How to calculate sample size for quantitative research: formulas, examples and choosing correctly
There is no single sample-size formula for every study. This article works through each common formula with concrete examples and shows how to settle on a number you can defend before a committee.
Quick summary
- Cochran for estimating a proportion (Z = 1,96; p = 0,5; e = 0,05): n₀ = 384,16, rounded up to 385; with a population N = 2.000, it corrects to 323.
- Slovin with N = 2.000, e = 0,05 gives 334, but this formula ignores the analysis method, so it is no basis for EFA, regression or SEM.
- EFA: at least 50, preferably 100 or more, and 5–10 observations per item; 25 items need 125 (5:1) to 250 (10:1).
- Regression with 5 independent variables: 50 + 8m = 90 and 104 + m = 109; G*Power (f² = 0,15; α = 0,05; power 0,80; 3 predictors) gives 77.
- PLS-SEM by inverse square root: β_min = 0,2 needs 155, β_min = 0,15 needs 275. Take the largest requirement, then divide by the expected valid-response rate.
What sample size depends on
"How many responses do I need" only has an answer when you know what you will do with the data. There are two different kinds of goal:
- Estimating a characteristic of the population, e.g. the share of students using e-wallets. Here sample size depends on the margin of error (e), confidence (Z) and the variability of the characteristic. This is the territory of Cochran, Slovin, Krejcie & Morgan.
- Testing a multi-variable model, e.g. EFA, multiple regression, PLS-SEM. Here sample size depends on the number of items, number of predictors, the effect size you want to detect, the significance level and power (the probability of detecting the effect if it really exists).
Most economics and management theses belong to the second group yet cite only the formulas of the first. Note too: every formula here assumes random sampling. If you take a convenience sample via a Google Form shared on social media, no sample size makes up for representativeness; you can only generalise with caution.
Cochran's formula and the finite population correction
When the goal is estimating a proportion, Cochran (1977) gives this formula for a large population:
- Z: the critical value for the confidence level; 95% corresponds to Z = 1,96.
- p: the estimated proportion in the population. Without information, choose p = 0,5, since p(1 − p) is largest there, giving the safest sample size.
- e: the margin of error, usually 0,05 (±5 percentage points).
Substituting: n0 = 1,962 × 0,5 × 0,5 / 0,052 = 3,8416 × 0,25 / 0,0025 = 384,16. Sample size is always rounded up, so n0 = 385.
When the population N is known and not too large
If the population is finite and known (e.g. all employees of a company), use the finite population correction:
For a population of several hundred thousand, the correction barely changes the figure of 385.
Slovin's formula: when it can be used
Slovin's formula is very common in theses because it needs only N and e:
With N = 2.000, e = 0,05: n = 2.000 / (1 + 2.000 × 0,0025) = 2.000 / 6 = 333,3, rounded up to 334.
Why Slovin is criticised
Tejada & Punzalan (2012) analyse the misuse of this formula in detail and show that:
- Slovin implicitly assumes p = 0,5 and about 95% confidence, but users usually don't know or state these assumptions.
- It only serves to estimate a proportion; it ignores the analysis method, number of variables and effect size entirely. So it is no basis for sample size in EFA, regression or SEM.
- When the population is unknown or very large, many people put a guessed N into the formula. That has no basis; in this case Cochran for a large population is more transparent.
Cross-checking with the Krejcie & Morgan table
Krejcie & Morgan (1970) tabulated sample sizes at 95% confidence, p = 0,5, and 5% error. Looking up N = 2.000 gives 322, matching the corrected Cochran result (a difference of 1 due to rounding). The table is useful as a cross-check but has the same limits as the proportion-estimation formulas.
Sample size for EFA, regression and power analysis
Exploratory factor analysis (EFA)
Hair et al. (2019) recommend a minimum sample of 50, preferably 100 or more, and at least 5 observations per item, preferably 10:1. In Vietnam the rule is often written as n ≥ 5 × the number of items. A questionnaire with 25 items needs at least 125 responses at 5:1, and 250 at 10:1. These are rules of thumb; what is enough also depends on the loadings and items per factor. How to read the output: see the article KMO, Bartlett and EFA.
Multiple regression
Tabachnick & Fidell (2013), based on Green (1991), give two rules, where m is the number of predictors:
- N ≥ 50 + 8m to test the R2 of the whole model.
- N ≥ 104 + m to test individual regression coefficients.
With m = 5: 50 + 8 × 5 = 90 and 104 + 5 = 109. Since theses almost always interpret each coefficient, take the larger, 109. Both rules assume a medium effect, α = 0,05 and power 0,80.
Power analysis with G*Power
A stricter approach is an a priori power analysis with the free G*Power software (Faul et al., 2007). Settings for multiple regression:
Test family: F tests Statistical test: Linear multiple regression: Fixed model, R² deviation from zero Type of power analysis: A priori Effect size f²: 0.15 α err prob: 0.05 Power (1-β err prob): 0.80 Number of predictors: 3 → Total sample size: 77Effect size f2 = R2 / (1 − R2); Cohen (1988) treats 0,02 as small, 0,15 as medium, 0,35 as large. α = 0,05 is the significance level; power 0,80 means a 80% chance of detecting a medium effect if it exists. If you expect a smaller effect, the required sample rises sharply. When running the regression, see the article reading SPSS regression output.
Sample size for PLS-SEM
The 10-times rule and why not to rely on it
The 10-times rule (Barclay, Higgins & Thompson, 1995) requires a sample at least 10 times the larger of: the largest number of formative indicators of a construct, and the largest number of structural paths pointing at a construct. If the dependent variable receives 4 arrows, n ≥ 40. The rule ignores effect size and power. Kock & Hadaya (2018) show it usually gives too small a sample, especially when path coefficients are small.
The inverse square root method
Kock & Hadaya (2018) propose a simple formula based on the smallest path coefficient you expect to be significant (βmin), at 5% significance and 80% power:
- βmin = 0,2: (2,486 / 0,2)2 = 12,432 = 154,5 → 155.
- βmin = 0,15: (2,486 / 0,15)2 ≈ 16,5732 ≈ 274,7 → 275.
βmin should come from earlier studies with similar models. Without a basis, choose a cautious value such as 0,15 rather than an optimistic guess. Once you have data, assessing the measurement model is covered in the article CR, AVE and HTMT.
Comparison of sample-size methods
| Method | When to use | Formula / rule | Example |
|---|---|---|---|
| Cochran (1977) | Estimating a proportion, large population | n0 = Z2p(1 − p)/e2 | 385 |
| Cochran with correction | Estimating a proportion, N known | n0 / (1 + (n0 − 1)/N) | N = 2.000 → 323 |
| Slovin | Estimating a proportion, N known, accepting the implicit assumptions | N / (1 + Ne2) | N = 2.000 → 334 |
| Krejcie & Morgan (1970) | Cross-check with a lookup table | Ready-made table | N = 2.000 → 322 |
| EFA (Hair et al., 2019) | Exploratory factor analysis | ≥ 100; 5:1, preferably 10:1 | 25 items → 125 / 250 |
| Tabachnick & Fidell (2013) | Multiple regression | 50 + 8m; 104 + m | m = 5 → 90; 109 |
| G*Power (Faul et al., 2007) | Regression, with an assumed effect | f2, α, power, number of predictors | f2 = 0,15; 3 predictors → 77 |
| The 10-times rule | Rough PLS-SEM estimate, not recommended | 10 × the largest number of arrows | 4 arrows → 40 |
| Inverse square root (Kock & Hadaya, 2018) | PLS-SEM | (2,486/|βmin|)2 | β = 0,2 → 155; β = 0,15 → 275 |
Settling the sample size for your study
- List every analysis you will run: Cronbach's Alpha, EFA, regression, PLS-SEM, group comparisons.
- Calculate the requirement for each analysis with the right method from the table above.
- Take the largest number, since the sample must suffice for the most demanding analysis.
- Add a buffer for invalid responses: responses to collect = valid responses needed / expected valid rate.
- State the sampling method and the limits on generalisation if the sample isn't random.
The expected valid rate should come from the pilot testing a questionnaire rather than a guess. To rehearse the whole analysis workflow at the planned sample size, you can use SPSS research module to generate simulated data and check the syntax and export steps. Simulated data is only for practising the workflow and must never be reported as real collected data.
Common mistakes
- Using Slovin as the only basis for a study running EFA and regression.
- Entering a guessed N when the population is actually unknown.
- Rounding down, or taking the exact minimum without a buffer for invalid responses.
- Writing "the sample is large, so it is representative" with a convenience sample.
Checklist and how to report it
- The requirement for each analysis is calculated, with the formula, inputs and source cited.
- Every result is rounded up.
- The largest number is chosen and the reason explained.
- A buffer is added from the expected valid rate in the pilot.
- The sampling method and limits on generalisation are stated.
- The questionnaire on Google Form has screening questions for the right population.
A sample paragraph: "The study uses PLS-SEM and EFA with 25 items. By the inverse square root method (Kock & Hadaya, 2018) with βmin = 0,15, the minimum sample size is 275; at a 10:1 ratio (Hair et al., 2019) it is 250. The author targets 300 valid responses and distributed 375 questionnaires with an expected valid rate of 80%."
Frequently asked questions
References
- Barclay, D., Higgins, C., & Thompson, R. (1995). The partial least squares (PLS) approach to causal modeling: Personal computer adoption and use as an illustration. Technology Studies, 2(2), 285–309.
- 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.
- 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.
- Green, S. B. (1991). How many subjects does it take to do a regression analysis? Multivariate Behavioral Research, 26(3), 499–510.
- Hair, J. F., Black, W. C., Babin, B. J., & Anderson, R. E. (2019). Multivariate data analysis (8th ed.). Cengage Learning.
- Kock, N., & Hadaya, P. (2018). Minimum sample size estimation in PLS-SEM: The inverse square root and gamma-exponential methods. Information Systems Journal, 28(1), 227–261.
- Krejcie, R. V., & Morgan, D. W. (1970). Determining sample size for research activities. Educational and Psychological Measurement, 30(3), 607–610.
- Tabachnick, B. G., & Fidell, L. S. (2013). Using multivariate statistics (6th ed.). Pearson.
- Tejada, J. J., & Punzalan, J. R. B. (2012). On the misuse of Slovin's formula. The Philippine Statistician, 61(1), 129–136.
The thresholds in this article are common conventions in the sources above. Check them against your supervisor's or target journal's requirements.