Sample Size & Power Calculator

What Is Statistical Power?

Statistical power is the probability that a hypothesis test will correctly reject the null hypothesis when a specific alternative hypothesis is true. In simpler terms, power tells you: "If there really is an effect of this size, what are the chances my test will detect it and declare it statistically significant?"

Power is mathematically defined as 1 − β, where:

• β (beta) is the Type II error rate—the probability of failing to reject the null hypothesis when the alternative is true (a "false negative").
• So if power = 0.80 (80%), then β = 0.20 (20% chance of missing the effect).

Common power thresholds:

• 0.80 (80%): The most common target in textbooks and introductory research. Means you have an 80% chance of detecting the effect if it exists, and a 20% chance of missing it.
• 0.90 (90%): Higher power, often used in more critical studies or when missing an effect would be especially costly. Requires larger sample sizes.
• 0.70 or below: Generally considered low power; studies with power < 0.70 have high risk of Type II errors and may be underpowered.

Ideally, you want high power (close to 1.0), but higher power requires larger sample sizes or detecting larger effect sizes, which may not always be feasible.

What Is Effect Size and Why Does It Matter?

Effect size quantifies the magnitude of the difference or relationship you're trying to detect. It answers: "How big is the effect?" Unlike statistical significance (which depends on sample size), effect size is a measure of practical importance.

Common effect size measures:

• Cohen's d (for means): Standardized difference between two means.
  • Formula: d = (μ₁ − μ₂) / σ
  • Interpretation: d ≈ 0.2 = small, d ≈ 0.5 = medium, d ≈ 0.8 = large (Cohen's rough guidelines)
• Difference in proportions (for proportions): p₁ − p₂ or Cohen's h (arcsine transformation).
  • Example: Conversion rate in control = 10%, treatment = 15% → difference = 0.05 (5 percentage points)
• Correlation coefficient (r): Expected strength of association between two variables.
  • r ≈ 0.1 = small, r ≈ 0.3 = medium, r ≈ 0.5 = large (rough guidelines)

Where do effect sizes come from?

• Prior studies: Look at similar research and see what effect sizes were observed.
• Domain knowledge: Experts may know what's considered a "meaningful" difference in their field.
• Minimum meaningful difference: In homework or planning, you decide: "What's the smallest effect I care about detecting?"
• Pilot data: A small preliminary study can give rough effect size estimates.

Larger effect sizes are easier to detect (require smaller samples); smaller effect sizes require larger samples to achieve the same power.

The Role of Significance Level (α)

Significance level (α, alpha) is the threshold for declaring a result statistically significant. It represents the maximum probability of a Type I error—rejecting the null hypothesis when it's actually true (a "false positive").

Common α values:

• 0.05 (5%): The most common default in many fields. Means you're willing to accept a 5% chance of a false positive.
• 0.01 (1%): Stricter threshold, used when false positives are especially costly. Requires larger sample sizes for the same power.
• 0.10 (10%): More lenient, sometimes used in exploratory or preliminary studies (less common in formal research).

How α affects sample size and power:

• Lower α (e.g., 0.01) → harder to declare significance → need more data or larger effects to achieve the same power.
• Higher α (e.g., 0.10) → easier to declare significance → can achieve target power with smaller samples, but more false positives.

The choice of α is often dictated by field conventions, assignment instructions, or the consequences of Type I vs Type II errors in your specific context.

How Sample Size, Power, Effect Size, and α Interact

These four parameters are interconnected. If you know any three, you can solve for the fourth:

• Larger sample size (n) → Higher power (holding effect size and α constant). More data gives you a better chance of detecting a true effect.
• Larger effect size → Smaller required sample size (for the same power and α). Big effects are easier to detect.
• Higher desired power → Larger required sample size (holding effect size and α constant). If you want to be more confident you'll detect the effect, you need more data.
• Stricter α (lower α) → Larger required sample size (for the same power and effect size). Being more conservative about false positives requires more evidence.

Example intuition: Suppose you want 80% power to detect a medium effect (Cohen's d = 0.5) at α = 0.05. The calculator might tell you that you need about 64 participants per group. If you only have 30 per group, your power drops to around 55%—you have a nearly 50-50 chance of missing the effect even if it exists. Conversely, if the effect is actually large (d = 0.8), 30 per group might give you 80% power. The calculator helps you explore these trade-offs quantitatively.

Type I and Type II Errors (α and β)

Understanding errors helps clarify why power matters:

• Type I Error (α): Rejecting H₀ when it's true (false positive). Controlled by significance level α.
• Type II Error (β): Failing to reject H₀ when H₁ is true (false negative). Power = 1 − β.

In study planning, you typically:

• Set α (e.g., 0.05) to limit false positives.
• Choose a target power (e.g., 0.80) to limit false negatives.
• Then determine the sample size needed to achieve both goals for a given effect size.

There's always a trade-off: lowering α (fewer false positives) tends to increase β (more false negatives) unless you compensate by increasing sample size.

Mode 1: Find Required Sample Size

Goal: Determine how many participants/observations you need to detect a specified effect with desired power.

1. Select Test Type: Choose the test that matches your study design:
   • Two-Sample Proportions: Comparing success rates between two groups (e.g., A/B test conversion rates, treatment vs control outcomes).
   • Two-Sample Means: Comparing average values between two independent groups (e.g., test scores between teaching methods, blood pressure between treatments).
   • Correlation: Testing whether two continuous variables are related (e.g., correlation between study hours and exam scores).
2. Set "Solve For" to "Sample Size" in the dropdown.
3. Enter Significance Level (α): Typically 0.05 (5%), or as specified by your assignment/field.
4. Enter Desired Power: Typically 0.80 (80%) or 0.90 (90%).
5. Specify Effect Size: This is critical and depends on test type:
   • For proportions: Enter p₁ (proportion in group 1) and p₂ (proportion in group 2). Example: p₁ = 0.10 (10% baseline), p₂ = 0.15 (15% treatment).
   • For means: Enter μ₁, μ₂ (expected means) and σ (standard deviation). Example: μ₁ = 100, μ₂ = 105, σ = 15.
   • For correlation: Enter target r (e.g., r = 0.3 for a medium correlation).
6. Optional: Set Allocation Ratio if groups have unequal sizes (default is 1:1).
7. Click Calculate.
8. Review Results: The calculator shows required sample size per group (n₁, n₂) and total sample size. It may also display Cohen's d or Cohen's h as standardized effect sizes.

Mode 2: Find Power for a Given Sample Size

Goal: Estimate what power you'll achieve with a fixed sample size (e.g., you have budget for 50 participants per group—what power will you have?).

1. Select Test Type as in Mode 1.
2. Set "Solve For" to "Power".
3. Enter α (e.g., 0.05).
4. Enter Sample Sizes: For two-sample tests, enter n₁ and n₂. For correlation, enter total n.
5. Specify Effect Size (p₁, p₂, μ₁, μ₂, σ, or r, depending on test type).
6. Click Calculate.
7. Review Results: The calculator displays "Achieved Power" as a percentage. If power ≥ 80%, your study is typically considered adequately powered. If power < 70%, you may have high risk of missing the effect (Type II error).

Mode 3: Find Minimum Detectable Effect (MDE)

Goal: Given a fixed sample size and desired power, determine the smallest effect you can reliably detect.

1. Select Test Type.
2. Set "Solve For" to "MDE".
3. Enter α and Power (e.g., α = 0.05, power = 0.80).
4. Enter Sample Sizes (n₁, n₂ for two-sample tests, or n for correlation).
5. For proportions: Enter baseline proportion p₀. For means, enter σ.
6. Click Calculate.
7. Review Results: The calculator shows the minimum detectable effect—the smallest difference or correlation you can detect with the specified power and sample. If the MDE is larger than what you consider practically meaningful, you may need a larger sample or accept lower power.

General Tips for Using the Calculator

• Align test type with study design: Use two-sample tests for independent groups, not paired data. If data are paired, standard formulas differ (the calculator assumes independent samples).
• One-tailed vs two-tailed: The calculator default is two-tailed (most common in homework and research). One-tailed tests require smaller samples but are only appropriate when you have a strong directional hypothesis and prior justification.
• Check assumptions: Power calculations assume approximate normality for means, large enough sample for proportion approximations, and that your effect size estimate is accurate. If assumptions are violated, actual power may differ.
• Use charts: If the calculator displays a power curve, use it to visualize how power changes with sample size—helpful for understanding diminishing returns.
• Iterate and explore: Try different scenarios: "What if effect is smaller?" "What if I can only get 30 participants?" This builds intuition and helps you make informed trade-offs.
• Remember the scope: This calculator is for learning and planning, not for final regulatory or clinical trial submissions. Always combine with expert consultation for real-world high-stakes studies.

General Idea of Power Calculations

Most power analyses involve:

• Critical value (based on α): The threshold for rejecting H₀, derived from the standard normal or t-distribution.
• Test statistic under H₁: Under the alternative hypothesis (assuming the specified effect exists), the test statistic follows a different distribution.
• Power: The probability that the test statistic falls in the "rejection region" when H₁ is true.

Conceptually, power is the area under the alternative distribution that exceeds the critical value. The calculator integrates these distributions using standard statistical formulas.

Example: Two-Sample t-Test for Means (Conceptual Formula)

For comparing two means with equal sample size n per group:

Intuition: Larger d → smaller n needed. Higher power or stricter α → larger n needed.

Example: Two-Sample Proportions (Conceptual Formula)

For comparing proportions p₁ and p₂ with equal n per group:

Example: Correlation Test (Conceptual Formula)

For testing correlation r ≠ 0:

Worked Example 1: Sample Size for Two-Sample Means (Illustrative)

Problem: You want 80% power (0.80) at α = 0.05 (two-tailed) to detect a medium effect size (Cohen's d = 0.5) comparing two independent groups. How many participants per group?

Solution (conceptual):

1. Identify critical values:
   Z_α/2 = 1.96 (for α = 0.05, two-tailed)
   Z_β = 0.84 (for power = 0.80, β = 0.20)
2. Apply the formula:
   n ≈ 2 × (1.96 + 0.84)² / (0.5)²
   n ≈ 2 × (2.80)² / 0.25
   n ≈ 2 × 7.84 / 0.25
   n ≈ 62.72 → round up to n = 63 per group
3. Total sample size: 63 × 2 = 126 participants

Interpretation: With 63 participants per group (126 total), you have an 80% chance of detecting a medium effect (d = 0.5) at α = 0.05. If your actual sample is smaller (e.g., 40 per group), power drops; if larger, power increases.

Worked Example 2: Power for a Fixed Sample Size (Illustrative)

Problem: You have n = 50 per group, α = 0.05, and expect d = 0.5. What power will you achieve?

Solution (conceptual):

1. Compute the noncentrality parameter:
   δ = d × √(n/2) = 0.5 × √(50/2) = 0.5 × 5 = 2.5
2. Find critical t-value: For α = 0.05 (two-tailed), df ≈ 98, critical t ≈ 1.98
3. Power is the probability that a non-central t-distribution (with δ = 2.5) exceeds the critical value. Using statistical tables or software, this yields power ≈ 0.71 (71%).

Interpretation: With 50 per group and d = 0.5, you have only 71% power—below the typical 80% target. You'd have a 29% chance of missing the effect (Type II error). To reach 80% power, you'd need about 63 per group (as in Example 1).

1. Thesis Proposal: Justifying Sample Size for Survey Study

Scenario: A master's student is proposing a survey comparing job satisfaction scores between two departments. The thesis committee requires sample size justification with power analysis.

How the calculator helps: The student estimates from prior research that the expected mean difference is about 5 points (μ₁ = 70, μ₂ = 75) with σ = 15. Using the calculator for two-sample means, α = 0.05, power = 0.80, they find they need about 143 participants per department (286 total). They report this in their proposal: "Based on an expected effect size of d = 0.33 and 80% power at α = 0.05, we require 143 participants per group." This demonstrates rigorous planning and increases proposal approval chances.

2. A/B Test Planning: Website Conversion Rate Optimization

Scenario: A product analyst wants to test whether a new checkout button increases conversion rate from baseline 10% to 12% (a 2 percentage point lift). They need to know how many visitors are required.

How the calculator helps: Using two-sample proportions, p₁ = 0.10, p₂ = 0.12, α = 0.05, power = 0.80, the calculator shows they need about 3,841 visitors per group (7,682 total). If traffic is low, they can explore trade-offs: "What if I accept 70% power?" or "What if I wait for a 3-point lift?" This helps communicate realistic timelines to stakeholders and avoid underpowered experiments.

3. Statistics Homework: Sample Size for Detecting Small Effect

Scenario: A homework problem asks: "How many participants per group are needed to detect Cohen's d = 0.3 with 90% power at α = 0.01?"

How the calculator helps: The student enters test type = two-sample means, d = 0.3, power = 0.90, α = 0.01 (stricter than usual), and finds n ≈ 468 per group. This illustrates how small effects and high power/low α dramatically increase sample requirements—a key learning objective. They can also check their manual calculations against the tool.

4. Research Methods Class: Exploring Power Trade-offs

Scenario: An assignment asks students to compare required sample sizes for Cohen's d = 0.2, 0.5, and 0.8, each at power = 0.80 and α = 0.05, then discuss implications.

How the calculator helps: Students run three calculations and find: d = 0.2 → n ≈ 394 per group, d = 0.5 → n ≈ 63, d = 0.8 → n ≈ 26. They learn that detecting small effects requires vastly more data, and that many real-world studies may be underpowered if effects are smaller than expected. This builds critical thinking about study feasibility and design.

5. Pilot Study: Estimating Power for Fixed Budget

Scenario: A researcher has funding for only 30 participants per group and wants to know what power they'll have to detect a medium effect (d = 0.5).

How the calculator helps: Using "Solve For: Power," they enter n₁ = 30, n₂ = 30, d = 0.5, α = 0.05, and find power ≈ 48%—very low. This helps them set realistic expectations: "With our budget, we have less than 50% chance of detecting the effect. We should treat this as a pilot study and seek additional funding for a properly powered follow-up." Transparent communication of power helps avoid overinterpreting pilot results.

6. Correlation Study: Planning Sample for Association Test

Scenario: A psychology student wants to test whether study hours and GPA are correlated, expecting r ≈ 0.25, with 80% power at α = 0.05.

How the calculator helps: Using test type = correlation, r = 0.25, power = 0.80, α = 0.05, the calculator shows n ≈ 123 participants. If the student can only recruit 60, power drops to about 52%. This informs their study design: they can aim for 123, adjust expectations, or explore detecting a larger correlation (r = 0.35) with 60 participants. The calculator makes trade-offs explicit and quantitative.

7. Class Project: Minimum Detectable Effect with Limited Data

Scenario: Students can collect data from 50 participants per group for a class project. They want to know the smallest effect they can detect with 80% power.

How the calculator helps: Using "Solve For: MDE," test type = two-sample means, n₁ = n₂ = 50, power = 0.80, α = 0.05, σ = 10, the calculator shows MDE ≈ 5.6 points (Cohen's d ≈ 0.56). This tells them: "We can reliably detect medium-to-large effects, but not small effects. Our study is appropriate for detecting meaningful differences, but won't catch subtle effects." This aligns project scope with statistical reality.

8. Exam Prep: Understanding α, β, and Power Relationships

Scenario: An AP Statistics exam asks students to explain how changing α from 0.05 to 0.01 affects required sample size for the same power and effect size.

How the calculator helps: Students use the calculator to run both scenarios and see that n increases when α decreases (for the same power and effect). This concrete numerical demonstration helps them explain: "Stricter α means we need more evidence to reject H₀, so sample size must increase to maintain power. The calculator shows n jumps from 64 to 99 per group when α changes from 0.05 to 0.01 for d = 0.5, power = 0.80." This builds deep understanding for test-taking and real-world applications.

1. Confusing Power with Significance Level (α)

Mistake: Thinking power = α, or confusing the two error rates.

Why it matters: α controls Type I error (false positive), while power = 1 − β controls Type II error (false negative). They are different concepts with different roles.

How to avoid: Remember: α is what you set before the study (typically 0.05); power is what you achieve based on sample size and effect size. α is about false positives; power is about correctly detecting true effects.

2. Choosing Unrealistic Effect Sizes to Force Small Samples

Mistake: Using very large effect sizes (e.g., d = 1.5) just to make the required sample size small, even though such effects are unlikely in your field.

Why it matters: If the true effect is smaller than assumed, your study will be severely underpowered. You'll likely fail to detect the effect and waste resources.

How to avoid: Base effect sizes on prior research, pilot data, or domain knowledge. Be conservative: if in doubt, assume a smaller effect. It's better to overpower slightly than to severely underpower.

3. Ignoring Directionality (One-Tailed vs Two-Tailed)

Mistake: Using a one-tailed test in the calculator to get smaller sample sizes, when the study question or homework clearly expects a two-tailed test.

Why it matters: One-tailed tests are only appropriate when you have strong prior justification for directionality and are willing to ignore effects in the opposite direction. Most homework and research use two-tailed by default.

How to avoid: Unless explicitly told to use one-tailed, or you have domain-specific justification, use two-tailed. If in doubt, ask your instructor or check assignment instructions.

4. Mismatching Test Type with Study Design

Mistake: Using a two-sample independent test when the design is actually paired (matched subjects, pre-post measurements), or vice versa.

Why it matters: Paired tests typically require smaller samples because they account for within-subject correlation. Using the wrong formula leads to incorrect sample size estimates.

How to avoid: Carefully read the study design or problem statement. If subjects are matched or measured twice, use paired formulas (note: this calculator assumes independent samples; paired tests require specialized formulas). If groups are independent, use two-sample tests.

5. Forgetting That Assumptions Matter

Mistake: Treating calculator output as "exact truth" without considering that formulas assume normality, equal variances, accurate effect size estimates, and other conditions.

Why it matters: If assumptions are violated (e.g., highly skewed data, wrong σ estimate), actual power may differ from calculated power.

How to avoid: Acknowledge assumptions in reports: "Assuming approximately normal distributions and σ = 15 based on prior data, we require n = 64 per group." If assumptions are questionable, consider simulation-based power analysis or nonparametric alternatives (beyond this calculator's scope, but worth mentioning).

6. Conducting Post-Hoc Power Analysis and Overinterpreting It

Mistake: After collecting data and finding a non-significant result, computing "observed power" and saying "our power was too low, so the null result doesn't count."

Why it matters: Post-hoc power (power calculated after the study using observed effect size) is controversial and uninformative—it's directly tied to the p-value you already have. It doesn't add new information.

How to avoid: Do power analysis before data collection (a priori power). If you must discuss power post-hoc, use the originally hypothesized effect size, not the observed effect. Better yet, report confidence intervals and effect sizes alongside p-values.

7. Not Accounting for Attrition or Missing Data

Mistake: Calculating that you need 100 participants, recruiting exactly 100, but ending up with only 80 complete cases due to dropouts or missing data.

Why it matters: Your effective sample size is smaller than planned, reducing power below your target.

How to avoid: If you expect attrition (common in longitudinal studies, surveys), inflate your recruitment target: Recruit n / (1 − attrition rate). For example, if you need 100 and expect 20% dropout, recruit 125. Mention this in your planning: "Accounting for ~20% attrition, we will recruit 125 to ensure 100 complete cases."

8. Using Calculator Results as Final Approval for High-Stakes Studies

Mistake: Relying solely on this educational calculator for clinical trials, regulatory submissions, or business-critical experiments without consulting statisticians or using specialized software.

Why it matters: Real-world studies often involve complexities (interim analyses, multiplicity adjustments, non-normal distributions) that require expert guidance and advanced methods.

How to avoid: Use this calculator for learning, homework, preliminary planning, and exploring trade-offs. For actual clinical trials, business experiments with financial implications, or regulatory work, consult professional statisticians and use validated, specialized software (e.g., nQuery, PASS, or G*Power for academic research).

9. Assuming Larger Samples Always Solve Everything

Mistake: Thinking "If I just make n huge, I can detect anything," without considering practical significance or cost.

Why it matters: With very large n, even tiny, meaningless effects become statistically significant. You can waste resources detecting effects too small to matter practically.

How to avoid: Balance statistical power with practical significance. Ask: "What's the smallest effect I care about?" and design for that. Don't over-recruit just to detect trivial effects. Report effect sizes alongside p-values to maintain focus on practical importance.

10. Not Reporting or Justifying Power Assumptions in Write-Ups

Mistake: Stating "we used 64 participants" without explaining why that number was chosen or what assumptions went into the power calculation.

Why it matters: Reviewers, instructors, and stakeholders need to understand your reasoning. Transparency about power assumptions builds trust and allows others to assess study quality.

How to avoid: In homework, thesis proposals, or reports, include a brief power justification: "We computed required sample size using two-sample t-test power analysis with α = 0.05, power = 0.80, and expected effect size d = 0.5 based on Smith et al. (2020), yielding n = 64 per group." This demonstrates rigorous planning.

1. Explore Trade-offs Systematically

Use the calculator to run multiple scenarios: How does required n change when you move from α = 0.05 to α = 0.01? From power = 0.80 to 0.90? From d = 0.3 to d = 0.5? Create a table of results and discuss implications. This builds intuition about which parameters have the biggest impact on sample size (hint: effect size and power are usually most influential).

2. Understand Diminishing Returns

Visualize (or mentally note) how power increases with sample size. Initially, small increases in n yield big power gains; beyond a certain point, doubling n only adds a few percentage points of power. For example, going from n = 20 to 40 per group might raise power from 50% to 80%, but going from 100 to 200 only raises it from 95% to 99%. This helps you decide when "enough is enough" and avoid over-recruiting.

3. Compare Independent vs Paired Designs (Conceptually)

Although this calculator focuses on independent samples, understand that paired/matched designs typically require smaller samples for the same power (because they reduce variability). If your study can use a paired design, note that you may need fewer participants. Consult paired t-test power formulas or specialized software for exact calculations.

4. Use Power Curves to Communicate with Stakeholders

If the calculator displays power curves (power vs sample size), use them in presentations to show decision-makers or committees why a certain sample size is needed. Saying "With 50 participants, we have only 65% power, but with 80, we reach 85%" is more convincing with a visual curve. This makes trade-offs transparent and helps secure resources.

5. Think About Allocation Ratios for Unbalanced Designs

If one group is more expensive or harder to recruit (e.g., rare disease patients), consider unequal allocation (e.g., 2:1 ratio, two controls per case). The calculator (if it supports allocation ratio) can show how this affects total sample size. Sometimes a 2:1 design is more efficient than trying to balance 1:1 when recruitment is asymmetric.

6. Connect Power to Confidence Interval Width

Higher power often corresponds to narrower confidence intervals around effect estimates. If precision (narrow CIs) is important in your study, you can use power analysis as a proxy for precision planning: higher n → higher power and narrower CIs. Some advanced planning uses CI width directly, but power is a good starting point.

7. Run Sensitivity Analyses

If you're uncertain about the true effect size or variability, compute required sample size for a range of plausible values (e.g., d = 0.3, 0.4, 0.5). Report: "We require 64–143 participants per group depending on whether the effect is medium (d = 0.5) or small-medium (d = 0.3)." This acknowledges uncertainty and prepares you for different scenarios.

8. Understand the Role of Pilot Studies

Pilot studies with low power (e.g., 30–50%) are useful for feasibility, refining methods, and getting rough effect size estimates—not for testing hypotheses. Use pilot data to inform power calculations for the main study, but don't claim "we found a significant effect in the pilot" if power was low. The pilot's role is to plan the main study, not to draw final conclusions.

9. Learn About Sequential and Adaptive Designs

For advanced planning, explore concepts like sequential testing (where you can stop early for efficacy or futility) and adaptive designs (where sample size can be adjusted mid-study). These are beyond this calculator's scope but are powerful tools in real research. Knowing they exist helps you appreciate that power analysis is just one piece of study design.

10. Always Combine Power Analysis with Critical Thinking

The calculator provides numbers, but you provide judgment. Ask: Is this sample size feasible given time and budget? Does the effect size make sense for my field? Am I planning for the right test? Power analysis is a tool to inform decisions, not a substitute for thoughtful study design. Use it to support your reasoning, not to replace it.