Find the sample size you need, check your statistical power, or discover the smallest effect you can detect. Turn your assumptions into a practical recruitment plan.
Need a sample size for survey precision? Use the margin-of-error calculator
Choose a design and calculate to see the required sample, recruitment target, and power curves.
New to power analysis? Start hereGet properly formatted citations for academic work. Citing the tools you use keeps your methods reproducible and reviewable.
StatsCalculators Team. (2026). Sample Size & Power Analysis Calculator. StatsCalculators. Retrieved September 19, 2026 from https://statscalculators.com/calculators/hypothesis-testing/sample-size-and-power-analysis-calculator
Understand the choices behind your sample size. Choose a topic below for a practical explanation, worked examples, or formulas, R and Python code, and their outputs.
Choose based on your outcome and how observations are related.
Independent t-test · Continuous outcome · two groups
Paired t-test · Continuous outcome · paired measurements
Two-proportion test · Binary outcome · two independent groups
One-way ANOVA · Continuous outcome · equal-size groups
Chi-square test · Category frequencies · goodness of fit or association
Multiple regression · Continuous outcome · overall model R²
Planning a survey for a desired margin of error? Use the margin-of-error sample size calculator. Clustered, repeated-measures, and survival designs need their own planning methods.
Use a meaningful effect and make uncertainty explicit.
Use prior research, a realistic pilot estimate, or a practical threshold. For example, a 5-point difference with a standard deviation of 10 gives d = 0.5. The effect-size helper beside the input does this conversion for you.
“Small,” “medium,” and “large” conventions are starting points, not a justification. Compare several plausible effects: smaller effects usually need larger samples.
If the specified effect and model assumptions hold, power is the probability of rejecting the null hypothesis. At 80% power, the corresponding false-negative probability is 20%. It is not an 80% probability that your hypothesis is true.
Compare 80%, 90%, and 95% to see the extra recruitment each target requires.
Alpha controls the false-positive probability under the null. A common choice is 0.05; stricter levels can be appropriate. A two-sided test allows a difference in either direction.
Use a one-sided test only when its direction is justified in advance. Multiple testing and repeated looks at the data may require a different analysis plan.
The statistical minimum and your recruitment target answer different questions.
Independent groups show each group and the total. Paired tests count complete pairs. Regression and chi-square count total observations.
Round recruitment up within each group. A target of 64 complete participants per group with 10% dropout means recruiting 72 per group, or 144 overall.
Copy the sample-size justification into your draft, then explain why you selected the effect size. Keep the settings link with your study plan.
A two-percentage-point improvement may sound small, but detecting it reliably can require thousands of observations per group. Statistical significance in one observed dataset is not the same as adequate power when planning a study.
Use Power to evaluate a specified, meaningful effect, or Detectable effect to find the smallest effect your available sample can detect at your target power. Avoid interpreting power calculated from the observed effect after a study as additional evidence about its result.
Open a test to see its assumptions and verify the calculation with R or Python. Every example includes its output for sample size, power, or detectable effect.
Two independent groups, independent observations, a continuous outcome, and a common within-group variance. The calculation uses the pooled-variance t-test, not Welch’s test.
Here r = n₂/n₁. This normal approximation explains the relationship between inputs. The calculator uses the noncentral t distribution and checks rounded group sizes against the target power.
Run either implementation with the inputs shown. The output below was captured by running the code. Minor differences in the last decimal places can arise from numerical solvers.
Requires statsmodels, NumPy, and SciPy.
from statsmodels.stats.power import TTestIndPower
import numpy as np
# Parameters
effect_size = 0.5 # Cohen's d (medium effect)
alpha = 0.05 # Significance level
power = 0.8 # Desired power
# Create power analysis object
analysis = TTestIndPower()
# Calculate sample size (per group)
sample_size = analysis.solve_power(
effect_size=effect_size,
alpha=alpha,
power=power,
ratio=1.0, # Equal group sizes
alternative='two-sided'
)
print(f"Sample size per group: {np.ceil(sample_size):.0f}")
# For unequal group sizes, n2/n1 = 2
ratio = 2.0
sample_size_unequal = analysis.solve_power(
effect_size=effect_size,
alpha=alpha,
power=power,
ratio=ratio,
alternative='two-sided'
)
n1 = np.ceil(sample_size_unequal)
n2 = np.ceil(n1 * ratio)
print(f"Unequal groups (ratio 1:{ratio:.0f}):")
print(f"- Group 1 sample size: {n1:.0f}")
print(f"- Group 2 sample size: {n2:.0f}")
print(f"- Total sample size: {n1 + n2:.0f}")Sample size per group: 64 Unequal groups (ratio 1:2): - Group 1 sample size: 48 - Group 2 sample size: 96 - Total sample size: 144
Independent pairs, approximately normal within-pair differences, and equal marginal standard deviations at the two measurements. For repeated measurements on one person, one complete pair is one person with both measurements.
Enter d standardized by a single-measurement SD. The calculator uses the expected within-pair correlation ρ to convert it to dz. Do not enter a difference-score effect size dz and apply the correlation adjustment a second time.
Run either implementation with the inputs shown. The output below was captured by running the code. Minor differences in the last decimal places can arise from numerical solvers.
Requires statsmodels, NumPy, and SciPy.
from statsmodels.stats.power import TTestPower, TTestIndPower
import numpy as np
# Parameters
d = 0.5 # Effect size (Cohen's d)
alpha = 0.05 # Significance level
power = 0.8 # Desired power
rho = 0.6 # Correlation between pairs
# Adjusted effect size for paired design
# For paired t-test, effective effect size is larger due to correlation
d_adj = d / np.sqrt(2 * (1 - rho))
# Create power analysis object
analysis = TTestPower()
# Calculate sample size (number of pairs)
sample_size = analysis.solve_power(
effect_size=d_adj,
alpha=alpha,
power=power,
alternative='two-sided'
)
print(f"Sample size (number of pairs): {np.ceil(sample_size):.0f}")
# Compare designs with the same single-measurement d.
independent_n = TTestIndPower().solve_power(
effect_size=d, alpha=alpha, power=power, ratio=1.0,
alternative='two-sided'
)
n_each = int(np.ceil(independent_n))
print(f"Independent design: {n_each} per group ({2 * n_each} total)")
print(f"Paired design: {np.ceil(sample_size):.0f} complete pairs")Sample size (number of pairs): 28 Independent design: 64 per group (128 total) Paired design: 28 complete pairs
Two independent groups with equal allocation and binary outcomes. The normal approximation is intended for sufficiently large expected counts; rare events or very small groups may require an exact or simulation-based design.
Here p̄ = (p₁ + p₂)/2 and n is per group. The calculator solves the pooled-null, unpooled-alternative normal power equation, including both rejection tails for a two-sided test. The expression above is its usual sample-size approximation. A 10% → 12% change is 2 percentage points, or 20% relative growth.
Run either implementation with the inputs shown. The output below was captured by running the code. Minor differences in the last decimal places can arise from numerical solvers.
Requires statsmodels, NumPy, and SciPy.
from math import ceil
from scipy.optimize import brentq
from statsmodels.stats.proportion import power_proportions_2indep
# Equal groups; pooled-null, unpooled-alternative normal approximation.
p1 = 0.6 # Baseline proportion
p2 = 0.4 # Expected proportion
alpha = 0.05
power = 0.8
def power_at(n):
return power_proportions_2indep(
diff=p2 - p1, prop2=p1, nobs1=n, ratio=1,
alpha=alpha, alternative='two-sided', return_results=False
)
n = ceil(brentq(lambda n: power_at(n) - power, 2, 1000000))
print(f"Sample size per group: {n}")
print(f"Total sample size: {2 * n}")
print(f"Achieved power: {power_at(n):.4f}")Sample size per group: 97 Total sample size: 194 Achieved power: 0.8003
Independent observations, equal group sizes, a common within-group variance, and approximately normal errors. This is an omnibus test of the group means; it does not guarantee power for every post-hoc comparison.
N is the total number of observations and k is the number of groups. The calculator uses the noncentral F distribution, rounds each group up, and reports both per-group and total counts. Cohen’s f is a different effect-size measure from Cohen’s d.
Run either implementation with the inputs shown. The output below was captured by running the code. Minor differences in the last decimal places can arise from numerical solvers.
Requires statsmodels, NumPy, and SciPy.
from statsmodels.stats.power import FTestAnovaPower
import numpy as np
# Parameters
k = 3 # Number of groups
f = 0.25 # Cohen's f (medium effect size)
alpha = 0.05 # Significance level
power = 0.8 # Desired power
# Create power analysis object
analysis = FTestAnovaPower()
# Calculate TOTAL sample size first (statsmodels returns total, not per-group)
total_sample_size = analysis.solve_power(
effect_size=f,
nobs=None,
alpha=alpha,
power=power,
k_groups=k
)
# Handle array conversion (statsmodels may return array)
if isinstance(total_sample_size, np.ndarray):
total_sample_size = float(total_sample_size.item())
else:
total_sample_size = float(total_sample_size)
# Divide by k to get per-group size
per_group = total_sample_size / k
n_per_group = int(np.ceil(per_group))
balanced_total = n_per_group * k
print(f"Sample size per group: {n_per_group}")
print(f"Total sample size: {balanced_total}")
# Also calculate what power we'd have with different sample sizes
for n_per_group in [30, 50, 70, 100]:
# Note: nobs must be TOTAL sample size (per-group × k)
pwr = analysis.power(
effect_size=f,
nobs=n_per_group * k,
alpha=alpha,
k_groups=k
)
print(f"Power with n={n_per_group} per group: {pwr:.3f} ({pwr*100:.1f}%)")Sample size per group: 53 Total sample size: 159 Power with n=30 per group: 0.540 (54.0%) Power with n=50 per group: 0.780 (78.0%) Power with n=70 per group: 0.907 (90.7%) Power with n=100 per group: 0.978 (97.8%)
Independent observations and mutually exclusive categories. Use degrees of freedom for the intended test. Small expected cell counts can make the chi-square approximation inaccurate.
p₀ describes category proportions under the null and p₁ under the alternative. For a goodness-of-fit test with k categories and no estimated parameters, df = k − 1. For an r × c contingency table, df = (r − 1)(c − 1). N is always the total count.
Run either implementation with the inputs shown. The output below was captured by running the code. Minor differences in the last decimal places can arise from numerical solvers.
Requires statsmodels, NumPy, and SciPy.
from statsmodels.stats.power import GofChisquarePower
import numpy as np
# Parameters
df = 1 # Degrees of freedom
w = 0.3 # Cohen's w (medium effect size)
alpha = 0.05 # Significance level
power = 0.8 # Desired power
# Create power analysis object
analysis = GofChisquarePower()
# Calculate sample size
n = analysis.solve_power(
effect_size=w,
nobs=None,
alpha=alpha,
power=power,
n_bins=df + 1 # number of categories = df + 1
)
print(f"Sample size needed: {np.ceil(n):.0f}")
# Verify the power with this sample size
power_check = analysis.power(
effect_size=w,
nobs=np.ceil(n),
alpha=alpha,
n_bins=df + 1
)
print(f"Achieved power: {power_check:.4f} ({power_check*100:.2f}%)")Sample size needed: 88 Achieved power: 0.8035 (80.35%)
An overall F-test of R² against zero in a fixed multiple linear regression model, with independent, normally distributed, constant-variance errors. This does not calculate power for a single coefficient or an incremental block of predictors.
p is the number of predictor coefficients, excluding the intercept. Count the separate coefficients used to encode categorical variables. N is the total sample size. The noncentrality convention matches R’s pwr.f2.test.
Run either implementation with the inputs shown. The output below was captured by running the code. Minor differences in the last decimal places can arise from numerical solvers.
Requires statsmodels, NumPy, and SciPy.
from statsmodels.stats.power import FTestAnovaPower
import numpy as np
# Parameters
num_predictors = 5 # Number of independent variables
f_squared = 0.15 # Cohen's f² (medium effect size)
alpha = 0.05 # Significance level
power = 0.8 # Desired power
# Convert f² to f
cohens_f = np.sqrt(f_squared)
# Create power analysis object
analysis = FTestAnovaPower()
# For multiple regression: k_groups = num_predictors + 1
k_groups = num_predictors + 1
# Solve for required sample size
nobs_result = analysis.solve_power(
effect_size=cohens_f,
nobs=None, # This is what we want to solve for
alpha=alpha,
power=power,
k_groups=k_groups
)
# Round up to get total sample size
total_n = int(np.ceil(nobs_result))
print(f"Cohen's f: {cohens_f:.4f}")
print(f"Sample size needed: {total_n}")
print(f"(Minimum 10-15 observations per predictor recommended)")
# Verify the power with this sample size
power_check = analysis.power(
effect_size=cohens_f,
nobs=total_n,
alpha=alpha,
k_groups=k_groups
)
print(f"Achieved power: {power_check:.4f} ({power_check*100:.2f}%)")Cohen's f: 0.3873 Sample size needed: 92 (Minimum 10-15 observations per predictor recommended) Achieved power: 0.8042 (80.42%)
Method references: statsmodels power analysis, R two-proportion power, and the R pwr reference manual. Approximations and rounding conventions can produce small differences between tools.
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