Enter the effect size, alpha level, power, and number of predictors into the calculator to determine the required sample size for a regression analysis. This calculator helps in planning a study by estimating the minimum number of observations needed.

Regression Sample Size Calculator

Enter any 4 values to calculate the missing variable

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Regression Sample Size Formula

Use this calculator to estimate the number of observations needed for a multiple regression analysis from four core planning inputs: expected effect size, significance level, desired power, and number of predictors. It is most useful during study design, when you need to know whether a proposed dataset is large enough to detect a meaningful relationship.

N = \frac{(Z_{1-\alpha/2}+Z_{1-\beta})^2}{f^2} + k + 1
  • N = required sample size
  • Z1-α/2 = critical z-value for the selected two-sided alpha level
  • Z1-β = critical z-value associated with the desired statistical power
  • f² = Cohen’s regression effect size
  • k = number of predictor terms in the model, excluding the intercept

The formula shows the main tradeoff clearly: smaller effects require larger samples, higher power requires larger samples, stricter alpha levels require larger samples, and adding more predictors increases the total sample needed.

What the Inputs Mean

Input What it Represents Typical Planning Choice
Effect size (f²) How strong the regression signal is expected to be Estimated from prior studies, pilot data, or expected R²
Alpha (α) Probability of a Type I error 0.05 is common for a two-sided test
Power (1−β) Probability of detecting the effect if it truly exists 0.80 or 0.90 are common targets
Predictors (k) Total number of estimated predictor terms Count each dummy variable, interaction, and nonlinear term separately

How to Estimate Effect Size

If you know the expected model R² instead of f², convert it before using the calculator.

f^2 = \frac{R^2}{1-R^2}
R^2 = \frac{f^2}{1+f^2}

Common interpretation ranges for regression effect size are shown below.

f² General Interpretation Practical Meaning
0.02 Small Subtle effect; often needs a much larger sample
0.15 Medium Moderate signal; often used in planning examples
0.35 Large Strong relationship; usually needs fewer observations

Because f² is in the denominator of the formula, even a modest reduction in expected effect size can increase the required sample substantially.

How to Use the Calculator

  1. Estimate the expected effect size using prior evidence, pilot data, or an anticipated R².
  2. Select the alpha level for the hypothesis test.
  3. Choose the desired statistical power.
  4. Count the number of predictor terms in the planned regression model.
  5. Enter any four values to solve for the missing one.
  6. Round the resulting sample size up to the next whole number.

Example

Suppose you expect a medium effect size of 0.15, want a two-sided alpha of 0.05, target 80% power, and plan to include 3 predictors.

N = \frac{(1.96+0.84)^2}{0.15} + 3 + 1 \approx 56.27

You would plan for at least 57 observations. In practice, many analysts collect more than the minimum to account for missing data, unusable records, or later model refinement.

If Your Sample Size Is Fixed

When the dataset size is already known, the same relationship can be rearranged to estimate the smallest effect size the study is approximately powered to detect.

f^2 = \frac{(Z_{1-\alpha/2}+Z_{1-\beta})^2}{N-k-1}

This is helpful when evaluating whether an existing dataset is likely to detect only large effects or whether it is large enough to identify smaller, more realistic ones.

Planning Tips for Better Regression Studies

  • Count terms, not just variables. If a categorical variable uses 3 dummy variables, all 3 count toward k.
  • Include interactions and polynomial terms. Each added coefficient increases the number of predictors.
  • Round up, then add a buffer. A small oversample helps preserve power after exclusions and missing values.
  • Use realistic effect sizes. Overly optimistic assumptions make required sample sizes look smaller than they should be.
  • Watch for multicollinearity. Strong correlation among predictors can make estimates unstable even when total N appears adequate.
  • Match the method to the model. This calculator is best suited to standard multiple linear regression planning, not specialized models such as mixed models, penalized regression, or clustered designs.

Common Questions

Does k include the intercept?
No. The intercept is already accounted for by the +1 term in the formula.
Should interaction terms count as predictors?
Yes. Any estimated coefficient beyond the intercept should generally be counted in k, including interactions, spline terms, and polynomial terms.
Why is the required sample size sometimes much larger than expected?
Small effects, higher desired power, stricter alpha levels, and larger models all increase the required sample quickly. In many cases, effect size is the biggest driver.
Can this be used for logistic regression?
Not directly. Logistic regression and other non-linear models usually require model-specific sample size approaches.

When This Calculator Is Most Useful

This regression sample size calculator is especially helpful when designing surveys, experiments, observational studies, academic research projects, and business analyses where multiple predictors will be used to explain a continuous outcome. It provides a fast approximation for planning, checking feasibility, and comparing alternative study designs before data collection begins.