APA Statistical Notation
Standard Notation Conventions
| Statistic | Symbol | Format |
|---|---|---|
| Mean | M | M = 12.45 |
| Standard Deviation | SD | SD = 3.21 |
| Sample Size | n, N | n = 45 (group), N = 90 (total) |
| p-value | p | p = .023 or p < .001 |
| Correlation | r | r(48) = .45, p = .001 |
| Cohen's d | d | d = 0.65 |
APA style uses italics for statistical symbols, leading zeros only for values that can exceed 1 (so 'p = .05' not 'p = 0.05'), and specific decimal places (usually 2-3). These conventions ensure clarity and consistency.
Reporting t-tests
A complete t-test report includes descriptive statistics for both groups, the t-statistic with degrees of freedom, exact p-value (or < .001), effect size (Cohen's d), and a confidence interval around the effect.
Reporting ANOVA
ANOVA reports need the F-ratio with both degrees of freedom (between-groups, within-groups), p-value, effect size (eta-squared or partial eta-squared), and post-hoc comparisons when the omnibus test is significant.
Reporting Regression
| Predictor | B | SE | β | t | p |
|---|---|---|---|---|---|
| Experience | 2,450 | 432 | .45 | 5.67 | < .001 |
| Education | 8,320 | 2,364 | .28 | 3.52 | < .001 |
Regression reports should include overall model fit (F, R², adjusted R²), and for each predictor: unstandardized (B) and standardized (β) coefficients, standard errors, t-values, and p-values. Tables are preferred for multiple predictors.
Reporting Effect Sizes
Why Effect Sizes Matter
P-values depend on both effect size AND sample size. Two studies with similar p-values can have vastly different effect sizes. Always report effect sizes to show the magnitude of your findings, not just their statistical reliability.
Choosing Effect Size Measures
| Analysis | Effect Size | Small/Med/Large |
|---|---|---|
| t-test | Cohen's d | 0.2 / 0.5 / 0.8 |
| Correlation | r (itself is ES) | .1 / .3 / .5 |
| ANOVA | η² or η²ₚ | .01 / .06 / .14 |
| Regression | R², f², β | .02 / .13 / .26 (R²) |
| Chi-square | Cramér's V, φ | .1 / .3 / .5 |
| Non-parametric | Rank-biserial r | .1 / .3 / .5 |
Different analyses require different effect size measures. Cohen's d for comparing means, r for correlations, eta-squared for ANOVA, R² for regression. Use the appropriate measure and report conventional benchmarks for interpretation.
Confidence Intervals for Effect Sizes
Confidence intervals around effect sizes show both significance (if CI excludes zero) and precision. A wide CI means uncertainty about the true effect magnitude. CIs are more informative than p-values alone.
Transparency Practices
Pre-registration
Pre-registration means publicly documenting your hypotheses, methods, and analysis plan BEFORE collecting data. This prevents p-hacking, HARKing (Hypothesizing After Results are Known), and publication bias. It's becoming standard in many fields.
Open Science Practices
Open science makes research more transparent and reproducible. Sharing data allows replication and secondary analysis. Sharing code ensures analyses can be verified. These practices build trust and accelerate science.
Complete Reporting
• Try multiple DVs, report the best
• Remove outliers until p < .05
• Label exploratory analyses as such
• Pre-specify exclusion criteria
Selective reporting (only publishing significant results) inflates the false positive rate in the literature. Report all pre-planned analyses, clearly distinguish confirmatory from exploratory, and don't hide null results.
Data Visualization
Effective Statistical Graphics
• Include error bars (SE or CI)
• Start y-axis at zero (usually)
• Label axes clearly with units
• Use colorblind-friendly palettes
• Bar charts hiding distribution shape
• 3D effects that distort perception
• Unlabeled or ambiguous legends
• Pie charts for comparisons
Good visualizations accurately represent data without distortion. Show individual data points when possible, include uncertainty (error bars, CIs), and avoid visual tricks that exaggerate effects. Honesty in graphics matches honesty in statistics.
Show the Data, Not Just Summaries
Bar charts with error bars hide crucial information about distributions. Better alternatives include violin plots, box plots with points, or raincloud plots. When sample size permits, show individual data points alongside summaries.
Error Bar Types
| Type | Shows | Use When |
|---|---|---|
| SD | Data spread (variability in sample) | Describing the sample itself |
| SE | Precision of mean estimate | How precisely we estimated mean |
| 95% CI | Plausible range for true mean | Inference about population |
Different error bars mean different things! SD shows data spread, SE shows precision of the mean, 95% CI shows where the true population value likely lies. Always specify which you're using - never just say 'error bars.'
Common Reporting Issues
The p < .05 Problem
The sharp cutoff at p = .05 is arbitrary and misleading. Results just above or below .05 provide similar evidence. Report exact p-values, focus on effect sizes and confidence intervals, and interpret results on a continuum rather than dichotomously.
Language Matters
"We failed to find an effect"
"The results approached significance"
"The effect was small and imprecisely estimated, d = 0.15, 95% CI [-0.18, 0.48]"
Avoid language that overstates conclusions. 'Proved' is too strong - we have evidence, not proof. 'Failed to find' implies we should have - a null result might be true. 'Approached significance' is meaningless. Be precise about what the data show.
Reporting Checklist
Use a reporting checklist to ensure completeness. Guidelines like JARS (APA), CONSORT (trials), and STROBE (observational studies) provide detailed checklists for specific study types.
Real-World Examples
Apply your knowledge to practical scenarios
Improving a Thesis Results Section
A student writes: 'The t-test was significant (p < .05). This proves our hypothesis that the drug works.' The advisor asks them to revise according to APA guidelines and best practices.
Sample Data
{
"original": "t-test was significant (p < .05)",
"sample_sizes": "n = 25 per group",
"means": "Treatment M = 72.3, Control M = 65.8",
"sds": "Treatment SD = 14.2, Control SD = 13.8",
"actual_p": "p = .032"
}Key Insights
- Always include n, M, SD for each group
- Report exact p-values and effect sizes with CIs
- Interpret effect size magnitude in practical terms
Clinical Trial Reporting (CONSORT)
A pharmaceutical company reports trial results. They need to follow CONSORT guidelines: flow diagram showing participant flow, intention-to-treat analysis, all randomized participants accounted for, adverse events reported.
Sample Data
{
"enrolled": 500,
"randomized_treatment": 252,
"randomized_control": 248,
"completed_treatment": 218,
"completed_control": 231,
"analyzed_treatment": 252,
"analyzed_control": 248,
"analysis": "Intention-to-treat with multiple imputation"
}Key Insights
- CONSORT ensures complete, transparent trial reporting
- Flow diagrams show what happened to every participant
- ITT analysis preserves randomization benefits
Pre-Registered Replication
Researchers conduct a pre-registered replication of a famous psychology finding. Their pre-registration specifies: exact procedures, sample size (n = 200, based on power analysis), primary DV, analysis plan. Results: d = 0.12, 95% CI [-0.08, 0.32], compared to original d = 0.65.
Sample Data
{
"original_d": 0.65,
"replication_d": 0.12,
"replication_ci": "[-0.08, 0.32]",
"replication_p": ".24",
"replication_n": 200,
"original_n": 40
}Key Insights
- Pre-registration prevents p-hacking in replications
- Small original samples often overestimate effects
- Report effect sizes and CIs to compare original vs replication
Statistical Reporting Quiz
Test your understanding • Passing score: 70%