Inferential Statistics for Psychology
Learning Objectives
- Explain the logic of inferential statistics: using a sample to make claims about a population.
- Distinguish parametric from nonparametric inference and know when each applies.
- Interpret a confidence interval and explain what "95% confidence" actually means.
- Calculate a confidence interval for a sample mean using the standard error.
- Match a research question to the appropriate inferential test (t-test, ANOVA, correlation, chi-square).
Quick Answer
Inferential statistics let a researcher take data from a sample and make a justified, probability-based claim about the larger population that sample came from. Because psychologists can almost never measure everyone in a population (all anxious teenagers, all married couples, all working adults), they measure a manageable sample and use inferential techniques — hypothesis tests, confidence intervals, correlation, and regression — to estimate how likely it is that a pattern seen in the sample reflects a real pattern in the population, rather than random sampling noise. This is the statistical backbone that turns "we found a difference in our 60 participants" into "we have evidence this effect exists more broadly."
The Core Logic: Sample to Population
Imagine you want to know if a new mindfulness app reduces stress in university students generally. You cannot test every student on Earth, so you test 80 students. Inferential statistics asks: if there were truly no effect in the population, how likely is it that we'd see a stress reduction this large in our sample just by chance? If that probability is very low, you have evidence for a real effect. This single question — "how likely is chance to explain what I found?" — underlies every inferential technique in this chapter.
Parametric vs. Nonparametric Inference
Parametric tests assume the population data follow a known distribution (usually normal) and that variables are measured on interval/ratio scales. Examples: t-tests, ANOVA, Pearson correlation. They're more statistically powerful when their assumptions hold.
Nonparametric tests make fewer assumptions about the population's distribution and work well with ordinal data or when normality is violated. Examples: Mann-Whitney U test (nonparametric alternative to the independent t-test), Kruskal-Wallis H test (alternative to one-way ANOVA).
| Situation | Choose |
|---|---|
| Interval/ratio data, roughly normal distribution, adequate sample size | Parametric test |
| Ordinal data, or small sample with skewed/non-normal distribution | Nonparametric test |
Confidence Intervals: Estimating with Honesty
A confidence interval (CI) gives a range of plausible values for a population parameter, rather than a single guess — acknowledging that any sample estimate carries uncertainty.
Worked example. A researcher measures the height of 500 college students: sample mean = 175 cm, sample standard deviation = 6 cm.
Standard error (SE) = SD / √n = 6 / √500 = 6 / 22.36 ≈ 0.268
For a 95% confidence interval, the critical z-value is 1.96: Margin of error = 1.96 × 0.268 ≈ 0.525
95% CI = 175 ± 0.525 → [174.47, 175.53]
What this actually means: if we repeated this sampling process many times and built a CI each time, about 95% of those intervals would contain the true population mean. It does not mean "there's a 95% chance the true mean is in this specific interval" — a common and important distinction on exams.
Common Inferential Tests at a Glance
- t-test: compares means between two groups (independent-samples) or two conditions within the same people (paired-samples). Example: comparing IQ scores of children from single-parent vs. two-parent households (H0: μ1 = μ2).
- ANOVA (Analysis of Variance): compares means across three or more groups at once, avoiding the inflated error rate of running many separate t-tests. Example: comparing academic performance across three teaching methods.
- Correlation (Pearson's r): tests whether two continuous variables are related. Example: hours of TV watched vs. life satisfaction (H0: ρ = 0).
- Regression: predicts one variable's value from one or more others. Example: predicting math scores from study hours (H0: β1 = 0).
- Chi-square: tests relationships between categorical variables. Example: is there an association between gender and product preference?
- Mann-Whitney U / Kruskal-Wallis H: nonparametric alternatives to the t-test/ANOVA, used with ordinal data or non-normal distributions.
Real-World Application
Clinical trials for psychotherapy rely entirely on inferential statistics. Suppose researchers test whether Cognitive Behavioral Therapy (CBT) reduces depression more than a waitlist control. They can't treat every depressed person on the planet — they recruit, say, 120 participants, randomly assign them to CBT or waitlist, and use an independent-samples t-test to compare post-treatment depression scores. A statistically significant result, combined with a reported confidence interval and effect size, is what allows clinicians and health systems to trust that CBT will likely help future patients too — not just the 60 people who happened to be in the treatment group.
Key Terms
| Term | Definition |
|---|---|
| Inferential statistics | Methods for drawing conclusions about a population based on sample data |
| Parametric test | A test that assumes the population follows a specific distribution (usually normal) |
| Nonparametric test | A test that makes fewer assumptions about the population's distribution |
| Standard error | The standard deviation of the sampling distribution of a statistic; SD / √n for a mean |
| Confidence interval | A range of values likely to contain the true population parameter |
| t-test | A test comparing means between two groups or conditions |
| ANOVA | A test comparing means across three or more groups |
| Chi-square test | A test of association between categorical variables |
Common Mistakes
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Misconception: "A 95% confidence interval means there's a 95% chance the true population mean falls in this specific range." Why it's wrong: Once a specific interval is calculated, the true mean either is or isn't in it — there's no probability left to assign to that particular interval. Correct: 95% confidence describes the long-run success rate of the method: if you repeated the sampling and interval-construction process many times, about 95% of those intervals would capture the true mean.
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Misconception: "Inferential statistics prove a theory is correct." Why it's wrong: Inferential tests only estimate the probability that observed data would occur if the null hypothesis were true — they never confirm a theory with certainty. Correct: A significant result provides evidence against the null hypothesis; it does not "prove" the alternative hypothesis or the underlying theory.
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Misconception: You can use a t-test regardless of how many groups you're comparing, just running it multiple times. Why it's wrong: Running many pairwise t-tests inflates the overall Type I error rate (the more comparisons, the higher the chance of a false positive somewhere). Correct: Use ANOVA for three or more groups, which controls the overall error rate in a single test, and follow up with corrected post-hoc comparisons if needed.
Comparison and Connections
| Feature | Descriptive Statistics | Inferential Statistics |
|---|---|---|
| Goal | Summarize the sample | Generalize to the population |
| Certainty | Exact | Probabilistic |
| Key outputs | Mean, SD, mode | p-value, confidence interval, test statistic |
| Depends on | Just the data collected | Sampling assumptions, distribution shape |
| Feature | Parametric Tests | Nonparametric Tests |
|---|---|---|
| Distribution assumption | Normal (or near-normal) | None/minimal |
| Data type | Interval/ratio | Ordinal or non-normal interval/ratio |
| Statistical power | Higher (when assumptions met) | Lower, but more robust to violations |
| Examples | t-test, ANOVA, Pearson r | Mann-Whitney U, Kruskal-Wallis H |
Practice Questions
Recall
- What is the difference between a parametric and a nonparametric test? Answer guidance: Parametric tests assume a specific population distribution (usually normal) and interval/ratio data; nonparametric tests make fewer distributional assumptions and suit ordinal or non-normal data.
- Define standard error and give its formula for a sample mean. Answer guidance: SE = SD / √n; it estimates how much sample means would vary from sample to sample.
Understanding 3. Explain in your own words why a 95% CI does not mean "95% chance the true value is in this interval." Answer guidance: The true parameter is fixed, not random; the 95% refers to the long-run proportion of such intervals (across repeated sampling) that would contain the true value. 4. Why does running many separate t-tests instead of one ANOVA increase the risk of a false-positive finding? Answer guidance: Each individual test carries its own chance of Type I error (commonly 5%); running multiple tests multiplies the opportunities for a chance false positive, inflating the overall error rate (family-wise error rate).
Application 5. A sample of 100 students has a mean test anxiety score of 45 with SD = 10. Calculate the standard error and construct an approximate 95% CI. Answer guidance: SE = 10/√100 = 1.0; margin of error = 1.96 × 1.0 = 1.96; 95% CI ≈ [43.04, 46.96]. 6. A researcher wants to compare reading comprehension scores across four different teaching methods. Which test should they use, and why not run six separate t-tests instead? Answer guidance: ANOVA, because it compares all four group means in a single test while controlling the overall Type I error rate; six separate t-tests would inflate the chance of a false positive.
Analysis 7. A study's data on reaction times is heavily right-skewed with a small sample. Evaluate whether a parametric t-test is appropriate, and justify an alternative. Answer guidance: The skewness and small sample violate the normality assumption underlying the t-test; a nonparametric alternative like the Mann-Whitney U test would be more appropriate since it doesn't require a normal distribution. 8. Compare what a p-value and a confidence interval each tell you about a research finding, and explain why reporting both is considered best practice. Answer guidance: A p-value indicates how surprising the data would be if the null hypothesis were true (significance), while a CI gives a range of plausible effect sizes (precision and practical magnitude). Reporting both gives both a significance decision and a sense of the effect's real-world size and precision.
FAQ
Why can't we just test the whole population instead of a sample? Populations are often too large, dispersed, or inaccessible to measure completely (cost, time, and ethical constraints), so researchers sample a manageable, ideally representative, subset and use inferential statistics to generalize responsibly.
What does "statistically significant" actually mean? It means the observed result would be unlikely (conventionally, less than 5% probability) to occur if the null hypothesis were true — it does not automatically mean the effect is large or practically important.
How do I know if I should use a parametric or nonparametric test? Check your data's distribution shape (normal vs. skewed), sample size, and level of measurement. When in doubt with small or skewed samples, nonparametric tests are the safer, more conservative choice.
Is a wider confidence interval better or worse? Worse for precision — a wider CI means more uncertainty about the true population value, often due to a smaller sample size or more variable data. Narrower CIs (from larger, more consistent samples) give more precise estimates.
What's the practical difference between correlation and regression as inferential tools? Correlation tests whether two variables are related and how strongly; regression goes further, building an equation to predict one variable's value from another, which is useful for forecasting, not just describing association.
Quick Revision
- Inferential statistics generalize from a sample to a population using probability.
- Parametric tests assume normal distributions and interval/ratio data (t-test, ANOVA, Pearson r).
- Nonparametric tests make fewer assumptions, suited to ordinal or skewed data (Mann-Whitney U, Kruskal-Wallis H).
- Standard error = SD / √n; it shrinks as sample size grows.
- A confidence interval gives a range of plausible values for a population parameter, not a single point estimate.
- 95% CI means 95% of such intervals, across repeated sampling, would contain the true value — not a 95% chance for this one interval.
- t-test compares two group means; ANOVA compares three or more group means in one test.
- Chi-square tests association between categorical variables.
- Running multiple t-tests instead of one ANOVA inflates the false-positive (Type I error) rate.
- Significant ≠ proven and significant ≠ large — always consider effect size and confidence intervals alongside p-values.
Related Topics
Prerequisites: Introduction to Statistics, Descriptive Statistics.
Related: Hypothesis Testing, Correlation and Regression.
Next: Hypothesis Testing, which formalizes the decision process (null vs. alternative hypotheses, significance levels, p-values) introduced here.