Sample Size and Statistical Power
Learn why underpowered studies waste resources and mislead, how to calculate the sample size you need for any study design, and what effect size to anticipate.
You'll learn
How to calculate required sample size and justify it before data collection.
Use this when
You are planning a study and need to determine how many participants are needed.
Why Sample Size Matters
Sample size is one of the most critical decisions in research design. Too small a sample leads to underpowered studies that miss real effects, waste resources, and may produce misleading negative results. Too large a sample wastes money and exposes more participants to risk unnecessarily.
- ●Statistical power (1-β): Probability of detecting a real effect of a given size. Most studies aim for 80% power.
- ●A study with 50% power is like flipping a coin to decide whether the treatment works.
- ●Underpowered studies contribute to the replication crisis in science.
- ●Power is calculated BEFORE data collection, not after.
The Four Determinants of Power
Power depends on four interrelated factors. Change any one, and the others must compensate:
- 1.Effect size (δ): The size of the difference you expect to detect. Larger effects require smaller samples.
- 2.Alpha (α): Your significance threshold. α = 0.05 is standard; α = 0.01 requires a larger sample.
- 3.Power (1-β): The probability of detecting the effect. 80% is conventional; 90% is used for pivotal trials.
- 4.Sample size (N): More subjects = more power. This is the variable you solve for.
These four factors are mathematically linked. Fix three of them and you can calculate the fourth. That is what power analysis tools (G*Power, R's pwr package, online calculators) do.
Estimating Effect Size
The most uncertain input in a power analysis is the expected effect size. Estimate it from:
- ●Previous studies in your area (meta-analyses are the best source)
- ●Pilot study data (be cautious — pilot studies are often too small to estimate effect reliably)
- ●The minimum clinically important difference (MCID) — the smallest difference that is clinically meaningful
- ●If you have no estimate, use conventional small/medium/large thresholds as a last resort
Never inflate effect sizes to reduce the required sample size — this is research misconduct. Underpowered studies designed to find implausibly large effects are likely to produce false positives.
Sample Size for Common Designs
- ●Independent t-test (α=0.05, power=0.80, d=0.5): ~64 per group (128 total)
- ●ANOVA with 3 groups (α=0.05, power=0.80, f=0.25): ~53 per group (159 total)
- ●Chi-square (α=0.05, power=0.80, V=0.3): ~88 per cell (minimum)
- ●Logistic regression: ≥10 events per predictor variable
- ●Survival analysis: Similar to logistic regression based on expected number of events, not total N
Power Analysis Tools
- ●G*Power: Free desktop software covering nearly all study designs
- ●R pwr package: pwr.t.test(), pwr.anova.test(), pwr.chisq.test()
- ●Python statsmodels: TTestIndPower, FTestAnovaPower
- ●Online calculators: clinicaltrials.gov, openepi.com
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