Hypothesis Test Power Calculator
Explore statistical power and sample size for simple z/t tests on means. Compute power given sample size, or find the sample size needed to achieve a target power. Visualize power curves to understand tradeoffs.
Understanding Statistical Power: Detecting True Effects in Hypothesis Testing
Statistical power is the probability that a hypothesis test will correctly reject a false null hypothesis—in other words, the probability of detecting a true effect when one exists. Power equals 1 − β, where β is the Type II error rate (false negative). Before conducting a study, power analysis helps determine how large a sample you need to detect meaningful effects. After a study, understanding power helps interpret non-significant results—low power means you might have missed a real effect. This tool demonstrates power concepts for simple z-tests and t-tests on means, showing how power depends on effect size, sample size, significance level (α), and variability (σ). Whether you're a student learning statistical inference, a researcher designing experiments, a data analyst planning studies, or a business professional evaluating A/B tests, statistical power enables you to design studies that can detect meaningful effects and interpret results correctly.
For students and researchers, this tool demonstrates practical applications of statistical power, power analysis, and study design. The power calculations show how power depends on four factors (effect size, sample size, alpha, variability), how z-tests and t-tests differ, how one-sample and two-sample tests compare, how one-sided and two-sided tests affect power, and how to calculate required sample sizes. Students can use this tool to verify homework calculations, understand how power works, explore concepts like Type II errors and effect sizes, and see how different factors affect power. Researchers can apply power analysis to design experiments, determine sample sizes, assess study feasibility, and interpret non-significant results. The visualization helps students and researchers see how power changes with sample size or effect size.
For data analysts and business professionals, statistical power provides essential tools for designing experiments and A/B tests. Data analysts use power analysis to determine sample sizes for A/B tests, ensuring they can detect meaningful differences. Researchers use power analysis to design clinical trials, ensuring adequate sample sizes to detect treatment effects. Quality control engineers use power analysis to design experiments that can detect process improvements. Marketing professionals use power analysis to plan experiments that can detect campaign effects. Business analysts use power analysis to evaluate whether studies are adequately powered to detect business-relevant effects. These applications require understanding how to calculate power, determine sample sizes, and interpret power curves.
For the common person, this tool answers practical study design questions: How large should my sample be? Can I detect the effect I'm looking for? The tool calculates statistical power and required sample sizes, showing how study design affects the ability to detect effects. Taxpayers and budget-conscious individuals can use power analysis to understand study design, assess whether studies are adequately powered, and make informed decisions about research investments. These concepts help you understand how to design studies that can detect meaningful effects and interpret research results correctly, fundamental skills in modern data literacy.
⚠️ Educational Tool Only - Not for Clinical or Regulatory Design
This calculator is strictly for educational purposes to help understand how statistical power works mathematically. It is NOT designed for clinical trial design, regulatory submissions, or high-stakes research planning. Professional applications require: (1) Proper consideration of dropout rates, (2) Interim analyses and multiple comparisons, (3) Adaptive designs, (4) Regulatory requirements, and (5) Professional statistical software. This tool uses simplified formulas and normal approximations. For real studies, use dedicated power analysis software (G*Power, PASS, nQuery, Stata, R packages) and consult with biostatisticians.
Understanding the Basics
What Is Statistical Power?
Statistical power is the probability that a test will correctly reject a false null hypothesis—in other words, if there really is an effect, power tells you how likely you are to find it. Power equals 1 − β, where β is the Type II error rate (false negative). A power of 80% means that if there really is an effect, you have an 80% chance of detecting it with your test. Conventional target power is 80%, but important studies often aim for 90% or higher. Low power means you might miss real effects, leading to false negatives. High power means you're more likely to detect real effects, but requires larger samples or larger effects.
The Four Factors of Power
Four main factors affect power, and they're mathematically linked—fixing any three determines the fourth: (1) Effect size (δ = μ₁ − μ₀): Larger differences between null and alternative means are easier to detect, leading to higher power. (2) Sample size (n): More observations provide more precise estimates, reducing standard error and increasing power. (3) Significance level (α): Higher α makes it easier to reject H₀, increasing power but also increasing false positives. (4) Variability (σ): Less noise in data makes effects clearer, increasing power. Understanding these relationships helps you design studies with adequate power and interpret power calculations correctly.
Z-Test vs. T-Test: When to Use Each
Z-tests are used when the population standard deviation (σ) is known or when sample sizes are very large. Z-tests use the standard normal distribution. T-tests are used when σ is unknown and must be estimated from the sample (using sample standard deviation s), which is more common in practice. T-tests use the t-distribution with degrees of freedom (df), accounting for estimation uncertainty. For large samples (n > 30), z and t tests give very similar results. For small samples, t-tests are more appropriate because they account for the uncertainty in estimating σ. The choice affects power calculations—t-tests typically have slightly lower power than z-tests for the same sample size because they account for estimation uncertainty.
One-Sample vs. Two-Sample Tests
One-sample tests compare a sample mean to a known or hypothesized value (μ₀). The standard error is SE = σ/√n. Two-sample tests compare means of two independent groups with equal sample sizes. The standard error is SE = σ × √(2/n) for each group, where n is the sample size per group. Two-sample tests require larger total sample sizes (2n) to achieve the same power as one-sample tests because they're comparing two groups. For equal sample sizes per group, two-sample tests need approximately twice the total sample size to achieve the same power as a one-sample test with the same effect size and variability.
One-Sided vs. Two-Sided Tests: Power Implications
Two-sided tests detect effects in either direction (H₁: μ ≠ μ₀), while one-sided tests detect effects in a specific direction (H₁: μ > μ₀ or H₁: μ < μ₀). One-sided tests have more power than two-sided tests for the same effect size and sample size because they concentrate all rejection probability in one direction. However, one-sided tests risk missing effects in the unexpected direction. Use one-sided tests only when you have strong prior reason to expect the effect in a specific direction (e.g., a new treatment can only help, not hurt). Two-sided tests are more conservative and are generally preferred unless there's strong justification for one-sided testing.
Effect Size: The Magnitude of the Difference
Effect size (δ = μ₁ − μ₀) is the magnitude of the difference between the null and alternative means. Larger effect sizes are easier to detect, leading to higher power for the same sample size. Smaller effect sizes require larger samples to achieve adequate power. In practice, the expected effect size is often uncertain, so consider sensitivity analyses across a range of plausible effect sizes. Effect size is often standardized (e.g., Cohen's d = δ/σ) to make it comparable across different studies. Understanding effect size helps you design studies that can detect meaningful effects and interpret power calculations correctly.
Type I and Type II Errors: The Trade-Off
Type I error (α) is rejecting a true null hypothesis (false positive). Type II error (β) is failing to reject a false null hypothesis (false negative). Power = 1 − β is the probability of correctly rejecting a false null hypothesis. There's a trade-off: increasing α (making it easier to reject H₀) increases power but also increases false positives. Decreasing α (making it harder to reject H₀) decreases power but reduces false positives. The conventional α = 0.05 balances these concerns. For important studies, you might use α = 0.01 to reduce false positives, but this requires larger samples to maintain power. Understanding this trade-off helps you choose appropriate significance levels and interpret test results.
Power Curves: Visualizing Power Relationships
Power curves show how power changes as you vary sample size or effect size. For sample size curves: find where the curve crosses your target power (typically 80%) to determine required n. For effect size curves: see how sensitive your test is to detecting effects of different magnitudes. Power curves show diminishing returns—increasing sample size beyond a certain point provides little additional power. Power curves help you understand the relationship between power and study design factors, making it easier to design studies with adequate power and interpret power calculations.
Why Power Matters: Interpreting Non-Significant Results
Low power means you might miss real effects, leading to false negatives. If a study has low power and finds a non-significant result, you can't conclude there's no effect—you might simply have failed to detect it. High power means you're more likely to detect real effects, so non-significant results are more meaningful. Before conducting a study, power analysis helps ensure you can detect meaningful effects. After a study, understanding power helps interpret non-significant results correctly. Always report power when interpreting non-significant results to help readers understand whether the study was adequately powered.
Step-by-Step Guide: How to Use This Tool
Step 1: Choose Test Scenario
Select the test scenario: "One-sample mean" to compare a sample mean to a known or hypothesized value (e.g., is the average height different from 170cm?), or "Two-sample means (equal n)" to compare means of two independent groups with equal sample sizes (e.g., do treatment and control groups differ?). The scenario determines the standard error calculation and affects power calculations. Make sure your research question matches the scenario.
Step 2: Choose Test Type (z or t)
Select the test type: "z-test" if the population standard deviation (σ) is known or sample sizes are very large, or "t-test" if σ is unknown and must be estimated from the sample (more common in practice). Z-tests use the standard normal distribution, while t-tests use the t-distribution with degrees of freedom. For large samples (n > 30), z and t tests give very similar results. For small samples, t-tests are more appropriate.
Step 3: Set Significance Level (Alpha)
Set the significance level (α), typically 0.05 (5%) for standard tests, 0.01 (1%) for more stringent tests, or 0.10 (10%) for exploratory tests. Higher α increases power but also increases false positives. Lower α decreases power but reduces false positives. The conventional α = 0.05 balances these concerns. Choose α based on the importance of avoiding false positives in your context.
Step 4: Set Null and Alternative Means
Enter the null mean (μ₀) and alternative mean (μ₁). The effect size is δ = μ₁ − μ₀. Larger effect sizes are easier to detect, leading to higher power. Make sure the alternative mean represents the effect you want to detect. In practice, the expected effect size is often uncertain, so consider sensitivity analyses across a range of plausible effect sizes. The effect size cannot be zero (would give power = α).
Step 5: Set Standard Deviation
For z-tests, enter the population standard deviation (σ). For t-tests, enter the sample standard deviation estimate (s). Less variability (smaller σ) makes effects clearer, increasing power. More variability (larger σ) makes effects harder to detect, decreasing power. Use prior data or pilot studies to estimate σ. If uncertain, consider sensitivity analyses across a range of plausible values.
Step 6: Choose Test Tails
Select the test tails: "Two-sided" to detect effects in either direction (H₁: μ ≠ μ₀), "Upper one-sided" to detect effects in the positive direction (H₁: μ > μ₀), or "Lower one-sided" to detect effects in the negative direction (H₁: μ < μ₀). One-sided tests have more power than two-sided tests for the same effect size and sample size, but risk missing effects in the unexpected direction. Use one-sided tests only when you have strong prior reason to expect the effect in a specific direction.
Step 7: Choose What to Solve For
Select what to solve for: "Power" to calculate power given sample size, or "Sample size" to calculate required sample size given target power. If solving for power, enter the sample size (n for one-sample, n per group for two-sample). If solving for sample size, enter the target power (typically 0.80 or 80%). The tool will calculate the other quantity based on the four factors of power.
Step 8: Generate Power Curve (Optional)
Optionally generate a power curve to visualize how power changes with sample size or effect size. Select the curve type: "Sample size" to see how power changes with n, or "Effect size" to see how power changes with δ. Set the range and number of points. Power curves help you understand the relationship between power and study design factors, making it easier to design studies with adequate power.
Step 9: Calculate and Review Results
Click "Calculate" or submit the form to compute power or required sample size. The tool displays: (1) Achieved power or required sample size, (2) Effect size (δ), (3) Standard error, (4) Critical value, (5) Noncentrality parameter, (6) Power curve (if generated), (7) Interpretation summary. Review the results to understand power and design studies with adequate power to detect meaningful effects.
Formulas and Behind-the-Scenes Logic
Power Calculation Formula
Power is calculated using the noncentrality parameter:
Effect size: δ = μ₁ − μ₀
Standard error (one-sample): SE = σ/√n
Standard error (two-sample): SE = σ × √(2/n)
Noncentrality parameter: λ = δ/SE
Power: P(reject H₀ | H₁ is true) = 1 − β
Power is calculated using the noncentrality parameter (λ = δ/SE), which measures how far the alternative hypothesis is from the null in standard error units. For z-tests, power is calculated using the standard normal distribution. For t-tests, power is calculated using the t-distribution (or normal approximation for educational purposes). The noncentrality parameter depends on effect size, sample size, and variability. Larger λ means higher power. Power is the probability of rejecting H₀ when H₁ is true, calculated as 1 − β.
Required Sample Size Calculation
Required sample size is calculated by solving for n in the power formula:
Z-critical values: z_α (for α) and z_β (for β = 1 − power)
One-sample: n = ((z_α + z_β) × σ / δ)²
Two-sample: n_per_group = 2 × ((z_α + z_β) × σ / δ)²
Total sample: 2n for two-sample tests
Required sample size is calculated by solving for n in the power formula, given target power, effect size, alpha, and variability. The formula shows that n is proportional to (z_α + z_β)² and σ², and inversely proportional to δ². Larger effect sizes require smaller samples. Higher power (lower β) requires larger samples. Two-sample tests require approximately twice the total sample size to achieve the same power as one-sample tests. The calculation is approximate and may need adjustment for t-tests or unequal variances.
One-Sided vs. Two-Sided Power Calculation
Power calculation differs for one-sided and two-sided tests:
Two-sided: Power = P(Z + λ < -z_crit) + P(Z + λ > z_crit)
Upper one-sided: Power = P(Z + λ > z_crit)
Lower one-sided: Power = P(Z + λ < -z_crit)
Critical value: z_crit = z_{1-α/2} (two-sided) or z_{1-α} (one-sided)
Two-sided tests split the rejection region into two tails, while one-sided tests concentrate all rejection probability in one direction. One-sided tests have more power than two-sided tests for the same effect size and sample size because they use a larger critical value (z_(1-α) vs z_(1-α/2)). However, one-sided tests risk missing effects in the unexpected direction. The power calculation accounts for the test direction and critical value. Use one-sided tests only when you have strong prior reason to expect the effect in a specific direction.
Worked Example: Calculating Power for a One-Sample Test
Let's calculate power for a one-sample z-test:
Given: μ₀ = 100, μ₁ = 105, σ = 15, n = 30, α = 0.05, two-sided test
Calculate: Power to detect effect size δ = 5
Step 1: Calculate standard error
SE = σ/√n = 15/√30 = 15/5.477 = 2.738
Step 2: Calculate effect size
δ = μ₁ − μ₀ = 105 − 100 = 5
Step 3: Calculate noncentrality parameter
λ = δ/SE = 5/2.738 = 1.826
Step 4: Get critical value (two-sided, α = 0.05)
z_crit = z_(0.975) = 1.96
Step 5: Calculate power
Power = P(Z + 1.826 < -1.96) + P(Z + 1.826 > 1.96)
= P(Z < -3.786) + P(Z > 0.134)
≈ 0.0001 + 0.4467 = 0.4468 (44.7%)
Interpretation:
With n=30, power is only 44.7%, meaning there's less than a 50% chance of detecting a true effect of 5 units. To achieve 80% power, you would need a larger sample size (approximately n=71 for this effect size).
This example demonstrates how power is calculated using the noncentrality parameter. The power of 44.7% is relatively low, meaning the study has less than a 50% chance of detecting the true effect. To achieve conventional 80% power, you would need a larger sample size. This illustrates why power analysis is important—it helps ensure studies are adequately powered to detect meaningful effects.
Practical Use Cases
Student Homework: Calculating Power for a One-Sample Test
A student needs to calculate power for a one-sample z-test with μ₀ = 100, μ₁ = 105, σ = 15, n = 30, α = 0.05, two-sided. Using the tool, they find power = 44.7%. The student learns that power is relatively low, meaning there's less than a 50% chance of detecting the true effect. They can explore how increasing sample size affects power using the power curve. This helps them understand how power works and how to design studies with adequate power.
Researcher: Determining Sample Size for a Two-Sample Test
A researcher wants to compare treatment and control groups with effect size δ = 10, σ = 20, α = 0.05, target power = 0.80, two-sided t-test. Using the tool to solve for sample size, they find n = 64 per group (total n = 128). The researcher learns that two-sample tests require larger total sample sizes to achieve the same power as one-sample tests. They can explore how different effect sizes affect required sample sizes using the power curve. Note: This is for educational purposes—real research requires proper statistical methods.
Data Analyst: Planning an A/B Test
A data analyst wants to plan an A/B test to detect a 5% increase in conversion rate. Using the tool with effect size δ = 0.05, σ = 0.20, α = 0.05, target power = 0.80, two-sided z-test, they find n = 251 per group (total n = 502). The analyst learns that detecting small effects requires large samples. They can explore how different effect sizes affect required sample sizes using the power curve. Note: This is for educational purposes—real A/B testing requires proper statistical methods and considerations.
Common Person: Understanding Study Design
A person reads about a study that found no significant effect and wants to understand whether the study was adequately powered. Using the tool with the study's parameters, they calculate power = 35%. The person learns that low power means the study might have missed a real effect, so the non-significant result doesn't necessarily mean there's no effect. This helps them understand how to interpret research results and assess study quality.
Quality Control: Designing an Experiment
A quality control engineer wants to design an experiment to detect a process improvement of 2 units with σ = 5, α = 0.05, target power = 0.90, one-sample t-test. Using the tool to solve for sample size, they find n = 54. The engineer learns that higher power (90% vs 80%) requires larger samples. They can explore how different power targets affect required sample sizes using the power curve. Note: This is for educational purposes—real quality control requires proper statistical process control methods.
Researcher: Comparing One-Sided vs. Two-Sided Tests
A researcher compares power for one-sided and two-sided tests with the same parameters: μ₀ = 100, μ₁ = 105, σ = 15, n = 30, α = 0.05. Two-sided gives power = 44.7%, while upper one-sided gives power = 60.2%. The researcher learns that one-sided tests have more power but risk missing effects in the unexpected direction. This demonstrates the trade-off between power and test direction, helping them choose appropriate test types.
Understanding Sample Size Effects on Power
A user explores how sample size affects power: with δ = 5, σ = 15, α = 0.05, two-sided, n=20 gives power = 28.5%, n=40 gives power = 60.1%, n=80 gives power = 88.2%. The user learns that larger samples provide more power, but there are diminishing returns. Doubling sample size doesn't double power. This demonstrates the relationship between sample size and power, helping them understand how to design studies with adequate power.
Common Mistakes to Avoid
Using This Tool for Clinical Trial Design
Never use this tool for clinical trial design, regulatory submissions, or high-stakes research planning—it's strictly for educational purposes. Clinical trials require much more sophisticated power analysis accounting for dropout rates, interim analyses, multiple comparisons, adaptive designs, and regulatory requirements. This tool uses simplified formulas and normal approximations. For real studies, use dedicated power analysis software (G*Power, PASS, nQuery, Stata, R packages) and consult with biostatisticians.
Ignoring Effect Size Uncertainty
Don't ignore effect size uncertainty—in practice, the expected effect size is often uncertain. Consider sensitivity analyses across a range of plausible effect sizes. If you're uncertain about effect size, use a conservative estimate or explore a range of values using power curves. Don't assume a single effect size without justification. Understanding effect size uncertainty helps you design robust studies that can detect meaningful effects even if the true effect size differs from your estimate.
Using One-Sided Tests Without Justification
Don't use one-sided tests without strong prior justification—they have more power but risk missing effects in the unexpected direction. Use one-sided tests only when you have strong prior reason to expect the effect in a specific direction (e.g., a new treatment can only help, not hurt). Two-sided tests are more conservative and are generally preferred unless there's strong justification for one-sided testing. Don't choose one-sided tests just to increase power—this can lead to missing important effects.
Ignoring Assumptions
Don't ignore the assumptions—this calculator assumes normality (or large samples), independent observations, equal variances for two-sample tests, and known or well-estimated standard deviations. Violations of these assumptions may affect the accuracy of power calculations. For non-normal data, small samples, or unequal variances, use appropriate methods. Always check whether assumptions are met before trusting power calculations.
Confusing Power with Significance
Don't confuse power with significance—power is the probability of detecting a true effect, while significance is whether you reject the null hypothesis. High power doesn't guarantee significance—it just means you're more likely to detect real effects. Low power means you might miss real effects, leading to false negatives. Always understand what power means and how it relates to your test results. Don't interpret low power as evidence of no effect.
Not Considering Practical Significance
Don't focus only on statistical significance—consider practical significance as well. A study might have high power to detect a statistically significant but practically meaningless effect. Conversely, a study might have low power to detect a practically important but small effect. Always consider both statistical and practical significance when designing studies and interpreting results. Don't design studies to detect effects that aren't practically meaningful.
Using This Tool for Complex Designs
Don't use this tool for complex designs—it only handles simple one-sample and two-sample tests with equal sample sizes. For complex designs (unequal sample sizes, multiple groups, repeated measures, mixed models), use appropriate power analysis methods. This tool cannot handle complex designs or advanced statistical methods. For complex designs, consult with a statistician and use appropriate power analysis software.
Advanced Tips & Strategies
Use Power Curves to Explore Trade-Offs
Use power curves to explore trade-offs between power, sample size, and effect size. Power curves show how power changes as you vary sample size or effect size, helping you understand the relationship between these factors. Use curves to find where power crosses your target (typically 80%) to determine required sample sizes. Use curves to see how sensitive your test is to different effect sizes. Power curves help you design robust studies and understand power relationships.
Conduct Sensitivity Analyses
Conduct sensitivity analyses across a range of plausible effect sizes and variability estimates. Since effect size and variability are often uncertain, explore how power changes across different values. Use sensitivity analyses to design robust studies that can detect meaningful effects even if true values differ from estimates. Sensitivity analyses help you understand the robustness of your power calculations and design studies that work across a range of scenarios.
Consider Both Statistical and Practical Significance
Always consider both statistical and practical significance when designing studies. A study might have high power to detect a statistically significant but practically meaningless effect. Conversely, a study might have low power to detect a practically important but small effect. Design studies to detect effects that are both statistically and practically meaningful. Consider the minimum practically important effect size when designing studies.
Report Power When Interpreting Non-Significant Results
Always report power when interpreting non-significant results to help readers understand whether the study was adequately powered. Low power means you might have missed a real effect, so non-significant results don't necessarily mean there's no effect. High power means non-significant results are more meaningful. Reporting power helps readers interpret results correctly and assess study quality. Don't interpret non-significant results without considering power.
Understand the Four Factors Are Interconnected
Understand that the four factors of power (effect size, sample size, alpha, variability) are mathematically linked—fixing any three determines the fourth. This means you can't increase power by adjusting one factor without affecting others. For example, increasing sample size increases power, but also increases cost. Decreasing alpha decreases power but reduces false positives. Understanding these relationships helps you make informed trade-offs when designing studies.
Use Appropriate Test Type (z vs. t)
Use appropriate test type based on whether population standard deviation is known. Use z-tests when σ is known or sample sizes are very large. Use t-tests when σ is unknown and must be estimated from the sample (more common in practice). For large samples (n > 30), z and t tests give very similar results. For small samples, t-tests are more appropriate because they account for estimation uncertainty. The choice affects power calculations—t-tests typically have slightly lower power than z-tests for the same sample size.
Remember This Is Educational Only
Always remember that this tool is strictly for educational purposes. Professional applications require: (1) Proper consideration of dropout rates, (2) Interim analyses and multiple comparisons, (3) Adaptive designs, (4) Regulatory requirements, and (5) Professional statistical software. For clinical trial design, regulatory submissions, or high-stakes research planning, use dedicated power analysis software (G*Power, PASS, nQuery, Stata, R packages) and consult with biostatisticians. This tool cannot replace professional statistical analysis for real-world applications.
Limitations & Assumptions
• Simplified Power Formulas: This tool uses standard closed-form power calculations that assume idealized conditions. Real-world studies face complications including missing data, protocol deviations, measurement error, and non-compliance that reduce actual power below theoretical calculations.
• Normality and Equal Variance Assumptions: Power formulas assume normally distributed data or large samples where the central limit theorem applies. For two-sample tests, equal variances and equal sample sizes per group are typically assumed. Violations of these assumptions affect the accuracy of power estimates.
• Effect Size Uncertainty: Power calculations require specifying the expected effect size, which is often unknown or uncertain before conducting a study. Power estimates are highly sensitive to effect size assumptions—small changes in assumed effect size can dramatically change required sample sizes.
• No Consideration of Practical Constraints: This tool does not account for dropout rates, interim analyses, multiple comparison adjustments, adaptive designs, or regulatory requirements that are essential for real clinical trial planning and grant applications.
Important Note: This calculator is strictly for educational and informational purposes only. It demonstrates how statistical power analysis works mathematically, not for clinical trial design, regulatory submissions, grant applications, or high-stakes research planning. Professional power analysis requires proper consideration of dropout rates, interim analyses, multiple comparisons, adaptive designs, and regulatory requirements. For real studies, use dedicated power analysis software such as G*Power, PASS, nQuery, Stata, or R packages (pwr, powerSurvEpi) and always consult with qualified biostatisticians. Sample size determination for clinical trials should follow FDA/EMA guidance documents and involve regulatory expertise.
Important Limitations and Disclaimers
- •This calculator is an educational tool designed to help you understand statistical power and power analysis. While it provides accurate calculations, you should use it to learn the concepts and check your manual calculations, not as a substitute for understanding the material. Always verify important results independently.
- •This tool is NOT designed for clinical trial design, regulatory submissions, or high-stakes research planning. It is strictly for educational purposes to help understand how statistical power works mathematically. Professional applications require proper consideration of dropout rates, interim analyses, multiple comparisons, adaptive designs, regulatory requirements, and professional statistical software. For real studies, use dedicated power analysis software and consult with biostatisticians.
- •This calculator assumes: (1) Normally distributed data or large samples, (2) Independent observations, (3) For two-sample tests: equal variances and equal sample sizes per group, (4) Known or well-estimated standard deviation, (5) Simple random sampling. Violations of these assumptions may affect the accuracy of power calculations. For non-normal data, small samples, or unequal variances, use appropriate methods.
- •In practice, the expected effect size is often uncertain. Consider sensitivity analyses across a range of plausible effect sizes. If you're uncertain about effect size, use a conservative estimate or explore a range of values using power curves. Don't assume a single effect size without justification. Understanding effect size uncertainty helps you design robust studies.
- •This tool is for informational and educational purposes only. It should NOT be used for critical decision-making, clinical trial design, regulatory compliance, legal advice, or any professional/legal purposes without independent verification. Consult with appropriate professionals (biostatisticians, domain experts) for important decisions.
- •Results calculated by this tool are power estimates or required sample sizes based on your specified parameters and statistical methods. Actual power in real-world scenarios may differ due to additional factors, assumption violations, dropout rates, or data characteristics not captured in this simple demonstration tool. Use power calculations as guides for understanding study design, not guarantees of specific outcomes.
Sources & References
The mathematical formulas and statistical power concepts used in this calculator are based on established statistical theory and authoritative academic sources:
- •NIST/SEMATECH e-Handbook: Power and Sample Size - Authoritative reference from the National Institute of Standards and Technology.
- •G*Power Documentation: G*Power - Industry-standard power analysis software documentation.
- •Cohen (1988): Statistical Power Analysis for the Behavioral Sciences - Seminal book on effect sizes and power analysis.
- •Penn State STAT 500: Power - University course material on statistical power concepts.
- •Statistics By Jim: Statistical Power - Practical explanations of power and sample size calculations.
Frequently Asked Questions
Common questions about statistical power, power analysis, sample size calculation, z-test power, t-test power, effect size, type II error, and how to use this calculator for homework and study design practice.
What is statistical power?
Statistical power is the probability that a test will correctly reject a false null hypothesis (detect a true effect). It equals 1 − β, where β is the Type II error rate (false negative). Power of 80% means that if there really is an effect, you have an 80% chance of detecting it with your test.
What is the difference between z-tests and t-tests?
Z-tests are used when the population standard deviation (σ) is known or when sample sizes are very large. T-tests are used when σ is unknown and must be estimated from the sample, which is more common in practice. For large samples (n > 30), z and t tests give very similar results.
When should I use a one-sided vs two-sided test?
Use a two-sided test when you want to detect effects in either direction (μ ≠ μ₀). Use a one-sided test only when you have strong prior reason to expect the effect in a specific direction (e.g., a new treatment can only help, not hurt). One-sided tests have more power but risk missing effects in the unexpected direction.
Why does power increase with sample size?
Larger samples provide more precise estimates of the population mean, reducing the standard error. This makes it easier to distinguish the true effect from random noise. The relationship is nonlinear: doubling n doesn't double power, and there are diminishing returns as n gets very large.
What factors affect power?
Four main factors affect power: (1) Effect size - larger effects are easier to detect; (2) Sample size - more data means more precision; (3) Significance level (α) - higher α increases power but also false positives; (4) Variability (σ) - less noise makes effects clearer. These factors are interconnected: fixing three determines the fourth.
Why is this tool not enough for clinical trial planning?
Clinical trials require much more sophisticated power analysis accounting for dropout rates, interim analyses, multiple comparisons, adaptive designs, and regulatory requirements. This tool uses simplified formulas and normal approximations. For real studies, use professional software (G*Power, PASS, nQuery) and consult with biostatisticians.
What assumptions does this calculator make?
This calculator assumes: (1) Normally distributed data or large samples; (2) Independent observations; (3) For two-sample tests: equal variances and equal sample sizes per group; (4) Known or well-estimated standard deviation; (5) Simple random sampling. Violations of these assumptions may affect the accuracy of power calculations.
How should I interpret the power curve?
The power curve shows how power changes as you vary sample size or effect size. For sample size curves: find where the curve crosses your target power (typically 80%) to determine required n. For effect size curves: see how sensitive your test is to detecting effects of different magnitudes. The curve also shows diminishing returns.
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Build confidence intervals for means and proportions
T-Test Calculator
Run one-sample, two-sample, and paired t-tests
CI for Proportions
Compute confidence intervals for proportions using Wald and Wilson methods
Correlation Significance
Test whether a correlation coefficient is statistically significant