Sample size for a thesis

Three ways to get a number, and which one fits you

Mainly there are three ways scholars fix the number. Which one you use depends on what your thesis is doing with the data.

Some theses are mostly describing a population. For example, how many farmers in a block use crop insurance, or what is the average stress score of nurses in one district. For this kind of study your sample should estimate that figure, and you say beforehand how much error you will accept. Cochran’s formula does this, and the Krejcie and Morgan table is built on the same idea.

In some other theses the scholar is testing hypotheses. Maybe two groups are compared, or a few factors are checked as predictors of some outcome. Then the sample must be big enough to pick up an effect of a stated size. That is called power analysis, and most scholars do it in G*Power.

A third type of thesis builds a structural model in AMOS or SmartPLS. In that case the size of the model and the strength of its weakest path will decide the number. SEM rules are explained further down.

Many theses fall in two groups. Calculate both numbers, take the largest and mention it clearly in Chapter 3.

Cochran’s formula, worked through

Many scholars are asking us whether 385 is fixed number for every thesis. It comes from Cochran’s formula for a proportion, which is written like this.

n₀ = (Z² × p × q) / e², with q = 1 − p

At 95% confidence, Z is 1.96. p is the proportion you expect to find. When you have no idea of p, take 0.5. That gives the biggest sample, so it is the safe choice. e is the margin of error you accept, mostly 0.05.

Putting these values in, 1.96² × 0.5 × 0.5 / 0.05² = 0.9604 / 0.0025 = 384.16. Round it up and you get 385, seen in so many theses. The 385 is correct only for a large or unknown population, 95% confidence, 5% margin and p of 0.5. Change any one of these and the number changes. For example, with a 6% margin the sample comes down to 267.

For a known population that is not very big, use the finite population correction.

n = n₀ / (1 + (n₀ − 1) / N)

Suppose your study covers all 1,200 staff of a textile group in Tiruppur. Then n = 384.16 / (1 + 383.16 / 1,200) = 384.16 / 1.319 = 291.2, and rounding up gives 292. The Krejcie and Morgan table from 1970 shows 291 for a population of 1,200, so both methods are matching. Do the same working for a population of 500 and the answer you get is 218.

Our sample size calculator does these steps for you, with the finite population correction built in.

Power analysis for theses that test hypotheses

Cochran’s formula has no idea which test you will run later. Power analysis tells you how many people you need for a fair chance of finding an effect, if the effect is really there.

For a power analysis you have to fix four things. First is the test. Then the significance level α, which is .05 in almost every thesis. Then the power you want, usually .80. Power of .80 means you have an 80% chance to detect a true effect. Last is the effect size you expect, and this is where scholars get stuck. If the earlier researches on your topic report an effect size, use that one. If not, mostly Cohen’s medium values are used, d = 0.5 for t-tests, f = 0.25 for ANOVA, r = .30 for correlation and f² = 0.15 for regression.

G*Power is free software from Heinrich Heine University Düsseldorf. For a t-test, open Test family → t tests, and under Statistical test pick Means: Difference between two independent means (two groups). In the box for type of power analysis choose “A priori”. Type in your effect size, α and power, then press Calculate. We ran G*Power 3.1 for a few tests that come up in theses mostly, all at medium effect, and put those numbers in the table here. Take them only as an illustration and run it with your own settings.

You will see some of these numbers are quite small. Take regression with five predictors. For a medium effect G*Power asks for just 92 people. With 400 respondents, power analysis will not limit you. Still put it in Chapter 3 to show the sample was enough. Our power analysis calculator gives the same type of result inside your browser.

Sample size for SEM in AMOS or SmartPLS

For SEM you have to be more careful, and the rules are not as firm as students expect.

For covariance-based SEM in AMOS, two figures come up again and again, “at least 200” and “10 respondents per indicator”. Both are only rules of thumb from the textbooks. If a model has 40 items, 10 per item means 400 respondents. Many reviewers will accept 200 to 300 for a moderate model when the data is clean and roughly normal. With smaller samples the fit indices become unstable, and you can trust the estimates less.

For PLS-SEM in SmartPLS there is an old “ten-times rule”. As per that rule, you count the arrows coming into each construct, take the highest count and multiply by ten. Now this rule is widely criticised for giving too low a number. Kock and Hadaya (2018) gave a better option, called the inverse square root method. At α of .05 and 80% power, it says you need at least (2.486 / |p_min|)² responses. Here p_min is the weakest path in your model that you still want to detect. Suppose your weakest path is 0.2. Then (2.486 / 0.2)² comes to 154.5, and after rounding up you need 155 respondents. A smaller expected path needs a much bigger sample.

If you have not decided between AMOS and SmartPLS yet, read our note on AMOS vs SmartPLS first and then fix the sample.

Allow for the forms that never come back

The formula gives you usable responses, and the forms you send out must be more. Say you need 292 usable responses and you expect 7 out of 10 to come back complete. Then send around 292 / 0.7 = 418. When a staff survey goes through managers, a return of 60% to 80% is common. For an online link shared in WhatsApp groups the return can be much lower. It is hard to calculate also, since nobody knows how many people saw the link.

Some forms will come back half filled, and you have to plan for those also. If half the scale items are left blank, mostly that form can’t go into the main analysis. Collect more than the minimum for this reason. Keep a record of how many forms went out, how many came back and how many you rejected, and give all three counts in the thesis.

Mistakes examiners pick up

Cochran’s formula with a convenience sample. Cochran’s formula is made for random sampling. If you gave forms to anyone willing, the size from the formula is still reasonable. But you can’t claim the margin of error that goes with it. Say the sampling was non-probability and add it under limitations.

Forgetting the subgroups. Take an objective that compares five districts. Then 385 overall does not tell you much, because each district needs enough people. If one of the district has only 30 respondents and another has 150, that comparison becomes weak.

Deciding the number only after the data is in. Some scholars collect 212 forms first and later search for a formula that gives 212. Examiners catch this easily. Decide the number before data collection and keep the working in your synopsis itself.

Survey formulas for interview studies. Qualitative studies do not use Cochran or G*Power. In interviews you keep going till saturation, that is, till the new interviews are not giving any new codes. Guest, Bunce and Johnson (2006) reported that in a fairly uniform group most codes came up within 12 interviews. For showing saturation, see our guide on qualitative analysis in NVivo.

What to write in Chapter 3

In the method chapter, the sample size paragraph should have five things. Start with the population, its size if you know it, and the sampling method. After that give the formula or tool, every value you entered, and the final number along with the response rate.

A sample paragraph is given here for reference. “The population consisted of 1,200 permanent employees of the selected units. Using Cochran’s formula (95% confidence, 5% margin of error, p = 0.5) with the finite population correction, the required sample was 292. Since a response rate of about 70% was expected, 420 questionnaires were given out. Of these, 318 usable responses came back. An a priori power analysis in G*Power 3.1 (f² = 0.15, α = .05, power = .80, five predictors) showed a minimum of 92 for the regression model, so the sample was adequate.”

In that paragraph each number can be traced back. Suppose the examiner asks in the viva why you took 292. You can point to this paragraph, and the working is all there. After the sample is fixed, each objective has to be matched with its test. For that step, read our guide on choosing a statistical test.