How do I determine sample size for a t-test?

Sample size for a t-test depends on three factors: effect size (Cohen's d), significance level (alpha), and desired power (1-beta). For example, with a medium effect size (d=0.5), alpha=.05, and power=.80, an independent samples t-test requires N=128 (64 per group). Use StatMate's sample size calculator to compute this instantly.

What is statistical power?

Statistical power is the probability of correctly detecting a real effect when it exists. A power of .80 (80%) is the conventional minimum, meaning there is an 80% chance of finding a true effect. Low power increases the risk of Type II errors (failing to detect a real effect), which is why power analysis before data collection is essential.

What effect size should I use?

The best approach is to estimate effect size from prior research or pilot data. If no prior information is available, use Cohen's conventions: for t-tests, d=0.2 (small), 0.5 (medium), and 0.8 (large). StatMate provides built-in effect size presets so you can quickly select the appropriate value for your analysis.

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What is Sample Size Calculation?

Sample size calculation (also called power analysis) determines the minimum number of participants needed for a study to detect a meaningful effect. It is a critical step in research planning — too few participants and you risk missing a real effect (Type II error); too many and you waste time and resources. Most IRBs (Institutional Review Boards) and grant agencies require a formal power analysis as part of the research proposal.

The Four Components of Power Analysis

Every power analysis involves four interconnected parameters. If you know any three, you can solve for the fourth. In practice, researchers most commonly fix alpha, power, and effect size to solve for sample size (N).

1. Effect Size (d, f, r, or w)

The expected magnitude of the effect you want to detect. Larger effects require fewer participants. Use Cohen's d for t-tests, Cohen's f for ANOVA, r for correlation, and Cohen's w for chi-square tests. If you don't know the expected effect size, use conventions: small, medium, or large.

2. Significance Level (α)

The probability of a Type I error — rejecting the null hypothesis when it is actually true (a false positive). The conventional level is α = .05, meaning a 5% chance of incorrectly declaring significance.

3. Statistical Power (1 - β)

The probability of correctly detecting a real effect — avoiding a Type II error (a false negative). The conventional minimum is .80 (80%), meaning an 80% chance of finding the effect if it truly exists. Some fields recommend .90 or higher.

4. Sample Size (N)

The total number of participants required. This is typically the unknown you solve for. More participants increase power but also increase cost and time.

Effect Size Conventions by Test Type

Jacob Cohen (1988) established widely-used conventions for small, medium, and large effect sizes. Use these when you lack pilot data or prior research to estimate the expected effect.

Test	Measure	Small	Medium	Large
T-tests	Cohen's d	0.20	0.50	0.80
ANOVA	Cohen's f	0.10	0.25	0.40
Correlation	r	0.10	0.30	0.50
Chi-square	Cohen's w	0.10	0.30	0.50
Proportions	Cohen's h	0.20	0.50	0.80

Worked Example: Independent Samples T-Test

A researcher plans to compare exam scores between two teaching methods. Based on prior literature, they expect a medium effect size (Cohen's d = 0.50). They set α = .05 and power = .80.

Parameters

Test: Independent samples t-test (two-tailed)
Effect size: d = 0.50 (medium)
Significance level: α = .05
Power: 1 - β = .80

Result

Required sample size: N = 128 (64 per group)

A power analysis for an independent-samples t-test was conducted using an effect size of d = 0.50, α = .05, and power = .80. The required sample size is N = 128 (64 per group).

When to Conduct a Power Analysis

Before data collection: A priori power analysis is the most common and important use. It determines how many participants you need to recruit.
Grant proposals: Funding agencies require justification of your planned sample size. A power analysis provides this justification.
IRB applications: Ethical review boards want to ensure you are not recruiting more participants than necessary (wasting resources) or too few (making the study futile).
Thesis/dissertation proposals: Most committees expect a formal power analysis in the methods section.

Common Mistakes in Power Analysis

Using a large effect size to reduce N: Don't inflate the expected effect size just to get a smaller sample. Use realistic estimates from pilot data or published literature.
Ignoring attrition: The calculated N is the minimum for analysis. If you expect 20% dropout, recruit 20% more than the calculated N.
Post-hoc power analysis: Computing power after data collection using the observed effect size is circular and uninformative. Power analysis should be done a priori.
Wrong test type: Make sure the power analysis matches the statistical test you plan to use. A power analysis for a t-test does not apply to an ANOVA.
One-tailed vs. two-tailed: This calculator uses two-tailed tests by default. One-tailed tests require fewer participants but are only appropriate when the direction of the effect is known in advance.

How to Report Power Analysis in APA Format

Include the power analysis in the Participants or Method section of your paper. State the test type, effect size, alpha level, desired power, and resulting sample size.

T-Test Example

An a priori power analysis for an independent-samples t-test was conducted using StatMate (Cohen's d = 0.50, α = .05, power = .80). The required minimum sample size was determined to be N = 128 (64 per group).

ANOVA Example

A power analysis for a one-way ANOVA with 4 groups was conducted using Cohen's f = 0.25 (medium effect), α = .05, and power = .80. The minimum required sample size wasN = 180 (45 per group).

Calculation Method

StatMate's sample size calculations use the standard normal approximation method with exact z-scores from the jStat library. For t-tests, the formula is n = (z_α/2 + z_β)² × 2 / d². Achieved power is computed by back-solving using the calculated sample size. Results have been validated against G*Power and R's pwr package.

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t検定

2群の平均値を比較

分散分析

3群以上の平均値を比較

カイ二乗検定

カテゴリ変数の関連を検定

相関分析

関係の強さを測定

記述統計

データを要約

1標本t検定

既知の値との比較

マン・ホイットニーU

ノンパラメトリック群間比較

ウィルコクソン検定

ノンパラメトリック対応検定

回帰分析

X-Yの関係をモデル化

重回帰分析

複数の予測変数

クロンバックのα

尺度の信頼性

ロジスティック回帰

二値アウトカムの予測

因子分析

潜在因子構造の探索

クラスカル・ウォリス

ノンパラメトリック3群以上比較

反復測定

被験者内分散分析

二元配置分散分析

要因計画の分析

フリードマン検定

ノンパラメトリック反復測定

フィッシャーの正確検定

2×2表の正確検定

マクネマー検定

対応のある名義データの検定

What is Sample Size Calculation?

The Four Components of Power Analysis

1. Effect Size (d, f, r, or w)

2. Significance Level (α)

3. Statistical Power (1 - β)

4. Sample Size (N)

The total number of participants required. This is typically the unknown you solve for. More participants increase power but also increase cost and time.

Effect Size Conventions by Test Type

Jacob Cohen (1988) established widely-used conventions for small, medium, and large effect sizes. Use these when you lack pilot data or prior research to estimate the expected effect.

Test	Measure	Small	Medium	Large
T-tests	Cohen's d	0.20	0.50	0.80
ANOVA	Cohen's f	0.10	0.25	0.40
Correlation	r	0.10	0.30	0.50
Chi-square	Cohen's w	0.10	0.30	0.50
Proportions	Cohen's h	0.20	0.50	0.80

Worked Example: Independent Samples T-Test

A researcher plans to compare exam scores between two teaching methods. Based on prior literature, they expect a medium effect size (Cohen's d = 0.50). They set α = .05 and power = .80.

Parameters

Test: Independent samples t-test (two-tailed)
Effect size: d = 0.50 (medium)
Significance level: α = .05
Power: 1 - β = .80

Result

Required sample size: N = 128 (64 per group)

A power analysis for an independent-samples t-test was conducted using an effect size of d = 0.50, α = .05, and power = .80. The required sample size is N = 128 (64 per group).

When to Conduct a Power Analysis

Before data collection: A priori power analysis is the most common and important use. It determines how many participants you need to recruit.

Grant proposals: Funding agencies require justification of your planned sample size. A power analysis provides this justification.

IRB applications: Ethical review boards want to ensure you are not recruiting more participants than necessary (wasting resources) or too few (making the study futile).

Thesis/dissertation proposals: Most committees expect a formal power analysis in the methods section.

Common Mistakes in Power Analysis

Using a large effect size to reduce N: Don't inflate the expected effect size just to get a smaller sample. Use realistic estimates from pilot data or published literature.

Ignoring attrition: The calculated N is the minimum for analysis. If you expect 20% dropout, recruit 20% more than the calculated N.

Post-hoc power analysis: Computing power after data collection using the observed effect size is circular and uninformative. Power analysis should be done a priori.

Wrong test type: Make sure the power analysis matches the statistical test you plan to use. A power analysis for a t-test does not apply to an ANOVA.

One-tailed vs. two-tailed: This calculator uses two-tailed tests by default. One-tailed tests require fewer participants but are only appropriate when the direction of the effect is known in advance.

How to Report Power Analysis in APA Format

Include the power analysis in the Participants or Method section of your paper. State the test type, effect size, alpha level, desired power, and resulting sample size.

T-Test Example

ANOVA Example

A power analysis for a one-way ANOVA with 4 groups was conducted using Cohen's f = 0.25 (medium effect), α = .05, and power = .80. The minimum required sample size wasN = 180 (45 per group).