Solving Statistics Assignments: Choosing the Right Statistical Test
Statistics Student Guide
Solving Statistics Assignments: Choosing the Right Statistical Test
A complete decision framework covering t-tests, ANOVA, chi-square, regression, and non-parametric tests — with practical examples for US and UK university students using SPSS, R, and Python.
Order Statistics Help NowThe Foundation
Why Choosing the Right Statistical Test Matters
Choosing the right statistical test is not a technicality you learn once and forget. It is the analytical spine of every empirical assignment you will ever submit. The wrong test produces invalid results — not just a lower grade, but genuinely misleading conclusions. Knowing why this matters changes how you approach every dataset you touch.
Think about it this way: running a t-test on ordinal data (like a 1–5 Likert scale) violates the test’s core assumptions. The output will look legitimate — you’ll get a p-value, degrees of freedom, a t-statistic — but the result is statistically invalid. Your professor knows this. Peer reviewers know this. And in applied settings — clinical trials, policy research, business analytics — the consequences extend beyond a poor grade.
4
key questions to answer before selecting any statistical test
30+
distinct statistical tests students encounter in undergraduate and graduate programs
0.05
the conventional alpha threshold — but understanding what it actually means changes everything
The Four Questions That Drive Every Test Selection Decision
Question 1: What is your research question? Are you comparing groups? Measuring the relationship between variables? Predicting an outcome? Testing whether observed frequencies match expected ones? Each maps to a different family of tests.
Question 2: What type of data do you have? Nominal (unordered categories — gender, blood type), ordinal (ranked categories — satisfaction ratings), interval (equal spacing, no true zero — IQ scores), or ratio (equal spacing, true zero — height, weight, income)? This determines whether parametric or non-parametric approaches apply.
Question 3: How many groups or variables are involved? Two groups or three? One predictor or five? One dependent variable or multiple? Each answer shifts you to a different test.
Question 4: Are statistical assumptions met? For parametric tests — is your data normally distributed? Are group variances approximately equal? Are observations independent? Checking assumptions before running a test is not optional — it determines whether your chosen test is valid.
“The most common mistake students make is choosing a statistical test based on what they know how to run, not based on what their data and research question actually require. Statistical software makes it dangerously easy to produce wrong results that look professional.”
What Is Hypothesis Testing and Why Does It Structure Everything?
Hypothesis testing is the framework virtually all common statistical tests operate within. Before choosing a test, you formulate two hypotheses: the null hypothesis (H₀) — usually a statement of “no effect,” “no difference,” or “no association” — and the alternative hypothesis (H₁) — the effect you are testing for. The statistical test then evaluates whether the data provide sufficient evidence to reject H₀ in favor of H₁, based on a pre-specified significance level α (typically 0.05).
Step One
Understanding Data Types: The First Decision in Choosing the Right Test
Before you select a statistical test, you must classify your variables. Data type determines which tests are mathematically valid for your analysis. Applying a test designed for continuous data to categorical data — or vice versa — produces results that are, at best, misleading and, at worst, completely meaningless.
The classic framework classifies data into four levels of measurement, originally proposed by psychologist Stanley Smith Stevens in a landmark 1946 paper in Science. Knowing which level your variables occupy maps directly to which tests apply.
The Four Levels of Measurement
Nominal Data
Nominal data consists of unordered categories with no inherent rank or numerical meaning. Examples: eye color, country of birth, disease diagnosis (yes/no), political party affiliation. Valid tests: chi-square, logistic regression, some non-parametric tests.
Ordinal Data
Ordinal data has a meaningful order, but the intervals between categories are not necessarily equal. Examples: Likert scale responses, satisfaction ratings (1–10), academic grades, pain severity. Valid tests: Mann-Whitney, Kruskal-Wallis, Wilcoxon, Spearman correlation, chi-square.
Interval Data
Interval data has equal intervals between values, but no true zero point. Examples: temperature in Celsius, IQ scores, standardized test scores. Valid tests: t-tests, ANOVA, Pearson correlation, regression — all parametric tests, assuming normality.
Ratio Data
Ratio data has equal intervals AND a true zero point. Examples: height, weight, income, reaction time, number of items sold. All arithmetic operations apply. Ratio data supports the full range of parametric statistical tests.
Practical shortcut: For test selection purposes, interval and ratio data are treated identically — both support parametric tests when normality holds. If your data is continuous and could in principle range from zero upward, treat it as ratio. If measured on an arbitrary scale with no meaningful zero, treat it as interval. Either way, the same tests apply.
How Data Type Maps to Statistical Tests
- Nominal dependent variable → chi-square test, logistic regression, Fisher’s exact test
- Ordinal dependent variable → Mann-Whitney, Kruskal-Wallis, Wilcoxon, Spearman correlation
- Interval/Ratio dependent variable, normally distributed → t-tests, ANOVA, Pearson correlation, linear regression
- Interval/Ratio dependent variable, not normal → non-parametric equivalents or data transformation
Group Comparisons
Statistical Tests for Comparing Groups: t-Tests, ANOVA, and Their Variants
Comparing group means is the most common task in statistics assignments across psychology, business, education, medicine, and social science. The right test depends on how many groups you have, whether those groups are independent or related, and whether your data meets parametric assumptions.
The Independent Samples t-Test
The independent samples t-test compares the means of two separate, unrelated groups on a continuous dependent variable. Classic example: do male and female students score differently on a statistics exam? Group 1 and Group 2 are independent — a student belongs to one group or the other, not both.
Assumptions: (1) the dependent variable is continuous (interval or ratio), (2) observations are independent within and across groups, (3) the dependent variable is approximately normally distributed in each group — use Shapiro-Wilk to test this, (4) the groups have equal variances — use Levene’s test; if violated, use Welch’s t-test correction.
Reading t-Test Output: What to Report
Report: the t-statistic, degrees of freedom (df), the p-value, the mean difference, a 95% confidence interval for the mean difference, and Cohen’s d as the effect size. Example: “An independent samples t-test revealed that students who used tutoring services (M = 78.4, SD = 9.2) scored significantly higher than those who did not (M = 71.3, SD = 10.5), t(148) = 4.12, p < .001, d = 0.71.”
The Paired Samples t-Test
The paired samples t-test compares means from the same group measured under two different conditions — typically before and after an intervention, or the same participants tested in two matched conditions. Because you are comparing within participants rather than across them, you control for individual differences, making this test more powerful when the study design calls for it.
One-Way ANOVA: Comparing Three or More Groups
The moment you have three or more groups to compare, shift from t-tests to ANOVA (Analysis of Variance). Running three separate t-tests does not maintain your 5% error rate — it inflates it. With three comparisons at α = .05, the familywise Type I error rate rises to approximately 14%. ANOVA tests all groups simultaneously, maintaining the error rate at α.
A significant F-test tells you that differences exist somewhere among the groups. That requires post-hoc tests such as Tukey’s HSD, Bonferroni correction, or Games-Howell (when variances are unequal) to identify which specific groups differ.
Two-Way ANOVA and Interaction Effects
Two-way ANOVA examines the effect of two independent categorical variables (factors) on a continuous dependent variable, and crucially, tests whether there is an interaction effect — whether the effect of one factor depends on the level of the other. Example: does the effect of study method (lectures vs. online) differ for different student groups (undergrad vs. postgrad)? Report: main effects for each factor, the interaction effect, F-statistics, p-values, partial η² as effect size.
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Chi-Square Tests and Other Tests for Categorical Data
When your data consists of categories rather than measurements — frequencies, counts, proportions — you need a fundamentally different class of tests. The chi-square family is the foundation here, though Fisher’s exact test, McNemar’s test, and logistic regression also play essential roles.
Chi-Square Test of Independence
The chi-square test of independence assesses whether two categorical variables are associated in a population. It works by comparing the observed frequencies in each cell of a contingency table with the expected frequencies under independence. Classic example: Is there an association between smoking status (smoker/non-smoker) and lung disease diagnosis (yes/no)?
Key Assumptions of Chi-Square Tests
- Categorical variables: Both variables must be categorical (nominal or ordinal with few categories).
- Independent observations: Each participant contributes to exactly one cell.
- Expected frequency ≥ 5 in each cell: If violated, use Fisher’s exact test instead.
- Large enough sample: Chi-square is an asymptotic test — it becomes more accurate with larger samples.
Chi-Square Goodness of Fit Test
The chi-square goodness of fit test tests whether the observed distribution of one categorical variable matches a theoretical or expected distribution. Example: is a die fair? You roll it 60 times and expect 10 outcomes for each face — the test checks whether observed counts deviate significantly from expectations.
Effect Size for Chi-Square: Cramér’s V and Phi
A significant chi-square test tells you an association exists — it does not tell you how strong it is. Always supplement with an effect size: phi (φ) for 2×2 tables, Cramér’s V for larger tables. Conventions: 0.1 = small effect, 0.3 = medium, 0.5 = large.
Fisher’s Exact Test and McNemar’s Test
Fisher’s exact test is the precise alternative to chi-square when expected cell frequencies are below 5 — most commonly used with 2×2 tables in small samples. It calculates the exact probability of the observed frequency distribution rather than approximating it.
McNemar’s test is the paired version — used when the same participants are classified on a binary variable under two conditions. Example: did patients’ diagnosis status change after treatment? It is the categorical analogue of the paired t-test.
Relationships & Prediction
Correlation and Regression: Measuring and Predicting Relationships
While comparison tests ask “do groups differ?”, correlation and regression tests ask “how are variables related?” and “can I predict one variable from another?” These are the workhorses of social science research, economics, public health, and business analytics.
Pearson Correlation: Measuring Linear Association
The Pearson correlation coefficient (r) measures the strength and direction of the linear relationship between two continuous variables. It ranges from -1 (perfect negative linear relationship) through 0 (no linear relationship) to +1 (perfect positive linear relationship). Assumptions: both variables must be continuous, the relationship must be linear, both should be approximately normally distributed, and there should be no extreme outliers.
Spearman’s Rank Correlation: The Non-Parametric Alternative
Spearman’s rho (ρ) measures the strength of monotonic relationships and works on the ranks of values rather than the raw data. Use Spearman’s when: your data is ordinal, your continuous data violates normality, or you have significant outliers that would distort Pearson’s r. Interpreted identically to Pearson’s r: -1 to +1.
Simple Linear Regression: One Predictor, One Outcome
Simple linear regression fits a line (Ŷ = b₀ + b₁X) to the data that minimizes the sum of squared prediction errors (residuals). The slope b₁ tells you: for each one-unit increase in X, Y changes by b₁ units on average. R-squared (R²) tells you what proportion of the variance in Y is explained by X.
Multiple Regression: Multiple Predictors
Multiple regression extends simple regression to include two or more predictors. The model: Ŷ = b₀ + b₁X₁ + b₂X₂ + … + bₖXₖ. Each slope coefficient represents the effect of that predictor on Y while holding all other predictors constant. Key outputs: the model F-test, individual predictor t-tests, standardized betas, R² and Adjusted R², and confidence intervals for each coefficient. Check for multicollinearity using Variance Inflation Factors (VIF): VIF > 10 is problematic.
Correlation ≠ Causation: Both correlation and regression measure statistical association, not causation. A significant regression coefficient means X is a statistically significant predictor of Y in your sample — it does not mean X causes Y. Causal inference requires experimental design, natural experiments, or sophisticated causal modelling. Always acknowledge this distinction when interpreting results.
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Non-Parametric Statistical Tests: When and How to Use Them
Non-parametric tests do not assume that the data follow a specific parametric distribution. Instead, they work on the ranks of data values. This makes them more robust when normality is violated, sample sizes are small, data is ordinal, or outliers are severe.
Mann-Whitney U Test: The Non-Parametric Independent t-Test
The Mann-Whitney U test is the non-parametric alternative to the independent samples t-test. It tests whether one group tends to have higher values than the other by comparing ranks rather than means. Use Mann-Whitney when: your two-group continuous data violates normality (especially with n < 30 per group), or your data is ordinal. Note: Mann-Whitney does not directly compare medians — it compares the entire distribution of ranks between groups.
Wilcoxon Signed-Rank Test: The Non-Parametric Paired t-Test
The Wilcoxon signed-rank test is the non-parametric equivalent of the paired samples t-test. Use it when you have paired or repeated-measures data but the difference scores are not normally distributed. It ranks the absolute differences between pairs and tests whether positive and negative ranks are symmetrically distributed around zero.
Kruskal-Wallis Test: The Non-Parametric ANOVA
The Kruskal-Wallis test extends Mann-Whitney to three or more independent groups — it is the non-parametric equivalent of one-way ANOVA. A significant result tells you that at least one group differs from the others. Post-hoc tests include pairwise Mann-Whitney tests with Bonferroni correction or the Dunn test.
Friedman Test: Non-Parametric Repeated Measures ANOVA
The Friedman test is the non-parametric equivalent of repeated measures ANOVA — used when the same participants are measured under three or more conditions and the data is ordinal or non-normal. Post-hoc: pairwise Wilcoxon tests with Bonferroni correction.
| Parametric Test | Non-Parametric Equivalent | When to Switch |
|---|---|---|
| Independent t-test | Mann-Whitney U / Wilcoxon rank-sum | Non-normal data, ordinal DV, small samples |
| Paired t-test | Wilcoxon signed-rank test | Non-normal difference scores, ordinal paired data |
| One-way ANOVA | Kruskal-Wallis test | Non-normal data, 3+ groups, ordinal DV |
| Repeated measures ANOVA | Friedman test | Non-normal repeated data, 3+ conditions, ordinal DV |
| Pearson correlation | Spearman rank correlation | Ordinal data, non-linearity, outliers |
| One-sample t-test | Wilcoxon one-sample signed-rank | Non-normal single-group data vs. hypothesized median |
The Decision Tree
The Complete Statistical Test Selection Decision Framework
All of the above comes together in a systematic decision framework. Rather than memorizing dozens of individual tests in isolation, choosing the right statistical test becomes a structured decision process — move through the branches in order and you will arrive at the right test for any situation you encounter.
Branch 1: What Is Your Research Question?
Step 1 — Identify the type of question
Comparing group means → Go to Branch 2
Testing association between two variables → Go to Branch 3
Predicting one variable from others → Use Regression (Branch 4)
Testing frequencies or proportions → Use Chi-Square family
Comparing one group to a known value → One-sample t-test or Wilcoxon one-sample
Branch 2: Comparing Group Means
Step 2A — How many groups?
Two groups → Step 2B
Three or more groups → Step 2C
Step 2B — Are the two groups independent or related?
Independent (different people in each group) → Step 2D
Related (same people measured twice, or matched pairs) → Step 2E
Step 2D — Is the dependent variable continuous and approximately normal?
Yes → Independent samples t-test (check Levene’s for equal variances; if unequal, use Welch’s)
No → Mann-Whitney U test
Step 2E — Is the difference score approximately normally distributed?
Yes → Paired samples t-test
No → Wilcoxon signed-rank test
Step 2C — Three or more groups
Continuous DV, normal, independent groups → One-way ANOVA + post-hoc tests
Continuous DV, normal, two factors → Two-way ANOVA
Continuous DV, normal, same participants, 3+ conditions → Repeated measures ANOVA
Non-normal or ordinal, independent groups → Kruskal-Wallis test
Non-normal or ordinal, same participants → Friedman test
Branch 3: Testing Association
Step 3 — What types are the two variables?
Both continuous, linear relationship, normal → Pearson correlation
Both continuous or ordinal, non-linear or non-normal → Spearman correlation
Both categorical → Chi-square test of independence
Both categorical, small sample (expected freq < 5) → Fisher’s exact test
Same participant, binary categorical, two conditions → McNemar’s test
Branch 4: Prediction and Regression
Step 4 — What type is your outcome (dependent) variable?
Continuous outcome, one predictor → Simple linear regression
Continuous outcome, multiple predictors → Multiple linear regression
Binary outcome (yes/no) → Binary logistic regression
Ordinal outcome → Ordinal logistic regression
Count outcome → Poisson regression
Categorical outcome with 3+ categories → Multinomial logistic regression
Validity Checks
Checking Statistical Assumptions: The Step Most Students Skip
Choosing the right statistical test is half the battle. The other half is verifying that your chosen test’s assumptions are met before running it. Skipping assumption checks is the single most common methodological error in student statistics assignments — and one that examiners, supervisors, and journal reviewers are trained to look for.
How to Test Normality
1
Shapiro-Wilk Test
The Shapiro-Wilk test is the most powerful normality test for small to moderate samples (n < 50). A non-significant result (p > .05) indicates data is consistent with a normal distribution. Available in SPSS (Explore), R (shapiro.test()), and Python (scipy.stats.shapiro). With very large samples, Shapiro-Wilk detects trivial deviations — use it alongside visual checks.
2
Q-Q Plot (Quantile-Quantile Plot)
A Q-Q plot displays sample quantiles against theoretical normal quantiles. If the data is normally distributed, points fall roughly on a straight diagonal line. Systematic curvature indicates skewness; S-shapes indicate heavy tails. Use in conjunction with Shapiro-Wilk, not as a substitute.
3
Skewness and Kurtosis Values
Skewness near 0 (roughly -0.5 to +0.5) and kurtosis near 3 indicate approximate normality. With n > 100, parametric tests are generally robust to moderate non-normality by the Central Limit Theorem.
4
Histogram
A simple histogram with a normal curve overlay gives a quick visual impression. Does the distribution look roughly bell-shaped? Heavily skewed, bimodal, or flat distributions signal that normality may be violated. Use as a starting point, not a final verdict.
How to Test Homogeneity of Variance
Levene’s test checks homogeneity of variance for t-tests and ANOVA. A significant Levene’s test (p < .05) indicates that variances differ significantly across groups. For t-tests, report Welch’s t-test correction. For ANOVA, use Welch’s ANOVA or Brown-Forsythe ANOVA.
Checking Independence of Observations
Independence of observations is the most fundamental and most overlooked assumption. Independence is violated when: data is collected from the same participants at multiple time points, participants are nested in groups like classrooms or clinics, or data has spatial or temporal autocorrelation.
Red flag: If your data has any clustering structure — students within schools, patients within hospitals, employees within companies — and you run a standard t-test or ANOVA without accounting for this nesting, your standard errors are too small, your test statistics are inflated, and your p-values are misleadingly low.
Practical Application
Running Statistical Tests in SPSS, R, and Python: What to Report
Knowing which statistical test to use is one thing. Running it correctly in software — and reporting results in the format your institution expects — is another. Statistical reporting standards matter: APA format for psychology, Vancouver style for medicine, Chicago for economics.
Running Tests in SPSS
SPSS remains the dominant tool in psychology, education, social work, and health science programs. Key navigation paths:
- Independent t-test: Analyze → Compare Means → Independent Samples T Test
- Paired t-test: Analyze → Compare Means → Paired Samples T Test
- One-way ANOVA: Analyze → Compare Means → One-Way ANOVA (include post-hoc options)
- Chi-square: Analyze → Descriptive Statistics → Crosstabs → Statistics → Chi-square
- Pearson/Spearman correlation: Analyze → Correlate → Bivariate (select Pearson or Spearman)
- Regression: Analyze → Regression → Linear (or Logistic for binary outcomes)
- Mann-Whitney / Wilcoxon: Analyze → Nonparametric Tests → Legacy Dialogs → 2 Independent Samples
- Kruskal-Wallis: Analyze → Nonparametric Tests → Legacy Dialogs → K Independent Samples
Running Tests in R
R is increasingly required in quantitative social science, statistics, data science, and econometrics programs. Key R functions:
- t.test(x, y) — independent; t.test(x, y, paired=TRUE) — paired
- aov(y ~ group, data=df) — one-way ANOVA; TukeyHSD() for post-hoc
- chisq.test(table) — chi-square; fisher.test(table) — Fisher’s exact
- cor.test(x, y, method=”pearson”) or method=”spearman”
- lm(y ~ x, data=df) — linear regression; glm(y ~ x, family=binomial) — logistic
- wilcox.test(x, y) — Mann-Whitney; kruskal.test(y ~ group, data=df) — Kruskal-Wallis
APA Reporting Format for Common Statistical Tests
The APA Publication Manual (7th edition) prescribes specific formatting for statistical results:
- Independent t-test: t(df) = x.xx, p = .xxx, d = effect-size. Example: t(78) = 3.42, p = .001, d = 0.76
- ANOVA: F(df_between, df_within) = x.xx, p = .xxx, η² = effect-size. Example: F(2, 87) = 8.14, p < .001, η² = .16
- Chi-square: χ²(df, N = sample_size) = x.xx, p = .xxx. Example: χ²(1, N = 120) = 6.42, p = .011
- Pearson correlation: r(df) = .xxx, p = .xxx. Example: r(48) = .52, p = .003
- Regression: F(df_regression, df_residual) = x.xx, p = .xxx, R² = .xxx. Example: F(3, 96) = 12.44, p < .001, R² = .28
Always report exact p-values (p = .037, not p < .05) unless p < .001. Always include effect sizes and confidence intervals.
| Test | APA Reporting Format | Effect Size Measure | Software Output to Use |
|---|---|---|---|
| Independent t-test | t(df) = x.xx, p = .xxx | Cohen’s d | SPSS: Independent Samples Test table |
| Paired t-test | t(df) = x.xx, p = .xxx | Cohen’s d (paired) | SPSS: Paired Samples Test table |
| One-way ANOVA | F(df₁, df₂) = x.xx, p = .xxx | Partial η² or η² | SPSS: ANOVA table; R: anova(model) |
| Chi-square | χ²(df, N=n) = x.xx, p = .xxx | Cramér’s V or phi | SPSS: Chi-Square Tests table |
| Pearson r | r(df) = .xxx, p = .xxx | r itself is the effect size | SPSS: Correlations table |
| Multiple Regression | F(df₁, df₂) = x.xx, p = .xxx, R² = .xxx | R², f² (Cohen) | SPSS: Model Summary + ANOVA + Coefficients |
| Mann-Whitney | U = xxx, p = .xxx | Rank-biserial correlation r | SPSS: Test Statistics table |
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Common Mistakes When Choosing Statistical Tests — and How to Avoid Them
Mistake 1: Using a t-Test When You Have Three or More Groups
Running three separate t-tests to compare groups A, B, and C instead of using ANOVA is one of the most common test selection errors in student assignments. With three pairwise comparisons at α = .05, the experiment-wise error rate rises to approximately 1 – (0.95)³ = .143 — not 5%, but 14%. ANOVA controls this. If you find yourself running multiple t-tests on the same dataset for the same dependent variable, stop and use ANOVA.
Mistake 2: Ignoring Assumption Violations
Running a parametric test without checking assumptions — and without reporting assumption checks in your methods section — is a fundamental methodological gap. Examiners and supervisors expect to see: which normality test you used, what the result was, whether it was significant, and how you responded. If you violated normality with a small sample, they expect to see either a non-parametric alternative or a clear justification for proceeding with the parametric test.
Mistake 3: Confusing Statistical and Practical Significance
A statistically significant result (p < .05) does not automatically mean a meaningful or important result. With a sample of 10,000 participants, a difference of 0.3 points on a 100-point scale might be statistically significant but practically meaningless. Always report effect sizes. Cohen’s d < 0.2 is trivially small regardless of the p-value. Similarly, a non-significant result does not mean “no effect” — it may reflect insufficient statistical power.
Mistake 4: Treating Ordinal Data as Interval
Using a mean and running a t-test on Likert scale data is technically a violation of the interval measurement assumption. This is a contentious area in statistics — many researchers treat Likert data as interval because parametric tests are more powerful and robust with larger samples. Check what your course or supervisor requires. When in doubt, run both parametric and non-parametric versions and note if conclusions differ.
Mistake 5: Data Dredging and p-Hacking
Running many different tests on the same data and reporting only those with p < .05 is a serious methodological and ethical problem. With 20 statistical tests, you expect one to achieve p < .05 purely by chance even if there are no real effects. In student assignments: decide on your analysis plan before running any tests, report all tests you ran, and correct for multiple comparisons when appropriate (Bonferroni correction: divide α by the number of tests).
One more critical error: Failing to distinguish between one-tailed and two-tailed tests. A one-tailed test is only legitimate when you have a strong directional prediction established before seeing the data. Using a one-tailed test because it gives you a smaller p-value after peeking at your results is p-hacking — and examiners know to look for it.
Frequently Asked
Frequently Asked Questions About Choosing Statistical Tests
How do I choose the right statistical test for my assignment?
Choosing the right statistical test requires answering four questions in order: (1) What is your research question — comparing groups, testing association, predicting outcomes, or examining frequencies? (2) What type of data is your dependent variable — nominal, ordinal, interval, or ratio? (3) How many groups or variables are involved? (4) Do your data meet parametric assumptions (normality, equal variances, independence)? Match these answers to the decision framework: two groups, continuous normal data → t-test; three+ groups → ANOVA; two categorical variables → chi-square; two continuous variables → correlation/regression; violated normality → non-parametric equivalents.
What is the difference between a t-test and ANOVA?
A t-test compares the means of exactly two groups (independent or paired). ANOVA compares means across three or more groups simultaneously. The critical reason for using ANOVA instead of multiple t-tests: running multiple t-tests inflates the Type I error rate. With three pairwise t-tests at α = .05, the true error rate rises to approximately 14%. ANOVA maintains the error rate at the specified α by testing all groups in one omnibus F-test. Both require continuous normally distributed data and comparable group variances. After a significant ANOVA, post-hoc tests (Tukey, Bonferroni) identify which specific groups differ.
What is a p-value and what does it actually mean?
A p-value is the probability of observing data at least as extreme as what you found, assuming the null hypothesis is true. A p-value of 0.03 means: if there were truly no effect in the population, there is a 3% chance of getting results this extreme just by random sampling variation. It is NOT the probability that the null hypothesis is true. The conventional threshold α = 0.05 is arbitrary — established by statistician Ronald Fisher as a rough guideline, not a hard truth. Always interpret p-values alongside effect sizes and confidence intervals.
What is the difference between parametric and non-parametric tests?
Parametric tests assume the data follows a specific distribution (usually normal) and make inferences about population parameters (mean, variance). Examples: t-tests, ANOVA, Pearson correlation, regression. Non-parametric tests make no distributional assumptions and work on the ranks of data values rather than raw values. Examples: Mann-Whitney, Kruskal-Wallis, Wilcoxon, Spearman. Use non-parametric tests when: sample size is small (n < 30 per group), data is ordinal, normality is clearly violated, or there are extreme outliers. Parametric tests are more powerful when their assumptions are met.
When should I use a chi-square test?
Use a chi-square test when both variables are categorical (nominal or ordinal with few categories) and you want to test either: (1) independence — is there an association between the two categorical variables? Or (2) goodness of fit — do observed frequencies match expected frequencies from a theoretical distribution? You need frequency counts in each category, not means. Key assumptions: expected frequency ≥ 5 in every cell (if violated, use Fisher’s exact test), and observations must be independent. Always supplement with Cramér’s V or phi as an effect size measure.
What is the difference between correlation and regression?
Correlation measures the strength and direction of the linear relationship between two variables, producing a coefficient r between -1 and +1. Neither variable is treated as causing the other. Regression models the relationship to predict one outcome variable (Y) from one or more predictor variables (X), producing slope coefficients, an equation (Ŷ = b₀ + b₁X), and R-squared (proportion of variance explained). Regression allows multiple predictors, control for confounders, and prediction of new values. Use correlation to describe association. Use regression to predict, model, or control for variables.
What is Type I and Type II error, and why do they matter?
A Type I error (false positive) is rejecting the null hypothesis when it is actually true — concluding there is an effect when there isn’t one. Probability of Type I error = α (usually .05). A Type II error (false negative) is failing to reject the null when it is actually false — missing a real effect. Probability of Type II error = β; statistical power = 1 – β. Using a parametric test when assumptions are violated can inflate Type I error rates beyond the stated α. Using a non-parametric test when a parametric test was appropriate reduces power and increases Type II error.
Do I always need to check for normality before running a t-test?
Yes — you should always check normality and report the result in your methods section, even if you ultimately proceed with the parametric test. With sample sizes of n > 30 per group, the Central Limit Theorem makes parametric tests reasonably robust to moderate non-normality. With small samples (n < 30 per group), normality matters much more. Use Shapiro-Wilk test and Q-Q plots to check. If normality is clearly violated with a small sample, switch to Mann-Whitney (for two groups) or Kruskal-Wallis (for three+ groups). Always document what you checked and what you found.