Key takeaways
- Descriptive statistics summarise your sample; inferential statistics use the sample to draw conclusions about a wider population.
- Pick a test by answering three questions: what are you asking (difference, relationship, prediction), what type is your outcome variable, and are your groups independent or repeated?
- Parametric tests (t-tests, ANOVA, Pearson r, linear regression) have non-parametric or alternative counterparts for ranked, skewed or categorical data.
- Always report descriptive statistics first, then the inferential result with its effect size and confidence interval.
- Choose and justify the test before you look at the results, ideally in your proposal or methodology chapter.
How do you choose a statistical test?
You choose a statistical test by matching it to your research question (are you comparing groups, looking for a relationship, or predicting an outcome?), the measurement level of your outcome variable (continuous, ordinal or categorical), and your design (independent groups or repeated measurements of the same people). Once those three are fixed, the options usually narrow to one main test and one fallback for when its assumptions fail.
Before any test, though, comes description. Every results chapter starts by showing what the data look like, and that is where we begin.
Descriptive vs inferential statistics, with examples
Descriptive statistics summarise the data you actually collected. They answer “what does my sample look like?” Typical examples are the mean and standard deviation of exam scores, the median household income in a survey, or the percentage of respondents who chose each answer.
Inferential statistics use that sample to make a claim about the population it came from, with a stated level of uncertainty. They answer “is this pattern likely to hold beyond my sample?” Examples include a t-test showing that two teaching methods produce different average scores, a correlation between sleep and mood, or a regression predicting job satisfaction from workload and pay.
| Descriptive example | Inferential counterpart |
|---|---|
| Group A scored M = 72.4, Group B M = 66.8 | Independent-samples t-test: is the 5.6-point gap larger than chance would produce? |
| 58% of nurses report burnout | 95% confidence interval for the population percentage |
| Scatterplot of study hours and grades | Pearson correlation with a p-value and confidence interval |
Both matter. Description lets readers judge whether a result is meaningful in practice; inference tells them how far to trust it.
Step 1: name your question type
Almost every quantitative research question falls into one of four families:
- Difference: do groups or time points differ on an outcome? (e.g. do online and in-person students differ in final grades?)
- Relationship: do two variables move together? (e.g. is screen time related to sleep quality?)
- Prediction: how well do one or more variables predict an outcome, and which matter most once the others are controlled?
- Association between categories: is membership in one category linked to membership in another? (e.g. is faculty linked to preferred study mode?)
If you are not sure which family your question belongs to, rewrite it as a hypothesis. The verb gives it away: “differ”, “relate”, “predict” or “is associated with”. Our guide to hypothesis testing and p-values shows how to phrase null and alternative hypotheses for each.
Step 2: identify your outcome variable type
- Continuous (interval or ratio): numbers where differences are meaningful, such as test scores, reaction time, income or blood pressure. Usually suits parametric tests.
- Ordinal: ordered categories or ranks, such as a single five-point Likert item or class rank. Usually suits rank-based (non-parametric) tests. A scale averaged across several Likert items is commonly treated as continuous.
- Categorical (nominal): unordered groups, such as yes/no, pass/fail or blood type. Suits chi-square tests or logistic regression.
Then check your design. Independent groups means each person appears in only one group. Paired or repeated measures means the same people (or matched pairs) are measured more than once, such as before and after an intervention.
Want a second pair of eyes? Not sure which test fits your design? A statistician can review your question and data with you and explain the choice. See statistical consulting.
Decision table: which test fits your data
| Question | Outcome variable | Design | Main test | If assumptions fail |
|---|---|---|---|---|
| Difference, 1 group vs a known value | Continuous | One sample | One-sample t-test | Wilcoxon signed-rank |
| Difference, 2 groups | Continuous | Independent | Independent-samples t-test (Welch version if variances differ) | Mann–Whitney U |
| Difference, 2 time points | Continuous | Paired | Paired-samples t-test | Wilcoxon signed-rank |
| Difference, 3+ groups | Continuous | Independent | One-way ANOVA | Kruskal–Wallis |
| Difference, 3+ time points | Continuous | Repeated | Repeated-measures ANOVA | Friedman test |
| Relationship between 2 variables | Continuous, both | One sample | Pearson correlation | Spearman’s rho |
| Relationship | Ordinal, at least one | One sample | Spearman’s rho or Kendall’s tau | — |
| Prediction | Continuous | One or more predictors | Linear regression | Robust or transformed regression |
| Prediction | Binary (yes/no) | One or more predictors | Logistic regression | — |
| Association | Categorical, both | Independent | Chi-square test of independence | Fisher’s exact test (small expected counts) |
| Association | Categorical, 2 time points | Paired | McNemar’s test | — |
Two notes keep this table honest. First, “non-parametric” is not a free pass: rank-based tests answer a slightly different question (about distributions or ranks rather than means), so describe the result accordingly. Second, with complex designs, such as nested data (pupils within schools), several outcomes at once, or latent constructs, you will move beyond this table to mixed models, MANOVA or structural equation modelling.
Checking assumptions before you commit
Parametric tests share a short list of assumptions. Check them in this order:
- Independence: one person’s score does not influence another’s. This comes from your design and sampling method, not from a test.
- Approximate normality: of the outcome within each group (t-test, ANOVA) or of the residuals (regression). Look at histograms and Q–Q plots. With roughly 30 or more per group, t-tests and ANOVA are fairly robust to moderate skew.
- Equal variances: for group comparisons. Levene’s test helps, but many statisticians simply default to Welch’s t-test, which does not need this assumption.
- Linearity and no extreme outliers: for correlation and regression. A scatterplot shows both.
Be cautious about formal normality tests such as Shapiro–Wilk with large samples: they flag trivial departures that do not matter. Plots and judgement usually serve you better.
A worked example of choosing a test
A master’s student asks: “Does a peer-mentoring programme improve first-year students’ sense of belonging?” She measures belonging on a 10-item scale (scores averaged, range 1–5) at the start and end of term for 40 mentored students.
- Question type: difference over time.
- Outcome: continuous (a multi-item scale mean).
- Design: paired, because the same students are measured twice.
The table points to a paired-samples t-test, with the Wilcoxon signed-rank test as a fallback if the difference scores are badly skewed. If she later adds a non-mentored comparison group, the question becomes “does change differ between groups?”, which calls for a mixed (between–within) ANOVA or a regression on post-test scores controlling for pre-test scores.
She would report descriptive statistics for both time points first, then the test result with an effect size and confidence interval.
Common mistakes when choosing a test
Most problems examiners raise about test choice come from a handful of avoidable errors:
- Running many t-tests instead of one ANOVA. Comparing four groups pair by pair means six tests, and the chance of at least one false positive climbs well above 5%. Run the omnibus test first, then planned or corrected post hoc comparisons.
- Treating repeated measurements as independent. If the same participants appear in both conditions, an independent-samples test ignores the pairing and misstates the standard error.
- Switching tests after seeing the results. Trying a parametric test, then a non-parametric one, and reporting whichever gives p < .05 inflates false positives. Decide your main test and fallback rule in advance.
- Confusing a significant result with an important one. With thousands of cases, trivial differences become significant. Always pair the test with an effect size.
- Ignoring the design behind the data. Pupils nested in classes, or repeated survey waves, need models that respect that structure; a simple test will overstate certainty.
- Using correlation for a causal question. A correlation or regression on observational data can show association, not cause, however strong it is.
A one-paragraph justification in your methodology chapter, naming the question type, outcome variable, design and assumption checks, prevents most of these issues and answers the question before your supervisor or viva panel asks it.
Reporting descriptive and inferential statistics in APA style
APA 7 conventions that apply to every test:
- Italicise statistical symbols: M, SD, t, F, r, p, N.
- Round most statistics to two decimals; report exact p values to two or three decimals, and write p < .001 for anything smaller.
- Drop the leading zero for values that cannot exceed 1 (p = .025, r = .42), but keep it otherwise (d = 0.65).
- Give degrees of freedom, an effect size and, where possible, a 95% confidence interval.
Mentored students reported a greater sense of belonging at the end of term (M = 3.86, SD = 0.62) than at the start (M = 3.41, SD = 0.70), t(39) = 4.12, p < .001, d = 0.65, 95% CI [0.23, 0.67].
The confidence interval here is for the mean difference of 0.45 scale points, and d is the mean difference divided by the standard deviation of the difference scores (0.69). If you want someone to check your choice before you run the analysis, our statistics help explains the reasoning so you can defend it yourself.