Inferential statistics is the branch of statistics that uses data from a sample to draw conclusions, or inferences, about the wider population that the sample came from. Because it is rarely practical to measure an entire population, researchers study a representative sample and then use two core tools — estimation (such as confidence intervals) and hypothesis testing — to generalise their findings, while quantifying how much uncertainty that generalisation carries.
This is exactly what separates inferential statistics from descriptive statistics. Descriptive statistics simply summarise the data you actually have; inferential statistics use that data to reason about data you do not have. In short: descriptive statistics describe, inferential statistics infer.
This guide explains what inferential statistics is, how it differs from descriptive statistics, the two pillars of estimation and hypothesis testing, which tests to choose for which data, and a fully worked example you can follow step by step.
“Inferential statistics… allow you to make inferences about a population based on data gathered from a sample.” — Australian Bureau of Statistics (ABS), Statistical Language
Inferential Statistics: An Introduction
To understand inferential statistics, it helps to place it within the wider field of statistics — the branch of mathematics concerned with collecting, organising, analysing, interpreting and presenting data.
Statistics is broadly split into two areas:
- Descriptive statistics — methods for summarising and describing the features of a dataset (for example, the mean, median, standard deviation or a bar chart).
- Inferential statistics — methods for using sample data to draw conclusions about a population.
Inferential statistics rests on two main activities:
- Estimating parameters: using a statistic calculated from a sample (such as the sample mean) to estimate the corresponding, unknown value in the population (such as the population mean).
- Hypothesis testing: using sample data to test a specific claim about the population, so you can answer research questions and decide whether an observed effect is likely to be real or simply due to chance.
The bridge between the two is probability. Because a sample is only a slice of the population, every inference carries a degree of uncertainty, and probability is what lets us measure that uncertainty rather than ignore it.
Descriptive vs inferential statistics
- Summarise the data you have
- Mean, SD, charts
- Generalise to a population
- Estimation & hypothesis tests
Inferential Statistics vs. Descriptive Statistics
Descriptive statistics let you summarise and present the data you have collected; inferential statistics let you go beyond that data to make generalisations and predictions about a population. The key differences are set out below.
| Feature | Descriptive Statistics | Inferential Statistics |
|---|---|---|
| Purpose | Describe and summarise the data you actually collected | Draw conclusions about a population from a sample |
| Scope | Limited to the dataset in hand | Generalises beyond the dataset to the population |
| Typical tools | Mean, median, mode, range, standard deviation, charts and tables | Confidence intervals, hypothesis tests, regression, p-values |
| Output | Summary values and visual displays | Estimates, probabilities and decisions about hypotheses |
| Uncertainty | None — it reports exactly what was measured | Built in — every inference is reported with a margin of error or significance level |
A simple way to remember it: if you measure the exam marks of 30 students and report their average, that is descriptive. If you then use that average to estimate the average mark of all students on the course, that is inferential.
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Sampling Error and Inferential Statistics
Before going further, here is a quick recap of four terms that sit at the centre of inferential statistics.
Population: the entire group you want to draw conclusions about.
Statistic: a number that describes a sample — for example, the sample mean (x̄) or sample proportion.
Parameter: a number that describes the whole population — for example, the population mean (μ) or population proportion.
Because a sample is smaller than the population, it will almost never capture the population perfectly. The difference between the true population value (the parameter) and the value calculated from your sample (the statistic) is called sampling error.
Sampling error can occur with any sampling method, whether random or systematic, which is why some uncertainty is unavoidable in inferential statistics. You cannot eliminate it, but you can reduce and quantify it: larger samples and well-designed probability (random) sampling both shrink sampling error and make your estimates more reliable. The standard error is the statistic that measures how much a sample estimate is expected to vary from sample to sample.
When you estimate a population parameter, you can do so in two ways:
- Point estimate — a single value that is your best guess for the parameter. For example, using a sample mean of 68 kg as the estimate of the population mean.
- Interval estimate — a range of values within which the parameter is likely to lie. The most widely used interval estimate is the confidence interval.
Confidence Intervals
A confidence interval (CI) uses the variability around a sample statistic to produce a range of plausible values for the population parameter. Where a point estimate gives a single number, a confidence interval also tells you how precise that number is by accounting for sampling error.
Every confidence interval has an associated confidence level, such as 95%. The correct interpretation is about the procedure, not a single interval: if you repeated the study many times and built a 95% confidence interval each time, about 95% of those intervals would contain the true population parameter. A common 95% confidence interval for a mean takes the form:
You cannot be 100% certain that any single interval contains the true parameter, because the only way to know a parameter exactly is to measure the entire population. But with an adequate sample size and random sampling, the confidence-interval method is reliable over the long run — and a narrower interval indicates a more precise estimate.
Hypothesis Testing and Inferential Statistics
Hypothesis testing is the second pillar of inferential statistics. It is a formal procedure for using sample data to decide between two competing claims about a population, so you can judge whether an observed effect is likely to be real or simply the result of chance.
A hypothesis test usually involves these steps:
- State the hypotheses. The null hypothesis (H₀) says there is no effect or no difference; the alternative hypothesis (H₁) says there is.
- Set the significance level (α), commonly 0.05. This is the risk you are willing to take of rejecting a true null hypothesis.
- Choose and run a suitable statistical test, which produces a test statistic.
- Find the p-value, the probability of obtaining a result at least as extreme as the observed one if the null hypothesis were true.
- Make a decision. If the p-value is less than or equal to α, you reject the null hypothesis and call the result statistically significant; otherwise you fail to reject it.
The statistical test you choose accounts for sampling error so that your inference is valid. Tests fall into two broad families:
- Parametric tests assume the data meet certain conditions (see below). When those assumptions hold, parametric tests are generally more powerful — that is, more likely to detect a real effect.
- Non-parametric tests make fewer assumptions about the shape of the population distribution and are used when parametric assumptions are violated, for example with skewed data or ordinal data. They are sometimes called “distribution-free” tests.
Common assumptions for parametric tests include:
- The data are drawn from a population that follows a normal distribution
- The sample is large enough and randomly drawn to represent the population
- The variances of the groups being compared are roughly equal (homogeneity of variance)
Forms of Statistical Tests
Inferential statistical tests generally serve one of three goals: comparing groups, measuring association, or predicting an outcome.
Comparison Tests
Comparison tests examine whether the means, medians or rankings of two or more groups differ. Examples include the t-test (two groups) and ANOVA (three or more groups).
Correlation Tests
Correlation tests measure the strength and direction of the association between two or more variables — for example, Pearson’s r for linear relationships.
Regression Tests
Regression tests model how changes in one or more predictor variables relate to changes in an outcome variable, and can be used for prediction. The right regression model depends on the number and type of predictor and outcome variables.
Common Inferential Statistics Tests
The table below summarises widely used inferential tests and when to apply each. Use it alongside our guide on which statistical test you should use.
| Test | Purpose | When to use | Type |
|---|---|---|---|
| Independent t-test | Compare the means of two independent groups | One continuous outcome, two unrelated groups, roughly normal data | Parametric |
| Paired t-test | Compare two related means (e.g. before vs after) | Paired/repeated measurements on the same subjects | Parametric |
| One-way ANOVA | Compare the means of three or more groups | One continuous outcome, three+ independent groups | Parametric |
| Mann–Whitney U | Compare two groups when data are not normal/ordinal | Non-normal or ordinal data, two independent groups | Non-parametric |
| Kruskal–Wallis | Compare three or more groups on ranks | Non-normal or ordinal data, three+ groups | Non-parametric |
| Pearson’s r | Measure linear association between two variables | Two continuous, normally distributed variables | Parametric |
| Spearman’s rho | Measure monotonic association | Ordinal data or non-linear monotonic relationships | Non-parametric |
| Chi-square test | Test association between categorical variables | Two categorical variables, frequency/count data | Non-parametric |
| Linear regression | Predict a continuous outcome from predictors | Continuous outcome, one or more predictors | Parametric |
| Logistic regression | Predict a binary outcome from predictors | Binary (yes/no) outcome, one or more predictors | Parametric |
Choosing an inferential test
Inferential Statistics: A Worked Example
The example below shows how estimation and hypothesis testing work in practice using a one-sample test.
Step 1 — State the hypotheses.
H₀: μ = 7 (the population mean is 7 hours).
H₁: μ < 7 (the population mean is less than 7 hours). This is a one-tailed test.
Step 2 — Set the significance level. Use α = 0.05.
Step 3 — Compute the standard error.
SE = s ÷ √n = 1.5 ÷ √100 = 1.5 ÷ 10 = 0.15.
Step 4 — Compute the test statistic.
t = (x̄ − μ₀) ÷ SE = (6.6 − 7) ÷ 0.15 = −0.4 ÷ 0.15 ≈ −2.67.
Step 5 — Make a decision. With 99 degrees of freedom, the one-tailed critical value at α = 0.05 is about −1.66. Because −2.67 is more extreme than −1.66 (the corresponding p-value is roughly 0.004, well below 0.05), we reject H₀. There is statistically significant evidence that students sleep fewer than 7 hours on average.
Step 6 — Estimate with a confidence interval. A 95% CI for the mean is x̄ ± 1.984 × SE = 6.6 ± 1.984 × 0.15 = 6.6 ± 0.30, i.e. about 6.30 to 6.90 hours. Note that 7 hours falls outside this interval, which agrees with the test result.
This single example shows the two pillars working together: the hypothesis test answers “is there an effect?”, while the confidence interval answers “how large is it, and how precisely have we measured it?”
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