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Clinical Tools · Biostatistics · ICH E9

Clinical Trial Sample Size Calculator

Calculate required sample size for parallel group, two-proportion, and crossover clinical trials. Includes power analysis, dropout adjustment, and power curve table.

Quick Answer

Clinical trial sample size for a two-arm parallel trial with a continuous endpoint uses n = 2 × [(zα/2 + zβ)² × σ²] / δ² per group, where δ is the minimum clinically important difference and σ is the pooled standard deviation. With α = 0.05 (two-sided) and 80% power, z values are 1.96 and 0.842. Inflate for expected dropout before finalizing the protocol synopsis and statistical analysis plan (SAP). Confirm with a qualified biostatistician per ICH E9.

Core Formula — Two-Group Parallel (Continuous)
n = 2 × [(zα/2 + zβ)² × σ²] / δ²
δ = minimum difference to detect    σ = pooled SD    n = per group
zα/2 = 1.96 (α=0.05)   zβ = 0.842 (power=0.80)   zβ = 1.282 (power=0.90)

Two-Group Parallel — Continuous Outcome

Calculate sample size per group for a continuous endpoint using alpha, power, effect size, and dropout adjustment.

Significance and power
Effect size
Trial design
n per group
—
subjects
Total N
—
subjects
After Dropout Adj.
—
to enroll

Two Proportions — Binary / Event-Rate Outcome

Calculate sample size per group for binary endpoints using control and treatment event rates.

Significance and power
Event rates
Trial design
n per group
—
subjects
Total N
—
subjects
NNT
—
number needed to treat
After Dropout Adj.
—
to enroll

Crossover Trial — Within-Subject Design

In a crossover trial each subject receives both treatments. This halves the required sample size by eliminating between-subject variability, but requires a stable condition and an adequate washout period.

Significance and power
Effect size
Trial design
n (crossover)
—
subjects (each receives both)
Parallel equivalent
—
total (parallel design)
After Dropout Adj.
—
to enroll

How to Use the Sample Size Calculator

1
Choose the tab matching your primary endpoint: Two-Group Parallel for continuous outcomes, Two Proportions for binary event rates, or Crossover for within-subject designs.
2
Set two-sided alpha (default 0.05) and desired power (default 80%). These determine zα/2 and zβ in the sample size formula.
3
Enter effect size inputs: δ and σ for continuous endpoints, or p1 and p2 for proportions. Use protocol-prespecified MCID and variability from prior studies when available.
4
Apply dropout inflation (typically 10–20%) to convert evaluable-subject n into planned enrollment. Review the power curve table for sensitivity to power assumptions.
5
Document assumptions in the protocol synopsis and statistical analysis plan (SAP), then confirm final sample size with a qualified biostatistician per ICH E9.

Worked Example — Two-Arm Continuous Endpoint

Example calculation

Design: Two-group parallel, continuous primary endpoint, two-sided α = 0.05, power = 80%.

Inputs: δ = 5 (minimum difference to detect), σ = 10 (pooled SD), dropout = 15%.

Formula: n per group = 2 × (1.96 + 0.842)² × 10² / 5² = 2 × 7.84 × 100 / 25 ≈ 63 evaluable subjects per group (total N ≈ 126).

Dropout adjustment: 63 / 0.85 ≈ 75 per group → enroll approximately 150 subjects total.

Interpretation: With these assumptions, the trial has 80% power to detect a 5-unit mean difference. Validate σ and δ against historical data before protocol lock.

Common Scenario Reference Table

Minimum sample sizes per group for common effect sizes (two-sided α=0.05, power=0.80, σ=10, allocation 1:1):

Standardized Effect Size (δ/σ) Small (0.2) Medium (0.5) Large (0.8) Very Large (1.0)
Power 70%1242196
Power 80%19732139
Power 85%246401611
Power 90%313512114
Educational purpose only: These calculations are for planning and educational use. Final sample size must be confirmed with a qualified biostatistician and should account for interim analyses, adaptive design features, multiplicity corrections, and protocol-specific considerations (ICH E9, ICH E9(R1)).

Pharma & Clinical Trial Context

Sample size is a core protocol decision documented in the protocol synopsis and fully justified in the statistical analysis plan (SAP). Sponsors must prespecify the primary estimand, alpha, power, expected effect size, variability assumptions, dropout handling, and any interim analysis or adaptation rules before first patient in. Regulators and ethics committees expect transparent, reproducible sizing rationale aligned with ICH E9.

For binary proportion endpoints, the Two Proportions tab reports NNT alongside sample size — use our NNT Calculator for deeper benefit interpretation. After sizing, translate expected treatment effects into confidence intervals with the Confidence Interval Calculator. Operationalize enrollment with the Randomization Generator and draft protocol sections via the Protocol Synopsis tool.

When the SAP specifies Bayesian posterior thresholds or historical borrowing, compare frequentist assumptions with our Bayesian Sample Size Calculator. Adaptive designs with sample size re-estimation require simulation beyond static formulas — see FDA adaptive design guidance in the evidence section below.

Evidence & Sources

Frequently Asked Questions

For a two-group parallel trial with a continuous outcome, use n = 2 × [(zα/2 + zβ)² × σ²] / δ² per group, where zα/2 is the critical value for significance (1.96 for α = 0.05 two-sided), zβ is the critical value for power (0.842 for 80% power), σ is the pooled standard deviation, and δ is the minimum clinically important difference. Always inflate for expected dropout before locking enrollment targets.
Statistical power (1 − β) is the probability that the trial will detect a true treatment effect if one exists. Conventionally set at 0.80 (80%) or 0.90 (90%), it determines sample size. A power of 80% means there is a 20% chance of a false-negative result (Type II error). Higher power requires larger sample sizes and should be justified in the statistical analysis plan.
Alpha (α) is the probability of a false-positive result — rejecting the null hypothesis when no true effect exists. Two-sided α = 0.05 is the most common confirmatory standard (zα/2 = 1.96). Sample size increases as α decreases because stricter significance requires more evidence. Multiplicity adjustments for interim analyses or secondary endpoints may require a smaller effective α.
Effect size should reflect the minimum clinically important difference (MCID) or minimum detectable effect prespecified in the protocol, not the effect you hope to see. For continuous endpoints, enter δ (difference to detect) and σ (pooled SD, often from prior studies or meta-analysis). For binary endpoints, enter control and treatment event rates p1 and p2. Overly optimistic effect sizes underestimate required enrollment.
Sample size formulas return the number of evaluable subjects needed to achieve the desired power. Since some enrolled subjects will drop out, withdraw consent, or have missing primary endpoint data, planned enrollment must be inflated. A 10–20% dropout adjustment is typical: adjusted N = calculated N / (1 − dropout rate). For 15% dropout, divide by 0.85.
A crossover trial requires roughly half the subjects of a parallel trial because each subject serves as their own control, reducing within-subject variability. The crossover formula uses within-subject SD (σw) rather than pooled SD: n = [(zα/2 + zβ)² × 2σw²] / δ². Crossover designs are unsuitable when the condition changes over time, when carryover effects persist, or when treatment sequences cannot be blinded.
Superiority trials test whether a new treatment is better than control by a clinically meaningful margin. Non-inferiority trials test whether the new treatment is not unacceptably worse than an active comparator, using a prespecified non-inferiority margin (Δ). Non-inferiority designs often require larger samples because the null hypothesis differs and one-sided alpha is common. This calculator uses standard two-sided superiority-style formulas; non-inferiority sizing needs margin-specific methods.
Continuous endpoints use mean difference and pooled SD in the parallel-group formula. Binary or event-rate endpoints use proportions p1 and p2 with a variance term based on p(1 − p). Rare events need very large samples; continuous endpoints with high variability (large σ) also inflate n. Choose the tab matching your primary endpoint type and confirm estimand definitions in the SAP.
Adaptive trials may include pre-planned sample size re-estimation, arm dropping, or enrichment based on interim data. Initial sample size may be set with a range or re-estimation rule rather than a single fixed n. FDA adaptive design guidance expects pre-specified adaptation rules and control of Type I error or documented operating characteristics. Do not treat a static calculator result as sufficient for adaptive protocols without simulation.
Yes. Final sample size must be confirmed by a qualified biostatistician before protocol finalization. They account for stratification, covariate adjustment, missing data assumptions, multiplicity, interim analyses, cluster designs, and regulatory expectations. This calculator provides educational planning estimates only and does not replace SAP-ready sample size justification.
ICH E9 (Statistical Principles for Clinical Trials) requires that sample size be scientifically justified in the protocol with explicit assumptions about the primary endpoint, effect size, variability, alpha, and power. ICH E9(R1) adds estimand framework guidance — the treatment effect of interest must align with how missing data and intercurrent events are handled. Sample size should be documented before unblinding and unchanging unless pre-specified adaptation rules apply.
Use Bayesian sample size when the SAP specifies a posterior probability threshold, when borrowing historical control data is justified, or when planning adaptive Phase II designs with Bayesian interim rules. For standard confirmatory superiority trials with fixed alpha and power, frequentist sizing (this calculator) is usually primary. Compare planning assumptions with our Bayesian Sample Size Calculator for binary posterior-probability targets.

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β Bayesian Sample Size NNT NNT Calculator CI Confidence Interval Calculator RND Randomization Generator SAP Protocol Synopsis

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