 217 | 218 | 223 |
| 1,000 | 278 | 278 | 286 |
| 2,000 | 322 | 323 | 334 |
| 5,000 | 357 | 357 | 371 |
| 10,000 | 370 | 370 | 385 |
| 20,000 | 377 | 377 | 393 |
| 50,000 | 381 | 382 | 397 |
| 1,000,000 | 384 | 384 | 400 |
The table was printed with values rounded to the nearest whole number, so it can be one below the rounded-up Cochran or Raosoft figure (for example 217 and 218 at N = 500). If you cite the table, use the table value. If you cite Raosoft or Cochran, use the rounded-up value. The table does not go above N = 1,000,000 (s = 384).
Comparing groups, correlation or regression? You need a power analysis
If your hypotheses test a difference between groups, a correlation, or the effect of predictors in a regression, the question is how many participants you need to detect the effect you expect. Survey formulas and the Morgan table do not answer that. G*Power does. It is free and runs on Windows and macOS.
Test family and statistical test
Choose the test you will actually run: t test for two groups, F test for ANOVA or multiple regression, exact test for a correlation, χ² for a contingency table. The analysis in your methods chapter and the test in G*Power must match.
Effect size
The size of the effect you expect or the smallest effect that would matter. Take it from earlier studies or a pilot where you can. Cohen’s small, medium and large values (table below) are a fallback, and you should say that you used them.
Alpha (α)
The risk of a false positive you accept, conventionally .05. Use one-tailed tests only when your hypothesis is directional and your field accepts it.
Power (1 − β)
The chance of detecting the effect if it is real. Cohen proposed .80 as a general convention, which pairs a .20 risk of missing the effect with a .05 risk of a false positive (Cohen, 1992). Some committees ask for .90.
Design details
The number of groups, the number of predictors, the allocation ratio between groups, or the degrees of freedom for χ². G*Power then reports the total sample size and the actual power.
Get G*Power
Download it only from the official G*Power page at Heinrich Heine University Düsseldorf. Its authors ask you to cite Faul et al. (2007) and, for correlation and regression, Faul et al. (2009).
Cohen’s conventional effect sizes
| Analysis | Effect size index | Small | Medium | Large |
|---|---|---|---|---|
| Two independent means (t test) | d | .20 | .50 | .80 |
| Correlation | r | .10 | .30 | .50 |
| One-way ANOVA | f | .10 | .25 | .40 |
| Multiple regression | f² | .02 | .15 | .35 |
| Chi-square | w | .10 | .30 | .50 |
Source: Cohen (1988), summarised in Table 1 of Cohen (1992).
What those conventions mean in practice (α = .05, power = .80)
- Two groups, medium difference (d = .50): 64 participants per group.
- Correlation, medium (r = .30): 85 participants.
- One-way ANOVA with three groups, medium (f = .25): 52 per group, 156 in total.
- Multiple regression with two predictors, medium (f² = .15): 67 participants.
- Chi-square with 1 degree of freedom, medium (w = .30): 87 participants.
From Table 2 of Cohen (1992). G*Power calculates exactly and can differ from these figures by a participant or two. Report the G*Power output you actually obtained, with its inputs. For CFA and SEM, sample size is a separate question. See our CFA and SEM fit thresholds page.
What examiners and reviewers will ask about your sample size
- Which formula, and why it fits your question. Name it, cite the primary source, and show the numbers you put in. A survey formula fits a descriptive aim. Hypothesis tests need a power analysis.
- What is your population and sampling frame? State who is included, where N comes from (a ministry statistic, a staff list, a registry) and the date of that figure.
- Why these settings? Justify the confidence level, the margin of error and the expected proportion. If you used 50%, say it gives the largest, most cautious sample. If you used a prevalence from an earlier study, cite it.
- How was the sample selected? The formulas assume random selection from the frame. If you used convenience or snowball sampling, say so and discuss what it means for generalising your results.
- What response rate did you expect, and what did you get? Show the allowance you added, the number invited, the number returned and the number usable after cleaning.
- Is the sample big enough for the analyses you ran? Group comparisons, regression and SEM need a power analysis or a justified rule for that method. The Morgan table is not a power analysis.
- Are your subgroups large enough? If you compare nationalities, genders, hospitals or colleges, each group you compare needs enough cases, not only the total.
- Do the numbers agree across the thesis? The planned, achieved and analysed sample sizes should match between the abstract, the methods and the results, and any shortfall should be explained.
Want your sample size checked before your examiners ask? Share your design, population and planned tests. We check the calculation or power analysis and its justification with you, and explain what to write in your methods chapter. You remain the author.
Common mistakes in GCC theses
These are framed as good practice drawn from the sources on this page, not as the rule of any one university. Your own guide and your supervisor come first.
- Using the Morgan table or Raosoft to justify a sample for a t test, ANOVA, regression or SEM. Those formulas size an estimate of a proportion. For hypothesis tests, add a power analysis.
- Citing the table but not following it. For example, citing Krejcie and Morgan for N = 500 but collecting 150 responses, or taking N as a whole country when the frame is one hospital or one university.
- Mixing formula and settings. Reporting Yamane’s formula alongside 99% confidence or a 30% expected proportion, which it cannot use, or quoting a Raosoft figure that does not match the settings stated in the text.
- No allowance for non-response. Distributing exactly the calculated number, receiving fewer usable questionnaires, and not discussing the shortfall.
- Treating a formula as a substitute for random sampling. Reaching the calculated number through social-media or snowball recruitment does not make the sample representative. Say how respondents were reached and what that limits.
- Citing formulas second-hand. Citing a website or another thesis for Cochran, Krejcie and Morgan or Yamane instead of the original work, or giving the wrong year or edition. The full references are below.
Frequently asked questions
Should I use Raosoft or G*Power?
They answer different questions. Raosoft tells you how many responses you need to estimate a percentage within a margin of error, which suits a descriptive survey. G*Power tells you how many participants you need to detect an effect of a given size with a given power, which suits hypothesis tests such as t tests, ANOVA, correlation and regression. If your thesis has both aims, calculate both, use the larger number and explain each in your methods chapter.
Is the Krejcie and Morgan (Morgan) table acceptable for a thesis?
It is a peer-reviewed, widely cited table for estimating a proportion from a known, finite population at 95% confidence, a 5% margin of error and P = .50, assuming random sampling. It is not a power analysis, so on its own it does not justify a sample for hypothesis tests, and it does not apply to other settings. Whether it is acceptable for your study is decided by your supervisor and your committee, so check with them.
What is the Steven Thompson formula, and does it give a different answer?
It is the sample size formula for estimating a proportion in Steven K. Thompson’s textbook Sampling (3rd ed., 2012): n = Np(1 – p) / ((N – 1)(d²/z²) + p(1 – p)). It is algebraically the same as the Krejcie and Morgan formula and the Cochran formula with the finite population correction, so with the same settings it gives the same sample size. Small differences come only from rounding, or from using 3.841 instead of 1.96².
Why do Yamane and Raosoft give different numbers?
Yamane’s formula, n = N / (1 + Ne²), behaves as if z = 2 and p = 50%, and it uses N where the other formulas use N – 1. Raosoft uses the exact z for your confidence level (1.96 at 95%). For a population of 1,000 at a 5% margin of error, Yamane gives 286 and Raosoft gives 278. Yamane’s formula also cannot reflect a 90% or 99% confidence level or an expected proportion other than 50%.
What if I do not know my population size?
Use the formula for an unlimited population: at 95% confidence, a 5% margin of error and 50% expected proportion it gives 385. Raosoft suggests entering 20,000 when the size is unknown, which gives 377. Beyond a few thousand the required sample barely grows, but you should still describe your population and sampling frame, and say which approach you used.
How to cite this page (APA 7)
Labeeb Writing & Designs. (2026, September 28). Sample Size Calculator for Theses: Cochran, Krejcie & Morgan, Yamane and Thompson (2026). https://labeeb.ae/sample-size-calculator-thesis/
Sources
Last checked: 28 September 2026. Each DOI below was checked on that date through Crossref. Books without a DOI were checked against library catalogue records and are cited in full.
- Cochran, W. G. (1977). Sampling techniques (3rd ed.). John Wiley & Sons.
- Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum Associates. Reissued by Routledge (2013): https://doi.org/10.4324/9780203771587
- Cohen, J. (1992). A power primer. Psychological Bulletin, 112(1), 155–159. https://doi.org/10.1037/0033-2909.112.1.155
- Faul, F., Erdfelder, E., Lang, A.-G., & Buchner, A. (2007). G*Power 3: A flexible statistical power analysis program for the social, behavioral, and biomedical sciences. Behavior Research Methods, 39(2), 175–191. https://doi.org/10.3758/BF03193146
- Faul, F., Erdfelder, E., Buchner, A., & Lang, A.-G. (2009). Statistical power analyses using G*Power 3.1: Tests for correlation and regression analyses. Behavior Research Methods, 41(4), 1149–1160. https://doi.org/10.3758/BRM.41.4.1149
- Krejcie, R. V., & Morgan, D. W. (1970). Determining sample size for research activities. Educational and Psychological Measurement, 30(3), 607–610. https://doi.org/10.1177/001316447003000308
- Rao, S. R., & Rao, P. M. (2009). Sample size calculator [Online calculator]. Raosoft. http://www.raosoft.com/samplesize.html
- Thompson, S. K. (2012). Sampling (3rd ed.). John Wiley & Sons. https://doi.org/10.1002/9781118162934
- Yamane, T. (1967). Statistics: An introductory analysis (2nd ed.). Harper & Row.
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