A psychology student collects data from 18 participants, runs an independent-samples t-test, gets p = 0.12, and concludes the intervention had no effect. But the real problem was never the intervention. With only 9 people per group and a medium effect size of d = 0.5, that study had roughly 12% power. There was an 88% chance of missing a genuine effect even if one existed. The sample was too small before a single questionnaire was handed out.
Why Sample Size Matters
Statistical power is the probability that your test will detect a real effect when one actually exists. If your study has 80% power, there is a 20% chance you will miss a true effect and incorrectly conclude nothing is going on. That 80% threshold is the standard convention across psychology, education, medicine, and most social sciences. Some fields, like clinical trials testing new drugs, require 90%.
Power depends directly on sample size. More participants means more information, tighter confidence intervals, and a better chance of catching real differences between groups. Run the same independent-samples t-test with d = 0.5 on 20 participants per group and power reaches about 34%. With 64 per group it hits 80%. The relationship is not linear; doubling participants does not double power, but it always pushes it upward.
The connection works in both directions. Collect too few participants and you waste everyone's time, including your own, because the study was never capable of producing a meaningful result. Collect far more than necessary and you spend months recruiting when the answer was already clear at half the sample. Power analysis lets you find the number where additional effort stops paying off. You should run it before you collect a single data point.
The Four Inputs You Need
Every power analysis requires four values. You set three and solve for the fourth. In practice, you almost always solve for sample size, since that is the thing you need to plan before data collection starts.
Alpha level (significance threshold). This is the probability of a false positive, the chance you declare an effect exists when it does not. Convention sets it at 0.05. Some researchers use 0.01 for stricter control, which requires a larger sample to maintain the same power.
Power (1 minus beta). Beta is the probability of a false negative. Power of 0.80 means beta is 0.20. You accept a 20% risk of missing a real effect. Raising power to 0.90 reduces that risk but demands more participants.
Effect size. This quantifies how large the difference or relationship is that you expect to find. Cohen's d measures the standardized mean difference between two groups. Cohen's f does the same for ANOVA with three or more groups. Pearson's r serves as the effect size for correlations. If you do not know the expected effect size from prior literature, Cohen's conventions offer a starting point: small (d = 0.2), medium (d = 0.5), and large (d = 0.8). Our effect size guide covers how to choose and interpret these values in detail.
Test type. Different statistical tests have different mathematical properties and different sample size requirements. A paired-samples t-test needs fewer participants than an independent-samples t-test at the same effect size because within-subject designs reduce variability. A correlation requires a different formula entirely. You need to know which statistical test you will run before you can calculate how many participants that test needs.
Practical Examples With G*Power
G*Power is a free desktop application (available for Windows and Mac) that handles power analysis for most common tests. Open it, select your test family, pick "A priori: compute required sample size," enter your parameters, and read the result. Here are three scenarios we run into regularly in client projects.
Independent-samples t-test
A nursing student wants to compare pain scores between a control group and an intervention group. She expects a medium effect (d = 0.5) based on two prior studies in her literature review. In G*Power: test family is "t tests," statistical test is "Means: difference between two independent means (two groups)," tail is two-tailed, alpha is 0.05, power is 0.80, and effect size d is 0.5. The result: 64 participants per group, 128 total. With 50 per group the power drops to about 70%, meaning she faces a 30% chance of missing a real difference.
One-way ANOVA with three groups
An education researcher compares test scores across three teaching methods. He estimates a medium effect of f = 0.25 (equivalent to roughly d = 0.5 in ANOVA terms). In G*Power: test family is "F tests," statistical test is "ANOVA: fixed effects, omnibus, one-way," alpha is 0.05, power is 0.80, number of groups is 3, and effect size f is 0.25. The result: 53 participants per group, 159 total. If he can only recruit 30 per group (90 total), power drops to around 61%.
Pearson correlation
A marketing student hypothesizes a positive relationship between social media usage hours and brand recall scores. She expects a small-to-medium correlation of r = 0.30 based on similar studies. In G*Power: test family is "Exact," statistical test is "Correlation: bivariate normal model," tail is two-tailed, alpha is 0.05, power is 0.80, and the correlation under the alternative hypothesis is 0.30. The result: 84 participants total. If she expects a weaker correlation of r = 0.20, the required sample jumps to 193.
Rules of Thumb When Power Analysis Is Not Possible
Sometimes you cannot run a formal power analysis. Perhaps no prior studies exist in your area, so there is no reasonable effect size estimate. Perhaps your design is too complex for standard G*Power inputs. In those cases, field-tested rules of thumb can help you set a defensible minimum.
For multiple regression, a common guideline is 10 to 15 participants per predictor variable. A model with 4 predictors should have at least 40 to 60 cases. Some methodologists push the ratio to 20:1 for more stable coefficient estimates, especially when predictors are correlated with each other. With 5 predictors and only 25 participants, the regression coefficients become unreliable and small changes in the data can flip their sign.
For factor analysis (exploratory, as used in questionnaire validation), the minimum ratio is 5 participants per item, and many textbooks recommend 10:1. A 20-item scale needs at least 100 respondents. Below that threshold, factor loadings become unstable and spurious factors may appear. With 200 or more respondents, the factor structure is usually reproducible.
For chi-square tests, the expected frequency in each cell should be at least 5. If your contingency table has 6 cells, you need enough participants so that even the smallest cell has an expected count of 5 or more. With a 2x3 table and evenly distributed categories, that means roughly 30 participants minimum. Uneven distributions push the requirement higher.
These are floors, not targets. If you can run a formal power analysis, always prefer it over a rule of thumb.
What Goes Wrong With the Wrong Sample Size
Too few participants means an underpowered study. The consequences are specific and predictable. Your confidence intervals will be wide, sometimes so wide that they span both positive and negative values even when the true effect is clearly in one direction. Your p-values will hover above 0.05 for effects that genuinely exist. You will conclude "no significant difference" and write that into your thesis, but the correct conclusion would have been "not enough data to tell." Reviewers and supervisors who understand power analysis will notice. We have seen committee members reject thesis drafts specifically because the student collected 15 participants per group for an effect that required 50.
Too many participants creates the opposite problem. With a large enough sample, even trivially small effects become statistically significant. An independent-samples t-test with 5,000 per group will produce p < 0.001 for a difference of d = 0.05, which corresponds to roughly half a point on a 100-point scale. The result is statistically significant and practically meaningless. Resources that went into recruiting those 10,000 participants could have served a different study. The practical cost matters especially in clinical or field research where each participant involves screening, consent, and follow-up. Academic Stats Agent can help you determine the right sample size for your specific design before you start recruiting.
Sample Size Reference Table
The table below shows required sample sizes for the most common tests used in student theses, assuming alpha = 0.05 and power = 0.80. Effect sizes follow Cohen's conventions. Use it as a quick reference when planning your study.
| Statistical Test | Effect Size | Required N |
|---|---|---|
| Independent t-test (two groups) | d = 0.5 (medium) | 64 per group (128 total) |
| Independent t-test (two groups) | d = 0.8 (large) | 26 per group (52 total) |
| Paired t-test | d = 0.5 (medium) | 34 pairs |
| One-way ANOVA (3 groups) | f = 0.25 (medium) | 53 per group (159 total) |
| One-way ANOVA (4 groups) | f = 0.25 (medium) | 45 per group (180 total) |
| Pearson correlation | r = 0.30 (medium) | 84 total |
| Pearson correlation | r = 0.20 (small) | 193 total |
| Chi-square (2x2) | w = 0.30 (medium) | 88 total |
| Multiple regression (5 predictors) | f² = 0.15 (medium) | 92 total |
| Logistic regression (1 predictor) | OR = 2.0 | 69 per group (138 total) |
These numbers assume two-tailed tests. One-tailed tests require slightly smaller samples but are only appropriate when you have a strong directional hypothesis and can justify ignoring effects in the opposite direction. Most thesis committees expect two-tailed tests unless there is a clear reason to do otherwise.
Key takeaway: Run a power analysis in G*Power before you collect data. You need four inputs: alpha (usually 0.05), power (usually 0.80), the expected effect size from prior literature, and your planned statistical test. If no prior effect size exists, use Cohen's conventions or the rules of thumb above. A study with too few participants wastes your effort; a study with too many wastes your time. The numbers in the reference table give you a starting point for the most common thesis designs.
Frequently Asked Questions
What is an adequate sample size for a study?
It depends on the expected effect size, significance level, and desired power. As a benchmark, detecting a medium effect (d = 0.5) with a two-tailed independent t-test at 80% power and alpha = .05 requires about 64 participants per group, or 128 in total.
How do I calculate statistical power?
Run an a priori power analysis in free software such as G*Power: choose the statistical test, enter the expected effect size (from previous studies or pilot data), set alpha (usually .05) and desired power (usually .80), and the software returns the required sample size.
Is 30 participants enough for a thesis study?
Often not. The "rule of 30" relates to the Central Limit Theorem, not to statistical power. With 30 participants per group you can reliably detect only large effects; smaller, more realistic effects require substantially larger samples, which is why a formal power analysis is expected in most theses.