Free Effect Size Calculator Tool for Research Statistics

Calculate effect sizes for research findings with our free effect size calculator. No registration, no fees - just comprehensive effect size computation with interpretation guidelines.

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What Are Effect Sizes?

Effect sizes quantify the magnitude of research findings in standardized units, independent of sample size. While p-values tell whether effects are statistically significant, effect sizes reveal how large or meaningful effects are. A statistically significant finding may have trivial practical importance, while large effects in small samples may not reach significance.

Why Effect Sizes Matter

• Practical significance - Distinguish statistical from practical importance
• Standardized comparison - Compare effects across different studies and measures
• Meta-analysis - Essential for combining results across studies
• Power analysis - Required for sample size calculations
• Journal requirements - Most journals mandate effect size reporting
• Effect interpretation - Contextualizes findings beyond p-values

Cohen's d (Standardized Mean Difference)

What It Measures

Cohen's d expresses mean differences in standard deviation units. Used for comparing two group means (t-tests) or pre-post changes within groups.

Formula: d = (M₁ - M₂) / SD_pooled

Interpretation Guidelines

Cohen's Benchmarks:

• d = 0.20 - Small effect
• d = 0.50 - Medium effect
• d = 0.80 - Large effect

More Nuanced Interpretation:

• d < 0.20 - Trivial
• d = 0.20-0.49 - Small
• d = 0.50-0.79 - Medium
• d ≥ 0.80 - Large

When to Use

Calculate Cohen's d for:

• Independent samples t-tests
• Paired samples t-tests
• Pre-test/post-test designs
• Experimental vs. control comparisons

Example

Treatment group mean anxiety = 45, control group = 55, pooled SD = 12 d = (45 - 55) / 12 = -0.83 (large effect, treatment reduces anxiety)

Hedges' g (Bias-Corrected d)

What It Measures

Hedges' g corrects Cohen's d for small sample bias. When n < 50, d overestimates population effect size. Hedges' g applies correction factor producing more accurate estimates.

Formula: g = d × (1 - 3/(4(n₁ + n₂) - 9))

When to Use

Use Hedges' g instead of Cohen's d when:

• Total sample size < 50
• Reporting for meta-analysis (preferred metric)
• Need unbiased effect size estimate

Interpretation identical to Cohen's d benchmarks.

Glass's Δ (Delta)

What It Measures

Glass's Δ uses control group standard deviation only as standardizer, rather than pooled SD. Appropriate when treatment changes variability or when control group represents baseline.

Formula: Δ = (M_treatment - M_control) / SD_control

When to Use

Use Glass's Δ when:

• Treatment affects variability
• Control group provides natural baseline
• Groups have heterogeneous variances

Interpretation similar to Cohen's d, though values may differ when variances unequal.

Eta Squared (η²) for ANOVA

What It Measures

η² represents proportion of total variance explained by factor(s). Used with ANOVA designs having one or more factors.

Formula: η² = SS_effect / SS_total

Interpretation Guidelines

• η² = 0.01 - Small effect
• η² = 0.06 - Medium effect
• η² = 0.14 - Large effect

Limitation

η² is biased in complex designs. Increases artificially as more factors added. Consider partial η² or ω² for multi-factor ANOVA.

Omega Squared (ω²)

What It Measures

ω² is less biased estimate of variance explained than η². Accounts for error variance, providing more conservative and accurate effect size for population.

Formula: ω² = (SS_effect - (df_effect × MS_error)) / (SS_total + MS_error)

When to Use

Prefer ω² over η² for:

• Multi-factor ANOVA
• Unequal sample sizes
• Effect size estimation for populations
• More accurate variance explained estimates

Interpretation uses same benchmarks as η² (0.01, 0.06, 0.14 for small, medium, large).

Cohen's f for ANOVA

What It Measures

Cohen's f expresses ANOVA effect sizes in standardized form useful for power analysis. Unlike η² or ω² which are bounded (0-1), f can exceed 1.

Formula: f = √(η² / (1 - η²))

Interpretation Guidelines

• f = 0.10 - Small effect
• f = 0.25 - Medium effect
• f = 0.40 - Large effect

When to Use

Use Cohen's f for:

• ANOVA power analysis (required by G*Power)
• Comparing effects across different ANOVA designs
• Planning sample sizes for factorial designs

Pearson r (Correlation Effect Size)

What It Measures

Correlation coefficient r serves as effect size for relationships between continuous variables. Already standardized (-1 to +1).

Interpretation Guidelines

Cohen's Benchmarks:

• r = 0.10 - Small effect
• r = 0.30 - Medium effect
• r = 0.50 - Large effect

Field-Specific Context: In some fields (meteorology, economics), r = 0.30 represents strong relationships. Consider disciplinary norms alongside Cohen's benchmarks.

R² Interpretation

R² (coefficient of determination) shows variance explained:

• r = 0.30 means R² = 0.09 (9% variance explained)
• r = 0.50 means R² = 0.25 (25% variance explained)

Even "medium" correlations explain limited variance, highlighting complexity of human behavior.

Odds Ratio (OR)

What It Measures

Odds ratio quantifies association strength in 2×2 contingency tables (chi-square tests). Compares odds of outcome in one group vs. another.

Formula: OR = (a × d) / (b × c)

Where a, b, c, d are cell counts in 2×2 table.

Interpretation Guidelines

• OR = 1.00 - No association
• OR = 1.50 - Small effect
• OR = 2.50 - Medium effect
• OR = 4.00 - Large effect
• OR < 1.00 - Negative association

Example

Disease present: Treatment = 20, Control = 40 Disease absent: Treatment = 80, Control = 60 OR = (20 × 60) / (40 × 80) = 0.375 (treatment reduces disease odds by 62.5%)

Cohen's w for Chi-Square

What It Measures

Cohen's w quantifies effect size for chi-square tests of independence. Indicates degree of association between categorical variables.

Formula: w = √(χ² / N)

Interpretation Guidelines

• w = 0.10 - Small effect
• w = 0.30 - Medium effect
• w = 0.50 - Large effect

Use for chi-square goodness-of-fit and independence tests.

Converting Between Effect Sizes

Common Conversions

Many effect sizes can be converted approximately:

• d to r: r ≈ d / √(d² + 4)
• r to d: d ≈ 2r / √(1 - r²)
• η² to f: f = √(η² / (1 - η²))
• OR to d: d ≈ ln(OR) × √3 / π

Conversions enable comparing effects across studies using different metrics.

Confidence Intervals for Effect Sizes

Why CI Matters

Confidence intervals show effect size precision. Wide intervals indicate uncertainty; narrow intervals suggest precise estimates.

Example: d = 0.45, 95% CI [0.15, 0.75] Effect is medium, but CI includes small effects (0.15) and approaches large (0.75), indicating uncertainty.

Reporting

Report effect sizes with confidence intervals: "Treatment significantly reduced anxiety, t(98) = 3.24, p = .002, d = 0.65, 95% CI [0.25, 1.05]."

Context-Specific Interpretation

Don't Over-Rely on Benchmarks

Cohen's benchmarks are rough guidelines. Consider:

• Field norms: What effect sizes are typical in your discipline?
• Practical significance: Does this effect matter for real-world applications?
• Cost-benefit: Even small effects may be valuable if interventions are inexpensive
• Baseline rates: Small effects on rare outcomes may be important

Example Contexts

Medical interventions: Small effects (d = 0.20) reducing mortality are extremely valuable despite being "small" by Cohen's standards.

Educational interventions: Medium effects (d = 0.50) represent meaningful learning gains equivalent to several months' progress.

Psychological interventions: Large effects (d = 0.80) for depression treatment represent substantial symptom reduction.

Reporting Effect Sizes

APA Style Requirements

Report effect sizes alongside significance tests:

• State which effect size measure used
• Provide effect size values
• Include interpretation when helpful
• Consider confidence intervals

Example: "The intervention significantly improved test scores, t(120) = 4.56, p < .001, Cohen's d = 0.83, 95% CI [0.46, 1.20], representing a large effect."

Meta-Analysis Preparation

When conducting research for potential meta-analysis:

• Report multiple effect size types
• Provide means, SDs, and sample sizes
• Calculate Hedges' g (preferred for meta-analysis)
• Include sufficient detail for future meta-analysts

Transform Your Statistical Reporting

Stop reporting p-values without effect sizes. Calculate comprehensive, interpretable effect sizes that reveal the magnitude and practical importance of your findings.