Statistics Resources: Test Chooser, APA7 Cheat Sheet and Glossary

Quick references for choosing a test, checking assumptions, reporting results in APA7 style and understanding statistical terms. Everything here is free.

Choose a test Assumption checks APA7 cheat sheet Effect sizes Data preparation Sample size Glossary Further reading

Which analysis do I need?

Start with your question and your data, then follow the link to the tutorial. If an assumption fails, the last column suggests an alternative.

I want to…	Use	If assumptions fail
Describe one variable (centre, spread, shape)	Descriptive Statistics	—
Count categories and percentages	Frequency Analysis	—
Compare one sample mean with a known value	One-Sample t-Test	Wilcoxon Signed-Rank Test
Compare two different groups on a continuous outcome	Independent-Samples t-Test	Mann–Whitney U Test
Compare the same people measured twice	Paired-Samples t-Test	Wilcoxon Signed-Rank Test
Compare three or more different groups	One-Way ANOVA	Kruskal–Wallis Test
Compare the same people across three or more conditions or times	One-Way ANOVA with Repeated Measures	Friedman Test
Test two factors and their interaction (different people in each cell)	Two-Way ANOVA	—
Test two factors when at least one is repeated	Two-Way ANOVA with Repeated Measures	—
Measure the association between two continuous variables	Pearson Correlation	Spearman's Rank Correlation
Predict a continuous outcome from one or more predictors	Linear Regression	—
Predict a yes/no outcome	Logistic Regression	—
Test whether two categorical variables are associated	Chi-Squared Test	—
Compare one yes/no proportion with an expected value	Binomial Test	—
Test whether a variable carries the effect of X on Y	Mediation Analysis	—
Test whether the effect of X on Y depends on another variable	Moderation Analysis	—
Test a model with latent variables and several paths	SEM	—
Check whether items measure the factors you expect	CFA	—
Find out how many factors underlie your items	EFA	—
Reduce many correlated variables to a few components	PCA	—
Explore direct connections among many variables	Network Analysis	—
Show group means, trends, shares or a matrix of values	Bar Plot, Line Graph, Pie Chart, Heatmap	—

Rule of thumb: count your groups, ask whether the same people are measured more than once, and check whether the outcome is continuous, ordinal or categorical.

Assumption checks at a glance

Assumption	How to check	If it is violated
Normality	Histogram, Q–Q plot, skewness and kurtosis, Shapiro–Wilk test	With large samples most tests are robust. For small samples with clear skew use a rank-based test or transform the data.
Equal variances (homogeneity)	Levene's test, comparing group SDs	Use Welch's t-test or Welch's ANOVA, and Games–Howell post hoc tests.
Sphericity (repeated measures)	Mauchly's test	Apply the Greenhouse–Geisser or Huynh–Feldt correction, or use a multivariate test.
Linearity	Scatterplot, plot of residuals against predicted values	Add a transformation or a curve term, or use Spearman correlation.
Independence	Study design: one row per person, no clustering	Use repeated-measures or multilevel methods when observations are related.
Outliers	Box plots, standardised scores beyond ±3.29, Cook's distance	Check for data-entry errors first, then report analyses with and without extreme cases.
Multicollinearity	Correlations among predictors, variance inflation factor (VIF above about 5–10 is a concern)	Drop or combine highly overlapping predictors.
Sampling adequacy (factor analysis)	KMO of about .60 or higher, significant Bartlett's test	Remove items with low correlations or collect more data.
Expected counts (chi-squared)	At least 5 in about 80% of cells, none below 1	Combine categories or use Fisher's exact test for 2 × 2 tables.
Multivariate normality (SEM)	Mardia's test, item skewness and kurtosis	Use a robust estimator or an ordinal estimator for Likert items.

APA7 reporting cheat sheet

Statistics

Result	APA7 format (illustrative numbers)	Notes
t-test	t(48) = 2.31, p = .025, d = 0.65	Degrees of freedom in parentheses. Welch tests can have decimal df, reported to two decimals.
ANOVA	F(2, 72) = 6.91, p = .002, η² = .16	Give both degrees of freedom: between (effect) and within (error).
Correlation	r(48) = .63, 95% CI [.43, .77], p < .001	df = N − 2. No leading zero for r.
Spearman correlation	r_s(38) = −.77, p < .001	Subscript s after the italic r.
Regression	F(2, 97) = 38.66, p < .001, R² = .44; b = 1.67, 95% CI [1.29, 2.05]	Put all coefficients in a table.
Chi-squared	χ²(2, N = 180) = 6.50, p = .039	Include the sample size next to the degrees of freedom.
Mann–Whitney U	U = 40.0, z = −4.42, p < .001, r = −.70	Report medians and interquartile ranges for each group.
Kruskal–Wallis H	H(2) = 7.82, p = .020	df = number of groups − 1.
Means and standard deviations	M = 24.92, SD = 3.73	Usually two decimals; use the same number of decimals for a variable throughout.
Confidence interval	95% CI [1.29, 2.05]	Square brackets, lower and upper limits separated by a comma.
p-values	p = .032; p < .001	Exact value to two or three decimals. Never write p = .000.
SEM / CFA fit	χ²(24) = 25.83, p = .362, CFI = .997, RMSEA = .016, 90% CI [.000, .051], SRMR = .031	Report the estimator and how missing data were handled.

Rules to remember

Italicise statistical symbols such as M, SD, t, F, p, r, N, n and d. Greek letters (α, β, χ², η²) are not italicised.
Report exact p-values to two or three decimals. Use p < .001 for smaller values. Do not write p = .000.
Omit the leading zero for values that cannot exceed 1: correlations, proportions, p-values, R² and η² (.45, not 0.45). Keep it for values that can exceed 1, such as t or d.
Put a space on both sides of =, < and >, and use a true minus sign (−) for negative values.
Report degrees of freedom in parentheses after the statistic, and the sample size with χ².
Use two decimals for most statistics and keep the same precision for a variable throughout the paper.
Give a confidence interval and an effect size with every key result, and say which effect size you used.
Use numerals for numbers 10 and above, and for any number that is a statistic, a score or a measurement.

Tables and figures

Number tables and figures in the order they are mentioned. The table number is bold, and the title below it is in italic title case.
Use horizontal lines only: above and below the column headings, and at the bottom of the table. Avoid vertical lines.
Add a note below the table or figure that starts with Note. to explain abbreviations, error bars and significance markers.
Label every axis with the variable name and units, and make figures readable in black and white.

Source: Publication Manual of the American Psychological Association (7th ed.). Journals and institutions may add their own rules.

Effect size benchmarks

Effect sizes say how large an effect is, independent of sample size. The labels below are Cohen's conventions and should be treated as rough guides.

Measure	Used with	Small	Medium	Large
Cohen's d (and Hedges' g)	t-tests, difference between two means	0.2	0.5	0.8
r (correlation)	Correlation; also z/√N for rank tests	.10	.30	.50
η² / partial η²	ANOVA, Kruskal–Wallis (η²_H)	.01	.06	.14
R²	Regression (variance explained)	.02	.13	.26
f²	Regression, change in R²	.02	.15	.35
Cohen's f	ANOVA (power analysis)	.10	.25	.40
Cramér's V (df* = 1)	Chi-squared test of independence	.10	.30	.50
Cramér's V (df* = 2)	Chi-squared test, larger tables	.07	.21	.35
Odds ratio	Logistic regression, 2 × 2 tables (rough guide)	1.5	2.5	4.3
Kendall's W	Friedman test	.10	.30	.50
Cohen's h	Difference between two proportions	0.2	0.5	0.8

Preparing your data

One row per case. In most analyses each row is one participant and each column is one variable. Add an ID column.
Wide or long. Repeated measures are usually easiest in wide format (one column per time point). Group comparisons use a group column and an outcome column.
Short, unique names. Use names without spaces or special characters, for example anxiety_t1.
Consistent coding. Spell category labels identically, code yes/no outcomes as 0 and 1, and keep a codebook that explains each variable.
Numbers only in numeric columns. Remove units, symbols and comments from number cells and leave missing values empty.
Reverse-score items before computing scale scores, and check ranges for impossible values.
Look at missing data. Report how much is missing and how you handled it (for example listwise deletion or full-information maximum likelihood).
Save a CSV copy. CSV is the most portable format. Keep your original file untouched and work on a copy.

How big a sample do I need?

Power analysis gives the sample size needed to detect an effect of a given size. The figures below are for 80% power, α = .05, two-sided, and are rounded. Use a dedicated tool such as G*Power for your own design.

Design and expected effect	Approximate sample size
Two independent groups, medium effect (d = 0.5)	About 64 per group for 80% power at α = .05 (two-sided).
Two independent groups, large effect (d = 0.8)	About 26 per group.
Paired design, medium effect (d_z = 0.5)	About 34 pairs.
Correlation, medium (r = .30)	About 85 participants.
One-way ANOVA, 3 groups, medium (f = .25)	About 53 per group (159 in total).
Multiple regression, medium (f² = .15), 3 predictors	About 77 participants.
Mediation and moderation	Usually 100–200 or more; interactions need larger samples than main effects.
CFA, EFA, SEM	At least about 200, more for complex models.

Glossary

Alpha (α): The cut-off for statistical significance, usually .05. It is the long-run probability of a false positive (Type I error) when the null hypothesis is true.
Bonferroni correction: A way to control false positives across several tests by dividing α (or multiplying each p-value) by the number of tests.
Bootstrap: Estimating uncertainty by repeatedly resampling your own data with replacement, for example to build a confidence interval for an indirect effect.
Confidence interval (CI): A range of plausible values for a population quantity. A 95% CI means that, over many samples, about 95% of such intervals would contain the true value.
Covariate: A variable you control for in an analysis because it may influence the outcome.
Degrees of freedom (df): The number of values that are free to vary when estimating a statistic. They determine the shape of the test distribution.
Dependent variable (DV): The outcome that you measure.
Effect size: A standardised measure of how large an effect or association is, independent of sample size.
Eigenvalue: In factor analysis and PCA, the amount of variance captured by a component or factor.
Factor (ANOVA): A categorical independent variable, such as teaching method. Its categories are called levels.
Factor (EFA/CFA): A latent variable that explains why a group of items correlate.
Fit index: A number such as CFI, TLI, RMSEA or SRMR that summarises how well a structural model reproduces the data.
Homoscedasticity: Equal spread of residuals across all levels of the predicted values.
Independent variable (IV): The predictor or grouping variable whose effect you study.
Interaction: When the effect of one factor depends on the level of another.
Kurtosis: How heavy the tails of a distribution are compared with a normal distribution.
Latent variable: A construct that is not measured directly but inferred from several observed items.
Likert scale: A rating scale with ordered response options, for example from strongly disagree to strongly agree.
Listwise deletion: Dropping every case that has a missing value on any variable used in the analysis.
Loading: The strength of the relationship between an item and a factor or component.
Mediator: A variable that carries part or all of the effect of a predictor on an outcome.
Moderator: A variable that changes the strength or direction of a relationship.
Multicollinearity: When predictors are so strongly correlated that their separate effects cannot be estimated reliably.
Normal distribution: The symmetric, bell-shaped distribution assumed by many parametric tests.
Null hypothesis: The default claim, usually of no difference or no effect, that a significance test tries to reject.
Odds ratio: How much the odds of an event change for a one-unit increase in a predictor; 1 means no effect.
Outlier: A value far from the rest of the data. Investigate it before deciding whether to keep it.
p-value: The probability of results at least as extreme as yours if the null hypothesis were true. It is not the probability that the null hypothesis is true.
Parametric / non-parametric: Parametric tests assume a specific distribution, usually normal. Non-parametric tests use ranks or counts and make fewer assumptions.
Post hoc test: A follow-up comparison between specific groups after a significant omnibus test.
Power: The probability of detecting a real effect of a given size; 1 − β. 80% is a common target.
Residual: The difference between an observed value and the value predicted by a model.
Skewness: The asymmetry of a distribution; a long tail to the right is positive skew.
Sphericity: In repeated-measures ANOVA, equal variances of the differences between all pairs of conditions.
Standard deviation (SD): The typical distance of scores from the mean.
Standard error (SE): The typical distance of a sample statistic, such as a mean, from the population value; it shrinks as the sample grows.
Type I / Type II error: A Type I error is a false positive (rejecting a true null). A Type II error is a false negative (missing a real effect).