Free Power and Sample Size Calculator for Clinical Research

This calculator covers two designs: comparing two independent proportions (e.g. an event rate in a treatment group vs. a control group) and comparing two independent means (e.g. a continuous outcome between two arms), each for a two-sided test at your chosen significance level and power.

When to use it

Use this calculator when you are planning a new prospective study with two independent, parallel groups and want to estimate how many participants per group you need to detect a specified difference. It is a planning aid for the design stage, not a substitute for a full protocol-level power analysis reviewed by a statistician, particularly for a funded or registered trial.

Two proportions

Uses the standard normal-approximation formula: n = (z_α/2√(2p̄(1−p̄)) + z_β√(p₁(1−p₁)+p₂(1−p₂)))² / (p₁−p₂)², where p̄ is the average of the two proportions. Suitable for planning a study comparing two independent event rates, such as a binary clinical outcome between a treatment and control arm.

Two independent means

Uses n = 2(z_α/2+z_β)²σ² / δ², assuming equal variance in both groups and a two-sided test - the standard formula behind most sample-size tables for a two-sample t-test. Suitable for planning a study comparing a continuous outcome (e.g. a score or lab value) between two independent groups.

What alpha, power, and effect size mean

• Significance level (α): the chance of a false positive (Type I error) you are willing to accept, usually 0.05.
• Power (1−β): the chance of detecting a true effect of the size you specified, usually 80% or 90%.
• Effect size: the difference between groups you want the study to be able to detect - the two proportions, or the two means and common standard deviation.

Sample size for a randomized controlled trial

For a two-arm parallel-group RCT with 1:1 allocation and a binary or continuous primary outcome, the calculator gives the number of participants needed in each arm. Work through it in this order:

1. Fix the primary outcome. Sample size is calculated for one outcome; pick the one your conclusions will rest on.
2. Choose the difference worth detecting. Use the smallest difference that would matter clinically, not the (often inflated) effect seen in a small pilot.
3. Take the control-arm value from prior data. That means the expected event rate for a binary outcome, or the standard deviation for a continuous one, ideally from a similar population.
4. Set α and power. Two-sided α = 0.05 and 80% or 90% power are the usual choices.
5. Inflate for dropout. Divide the per-group result by (1 − expected dropout rate) and round up.

Worked example for a binary outcome

Suppose you expect an event rate of 30% in one group and 50% in the other, with two-sided α = 0.05.

• At 80% power: 93 participants per group, 186 in total.
• At 90% power: 124 per group, 248 in total.

Enter 0.3 and 0.5 in Two Proportions mode to reproduce this. Asking for 90% instead of 80% power costs about a third more participants.

Worked example for a continuous outcome

Suppose two groups have means of 10 and 12 on a scale with a common standard deviation of 5, so the standardized difference is 2 / 5 = 0.4 (see the effect size calculator), with two-sided α = 0.05.

• At 80% power: 99 per group, 198 in total.
• At 90% power: 132 per group, 264 in total.

Enter 10, 12 and 5 in Two Means mode to reproduce this.

Worked example: RCT with dropout

A hypothetical trial expects 30% of control participants and 20% of intervention participants to have the event, with α = 0.05 and 80% power. The calculator gives 294 per group. If you expect 15% to drop out before the outcome is measured, recruit 294 / (1 − 0.15) = 345.9, rounded up to 346 per group, 692 in total. The dropout inflation is done by hand because the calculator does not apply it.

How to report a sample-size calculation in a protocol

State the design (two proportions or two means), the assumed values for each group, the significance level, the target power, the resulting per-group and total sample size, and any inflation applied for anticipated dropout or non-adherence - the last of which this calculator does not compute for you (see below).

When not to use this calculator

It also does not build in an allocation-ratio input (it assumes equal group sizes) or a dropout/loss-to-follow-up adjustment - inflate the per-group result yourself by your expected dropout rate (e.g. divide by (1 − dropout rate)) when planning actual recruitment, and consult a statistician for unequal allocation ratios or any of the designs listed above. Planning a non-inferiority trial specifically? Use the non-inferiority sample size calculator instead - it uses the correct one-sided test and margin-based formula.

References

Chow S-C, Shao J, Wang H, Lokhnygina Y. Sample Size Calculations in Clinical Research. 3rd ed. Chapman & Hall/CRC; 2017. Cochrane Handbook for Systematic Reviews of Interventions (current version), training.cochrane.org/handbook.