{"id":482,"date":"2023-04-05T13:34:07","date_gmt":"2023-04-05T13:34:07","guid":{"rendered":"https:\/\/www.editage.com\/blog\/?p=482"},"modified":"2026-06-08T06:14:32","modified_gmt":"2026-06-08T06:14:32","slug":"guide-to-types-of-inferential-statistics-for-biomedical-researchers","status":"publish","type":"post","link":"https:\/\/www.editage.com\/blog\/guide-to-types-of-inferential-statistics-for-biomedical-researchers\/","title":{"rendered":"What are Inferential Statistics? Calculations, Examples, Tips"},"content":{"rendered":"\n<p>Contents<\/p>\n\n\n\n<ul><li><a href=\"#_Toc231811362\">What Is Inferential Statistics?<\/a><\/li><li><a href=\"#_Toc231811363\">Why Is Inferential Statistics Needed?<\/a><\/li><li><a href=\"#_Toc231811364\">Inferential Statistics vs Descriptive Statistics<\/a><\/li><li><a href=\"#_Toc231811365\">Key Concepts in Inferential Statistics<\/a><\/li><li><a href=\"#_Toc231811366\">Estimating Population Parameters<\/a><\/li><li><a href=\"#_Toc231811367\">Hypothesis Testing<\/a><\/li><li><a href=\"#_Toc231811368\">Types of Statistical Tests in Inferential Statistics<\/a><\/li><li><a href=\"#_Toc231811369\">Core Statistical Tests Explained<\/a><\/li><li><a href=\"#_Toc231811370\">Regression Analysis in Inferential Statistics<\/a><\/li><li><a href=\"#_Toc231811371\">Worked Example: Applying Inferential Statistics<\/a><\/li><li><a href=\"#_Toc231811372\">Assumptions of Inferential Statistics<\/a><\/li><li><a href=\"#_Toc231811373\">Real-World Applications of Inferential Statistics<\/a><\/li><li><a href=\"#_Toc231811374\">Key Takeaways<\/a><\/li><li><a href=\"#_Toc231811375\">Frequently Asked Questions<\/a><\/li><\/ul>\n\n\n\n<p>Inferential statistics is a branch of statistics that allows researchers, analysts, and data scientists to draw conclusions and make predictions about a <strong>population<\/strong> based on data collected from a <strong>sample<\/strong>. Rather than examining every individual in a group, which is often impossible or prohibitively expensive, inferential statistics uses probability and analytical tools to make reliable generalizations. It is central to fields as diverse as medicine, economics, social science, machine learning, and market research.<\/p>\n\n\n\n<p><strong>Glossary of Key Terms<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><td><strong>Term<\/strong><\/td><td><strong>Definition<\/strong><\/td><\/tr><\/thead><tbody><tr><td>Population<\/td><td>The entire group of individuals or data points that a study seeks to understand.<\/td><\/tr><tr><td>Sample<\/td><td>A subset of the population selected for analysis.<\/td><\/tr><tr><td>Parameter<\/td><td>A numerical value that describes a characteristic of the whole population (e.g., population mean).<\/td><\/tr><tr><td>Statistic<\/td><td>A numerical value that describes a characteristic of a sample (e.g., sample mean).<\/td><\/tr><tr><td>Sampling Error<\/td><td>The difference between a true population parameter and the corresponding sample statistic.<\/td><\/tr><tr><td>Null Hypothesis (H\u2080)<\/td><td>The default assumption that there is no effect or no difference between groups.<\/td><\/tr><tr><td>Alternative Hypothesis (H\u2081)<\/td><td>The claim being tested; asserts that an effect or difference exists.<\/td><\/tr><tr><td>p-value<\/td><td>The probability of obtaining the observed results if the null hypothesis is true. A small p-value indicates strong evidence against H\u2080.<\/td><\/tr><tr><td>Significance Level (\u03b1)<\/td><td>The threshold p-value (commonly 0.05) below which the null hypothesis is rejected.<\/td><\/tr><tr><td>Confidence Interval<\/td><td>A range of values within which the true population parameter is expected to lie with a stated probability (e.g., 95%).<\/td><\/tr><tr><td>Point Estimate<\/td><td>A single value used to estimate a population parameter.<\/td><\/tr><tr><td>Test Statistic<\/td><td>A numerical value computed from sample data used to decide whether to reject H\u2080.<\/td><\/tr><tr><td>Type I Error<\/td><td>Rejecting a true null hypothesis (false positive). Probability = \u03b1.<\/td><\/tr><tr><td>Type II Error<\/td><td>Failing to reject a false null hypothesis (false negative). Probability = \u03b2.<\/td><\/tr><tr><td>Statistical Power<\/td><td>The probability of correctly detecting a true effect (1 \u2212 \u03b2).<\/td><\/tr><tr><td>Normal Distribution<\/td><td>A symmetric, bell-shaped probability distribution important to many statistical tests.<\/td><\/tr><tr><td>Central Limit Theorem<\/td><td>States that sample means approximate a normal distribution as sample size increases, regardless of the population distribution.<\/td><\/tr><tr><td>Degrees of Freedom<\/td><td>The number of values in a calculation that are free to vary; affects the shape of many test distributions.<\/td><\/tr><tr><td>Regression Analysis<\/td><td>A method for quantifying the relationship between one or more predictor variables and an outcome variable.<\/td><\/tr><tr><td>ANOVA<\/td><td>Analysis of Variance; a test for comparing means across three or more groups.<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2><a id=\"_Toc231811362\">What Is Inferential Statistics?<\/a><\/h2>\n\n\n\n<p>Inferential statistics is a field of statistics that uses analytical tools to draw conclusions about a population by examining data from a representative sample. The goal is to make generalizations that extend beyond the data actually collected, using probability theory to account for the inherent uncertainty of working with samples rather than entire populations.<\/p>\n\n\n\n<h3>Example<\/h3>\n\n\n\n<p>Consider a pharmaceutical company testing a new drug. It cannot give the drug to every person on earth; instead, it gives the drug to several thousand participants and uses inferential statistics to determine whether the results are likely to hold across the broader population.<\/p>\n\n\n\n<h3>Purpose of inferential statistics<\/h3>\n\n\n\n<p>Inferential statistics has two core functions:<\/p>\n\n\n\n<ul><li>Estimating population parameters from sample data (e.g., estimating the average income of a country from a survey sample).<\/li><li><a href=\"https:\/\/www.editage.com\/blog\/hypothesis-testing-different-types-for-biomedical-researchers\/\">Testing hypotheses<\/a> to draw conclusions about relationships or differences in populations (e.g., determining whether a new teaching method improves exam scores).<\/li><\/ul>\n\n\n\n<h2><a id=\"_Toc231811363\">Why Is Inferential Statistics Needed?<\/a><\/h2>\n\n\n\n<p>In most real-world situations, studying an entire population is impractical. Censuses are expensive and time-consuming; clinical trials cannot enrol every patient; quality checks cannot test every product on a production line. Inferential statistics solves this problem systematically by:<\/p>\n\n\n\n<ul><li>Allowing conclusions about large, inaccessible populations from manageable samples.<\/li><li>Providing a mathematical framework for measuring and communicating uncertainty.<\/li><li>Enabling hypothesis testing so claims can be accepted or rejected with a calculable level of confidence.<\/li><li>Supporting data-driven decision-making in business, science, policy, and engineering.<\/li><li>Facilitating model evaluation and comparison in data science and machine learning.<\/li><\/ul>\n\n\n\n<h2><a id=\"_Toc231811364\">Inferential Statistics vs Descriptive Statistics<\/a><\/h2>\n\n\n\n<p>Statistics divides broadly into two branches: descriptive and inferential. Understanding the distinction is fundamental before applying either.<\/p>\n\n\n\n<p><strong><a href=\"https:\/\/cactusglobal-my.sharepoint.com\/personal\/marishaf_cactusglobal_com\/Documents\/Meetings\">Descriptive statistics<\/a><\/strong> summarise and describe the data you actually have. They do not involve uncertainty because they describe only the observed data set precisely. Measures include mean, median, mode, standard deviation, variance, range, and <a href=\"https:\/\/www.editage.com\/blog\/frequency-distributions-and-their-uses-in-biomedical-research\/\">frequency distributions<\/a>.<\/p>\n\n\n\n<p><strong>Inferential statistics<\/strong> go further by using the sample data to make probabilistic statements about a population that was not fully observed. There is always some uncertainty involved, quantified through concepts like <a href=\"https:\/\/www.editage.com\/blog\/what-is-confidence-intervals-and-why-is-it-important\/\">confidence intervals<\/a> and <a href=\"https:\/\/www.editage.com\/insights\/correct-way-report-p-values\">p-values<\/a>.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><td><strong>Dimension<\/strong><\/td><td><strong>Descriptive Statistics<\/strong><\/td><td><strong>Inferential Statistics<\/strong><\/td><\/tr><\/thead><tbody><tr><td>Purpose<\/td><td>Summarise and describe a known data set<\/td><td>Draw conclusions about an unknown population<\/td><\/tr><tr><td>Scope<\/td><td>The data at hand only<\/td><td>Extends beyond collected data to the wider population<\/td><\/tr><tr><td>Uncertainty<\/td><td>None \u2014 precisely describes observed data<\/td><td>Always present; quantified by probability<\/td><\/tr><tr><td>Key Tools<\/td><td>Mean, median, mode, standard deviation, charts<\/td><td>Hypothesis tests, confidence intervals, regression<\/td><\/tr><tr><td>Output<\/td><td>Summary figures and visualisations<\/td><td>Probability-based conclusions and predictions<\/td><\/tr><tr><td>Example<\/td><td>Average age of 100 survey respondents is 34 years<\/td><td>Estimating the average age of all adults in a country from 100 respondents<\/td><\/tr><tr><td>Inference Required?<\/td><td>No<\/td><td>Yes<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2><a id=\"_Toc231811365\">Key Concepts in Inferential Statistics<\/a><\/h2>\n\n\n\n<h3>Population and Sample<\/h3>\n\n\n\n<p>A population encompasses every individual or data point relevant to a research question. A sample is a subset of the population selected for practical measurement. The accuracy of inferential conclusions depends heavily on how representative the sample is.<\/p>\n\n\n\n<p>Common <a href=\"https:\/\/researcher.life\/blog\/article\/what-is-probability-sampling-techniques-types-examples\/\">probability sampling methods<\/a> used to achieve representative samples include:<\/p>\n\n\n\n<ul><li><strong><a href=\"https:\/\/researcher.life\/blog\/article\/simple-random-sampling-definition-methods-examples\/\">Simple random sampling<\/a><\/strong>: every member of the population has an equal chance of selection.<\/li><li><strong><a href=\"https:\/\/researcher.life\/blog\/article\/what-is-stratified-sampling-definition-types-examples\/\">Stratified sampling<\/a><\/strong>: the population is divided into subgroups (strata) and samples are drawn from each.<\/li><li><strong><a href=\"https:\/\/researcher.life\/blog\/article\/what-is-cluster-sampling-definition-method-and-examples\/\">Cluster sampling<\/a><\/strong>: naturally occurring groups are randomly selected and all members within the chosen groups are studied.<\/li><li><strong><a href=\"https:\/\/researcher.life\/blog\/article\/what-is-systematic-sampling-advantages-disadvantages-examples\/\">Systematic sampling<\/a><\/strong>: every nth member of a list is selected after a random starting point.<\/li><\/ul>\n\n\n\n<h3>Parameters and Statistics<\/h3>\n\n\n\n<p>A parameter describes a population characteristic and is usually unknown. A statistic describes a sample characteristic and is observable. Inferential statistics uses the known statistic to estimate the unknown parameter.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><td><strong>Measure<\/strong><\/td><td><strong>Sample (Statistic)<\/strong><\/td><td><strong>Population (Parameter)<\/strong><\/td><\/tr><\/thead><tbody><tr><td>Mean<\/td><td>x\u0304 (x-bar)<\/td><td>\u03bc (mu)<\/td><\/tr><tr><td><a href=\"https:\/\/www.editage.com\/insights\/what-is-standard-deviation-and-how-to-find-standard-deviation\">Standard Deviation<\/a><\/td><td>S<\/td><td>\u03c3 (sigma)<\/td><\/tr><tr><td>Variance<\/td><td>s\u00b2<\/td><td>\u03c3\u00b2<\/td><\/tr><tr><td>Proportion<\/td><td>p\u0302 (p-hat)<\/td><td>p<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h3>Sampling Error<\/h3>\n\n\n\n<p>Because a sample never fully captures the population, there is always a gap between the sample statistic and the true population parameter. This is called sampling error. Sampling error is not a mistake; it is an inevitable consequence of using a sample. It can be reduced by increasing sample size, but it can never be entirely eliminated. Inferential statistics accounts for sampling error explicitly when constructing estimates and testing hypotheses.<\/p>\n\n\n\n<h3>The Central Limit Theorem<\/h3>\n\n\n\n<p>The Central Limit Theorem (CLT) is a foundational principle underpinning much of inferential statistics. It states that the distribution of sample means will approximate a <a href=\"https:\/\/www.editage.com\/blog\/normality-test-methods-of-assessing-normality\/\">normal distribution<\/a> as the sample size grows, regardless of the shape of the original population distribution. This is important because:<\/p>\n\n\n\n<ul><li>It justifies applying normal-distribution-based methods even when the raw data is not normally distributed.<\/li><li>It explains why larger samples yield more reliable estimates.<\/li><li>It makes many statistical tests valid across a wide range of real-world data types (skewed income distributions, purchasing behaviour, biological measurements, and so on).<\/li><\/ul>\n\n\n\n<h2><a id=\"_Toc231811366\">Estimating Population Parameters<\/a><\/h2>\n\n\n\n<p>One major purpose of inferential statistics is estimation: using sample data to make informed guesses about unknown population parameters. There are two types of estimates.<\/p>\n\n\n\n<h3>Point Estimates<\/h3>\n\n\n\n<p>A point estimate is a single value calculated from sample data that serves as the best guess for the population parameter. For example, if a random sample of employees has an average of 19 paid vacation days, that figure is the point estimate for the population mean. While concise, a point estimate provides no information about the precision or reliability of the estimate.<\/p>\n\n\n\n<h3>Interval Estimates and Confidence Intervals<\/h3>\n\n\n\n<p>An interval estimate provides a range of plausible values for the population parameter, giving a sense of the estimate&#8217;s precision. The most widely used form is the confidence interval.<\/p>\n\n\n\n<p>A <strong>95% confidence interval<\/strong>, for instance, means that if the same study were repeated 100 times using different random samples under identical conditions, the confidence interval would capture the true population parameter on approximately 95 of those occasions. It does not mean there is a 95% probability that the specific calculated interval contains the parameter: the parameter is fixed and the interval is what varies across samples.<\/p>\n\n\n\n<p>The formula for a confidence interval for the mean is:<\/p>\n\n\n\n<p><strong>CI = x\u0304 \u00b1 Z\u03b1\/2 \u00d7 (\u03c3 \/ \u221an)<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><td><strong>Component<\/strong><\/td><td><strong>Symbol<\/strong><\/td><td><strong>Meaning<\/strong><\/td><\/tr><\/thead><tbody><tr><td>Sample mean<\/td><td>x\u0304<\/td><td>The average calculated from the sample<\/td><\/tr><tr><td>Critical Z-value<\/td><td>Z\u03b1\/2<\/td><td>Derived from the chosen confidence level (e.g., 1.96 for 95%)<\/td><\/tr><tr><td>Population standard deviation<\/td><td>\u03c3<\/td><td>A measure of population variability<\/td><\/tr><tr><td>Sample size<\/td><td>n<\/td><td>Number of observations in the sample<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>Point estimates and confidence intervals are complementary: the point estimate gives precision; the confidence interval gives context about uncertainty.<\/p>\n\n\n\n<h2><a id=\"_Toc231811367\">Hypothesis Testing<\/a><\/h2>\n\n\n\n<p>Hypothesis testing is a formal statistical procedure for evaluating claims about population parameters or relationships between variables. It provides a structured framework for deciding whether observed sample data provide sufficient evidence to reject a pre-specified assumption about the population.<\/p>\n\n\n\n<h3>Steps in Hypothesis Testing<\/h3>\n\n\n\n<ol type=\"1\"><li>State the <a href=\"https:\/\/www.editage.com\/insights\/the-null-hypothesis-what-researchers-often-get-wrong\">null hypothesis<\/a> (H\u2080) and the alternative hypothesis (H\u2081).<\/li><li>Choose a significance level (\u03b1), typically 0.05 or 0.01.<\/li><li><a href=\"https:\/\/www.editage.com\/insights\/3-simple-steps-to-help-you-pick-the-right-statistical-test\">Select the appropriate statistical test<\/a> based on the data type and research question.<\/li><li>Calculate the test statistic from the sample data.<\/li><li>Compare the test statistic to the critical value, or compute the p-value.<\/li><li>Make a decision: reject H\u2080 if the p-value &lt; \u03b1, or if the test statistic exceeds the critical value.<\/li><li>Draw a conclusion in the context of the <a href=\"https:\/\/www.editage.com\/insights\/how-to-choose-a-research-question\">research question<\/a>.<\/li><\/ol>\n\n\n\n<h3>The p-value<\/h3>\n\n\n\n<p>The p-value is the probability of observing a result at least as extreme as the one obtained, assuming the null hypothesis is true. A smaller p-value means the data are less consistent with H\u2080.<\/p>\n\n\n\n<ul><li><strong>p &lt; 0.05<\/strong> is conventionally considered statistically significant; reject H\u2080.<\/li><li><strong>p \u2265 0.05<\/strong> means that there is insufficient evidence to reject H\u2080 (this does not prove H\u2080 is true).<\/li><\/ul>\n\n\n\n<h3>Type I and Type II Errors<\/h3>\n\n\n\n<p>No hypothesis test is infallible. Two types of errors can occur:<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><td><strong>Error Type<\/strong><\/td><td><strong>What Happens<\/strong><\/td><td><strong>Probability<\/strong><\/td><td><strong>Also Called<\/strong><\/td><\/tr><\/thead><tbody><tr><td>Type I Error<\/td><td>Reject a true null hypothesis (false positive)<\/td><td>\u03b1 (significance level)<\/td><td>False positive<\/td><\/tr><tr><td>Type II Error<\/td><td>Fail to reject a false null hypothesis (false negative)<\/td><td>\u03b2<\/td><td>False negative<\/td><\/tr><tr><td>Correct decision (power)<\/td><td>Correctly reject a false null hypothesis<\/td><td>1 \u2212 \u03b2<\/td><td>Statistical power<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>Minimising both error types simultaneously is difficult. Increasing sample size is the most effective way to reduce both. Lowering \u03b1 reduces Type I errors but increases Type II errors, so a balance must be struck based on the consequences of each error type.<\/p>\n\n\n\n<h2><a id=\"_Toc231811368\">Types of Statistical Tests in Inferential Statistics<\/a><\/h2>\n\n\n\n<p>Statistical tests in inferential statistics fall into two broad families: <a href=\"https:\/\/www.editage.com\/insights\/an-introduction-to-non-parametric-tests-for-biomedical-researchers\">parametric and non-parametric<\/a>. They are further organised by purpose: comparison, correlation, or regression.<\/p>\n\n\n\n<h3>Parametric vs Non-Parametric Tests<\/h3>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><td><strong>Feature<\/strong><\/td><td><strong>Parametric Tests<\/strong><\/td><td><strong>Non-Parametric Tests<\/strong><\/td><\/tr><\/thead><tbody><tr><td>Assumption about distribution<\/td><td>Assumes data follow a known distribution (usually normal)<\/td><td>No distribution assumptions; distribution-free<\/td><\/tr><tr><td>Data level<\/td><td>Interval or ratio scale<\/td><td>Ordinal, nominal, or non-normal interval\/ratio<\/td><\/tr><tr><td>Sample size<\/td><td>Generally n \u2265 30 recommended<\/td><td>Suitable for small samples<\/td><\/tr><tr><td><a href=\"https:\/\/www.editage.com\/insights\/importance-of-statistical-power-in-research-design\">Statistical power<\/a><\/td><td>Higher (more likely to detect an effect)<\/td><td>Lower, but appropriate when assumptions are violated<\/td><\/tr><tr><td>Examples<\/td><td>Z-test, <a href=\"https:\/\/www.editage.com\/insights\/what-biomedical-researchers-need-to-know-about-t-tests\">t-test<\/a>, <a href=\"https:\/\/www.editage.com\/blog\/anova-types-uses-assumptions-a-quick-guide-for-biomedical-researchers\/\">ANOVA<\/a>, linear regression<\/td><td>Mann-Whitney U, Kruskal-Wallis, Chi-square, Wilcoxon<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h3>Comparison Tests<\/h3>\n\n\n\n<p>Comparison tests evaluate whether there are meaningful differences in means, medians, or distributions <a href=\"https:\/\/www.editage.com\/insights\/whats-the-best-way-to-check-for-differences-between-groups-t-test-chi-square-test-or-anova-a-quick-and-simple-guide\">across two or more groups<\/a>.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><td><strong>Test<\/strong><\/td><td><strong>Parametric?<\/strong><\/td><td><strong>What Is Compared<\/strong><\/td><td><strong>Number of Samples<\/strong><\/td><\/tr><\/thead><tbody><tr><td>Z-test<\/td><td>Yes<\/td><td>Means (population SD known, n \u2265 30)<\/td><td>1 or 2 samples<\/td><\/tr><tr><td>Independent t-test<\/td><td>Yes<\/td><td>Means of two separate groups<\/td><td>2 samples<\/td><\/tr><tr><td>Paired t-test<\/td><td>Yes<\/td><td>Means of related\/matched pairs<\/td><td>2 related samples<\/td><\/tr><tr><td>One-way ANOVA<\/td><td>Yes<\/td><td>Means across groups<\/td><td>3+ samples<\/td><\/tr><tr><td>Two-way ANOVA<\/td><td>Yes<\/td><td>Means with two independent variables<\/td><td>3+ samples<\/td><\/tr><tr><td>Wilcoxon signed-rank test<\/td><td>No<\/td><td>Distributions of matched pairs<\/td><td>2 related samples<\/td><\/tr><tr><td>Mann-Whitney U test<\/td><td>No<\/td><td>Sums of rankings<\/td><td>2 independent samples<\/td><\/tr><tr><td>Kruskal-Wallis H test<\/td><td>No<\/td><td>Mean rankings<\/td><td>3+ samples<\/td><\/tr><tr><td>Mood&#8217;s median test<\/td><td>No<\/td><td>Medians<\/td><td>2+ samples<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h3>Correlation Tests<\/h3>\n\n\n\n<p><a href=\"https:\/\/www.editage.com\/insights\/5-things-biomedical-researchers-need-to-know-about-correlation-analysis\">Correlation tests<\/a> measure the strength and direction of association between two variables. They <a href=\"https:\/\/www.editage.com\/insights\/correlation-vs-causation-key-differences-with-examples\">do not establish causation<\/a>.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><td><strong>Test<\/strong><\/td><td><strong>Parametric?<\/strong><\/td><td><strong>Variable Types<\/strong><\/td><td><strong>Notes<\/strong><\/td><\/tr><\/thead><tbody><tr><td>Pearson&#8217;s r<\/td><td>Yes<\/td><td>Two continuous (interval\/ratio) variables<\/td><td>Assumes linear relationship and normality<\/td><\/tr><tr><td>Spearman&#8217;s r<\/td><td>No<\/td><td>Ordinal or non-normally distributed continuous variables<\/td><td>Based on ranks<\/td><\/tr><tr><td><a href=\"https:\/\/www.editage.com\/blog\/chi-square-test-types-explained-for-biomedical-researchers\/\">Chi-square test<\/a> of independence<\/td><td>No<\/td><td>Two categorical (nominal\/ordinal) variables<\/td><td>Only test for nominal variables<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h3>Regression Tests<\/h3>\n\n\n\n<p><a href=\"https:\/\/www.editage.com\/insights\/choosing-the-right-regression-method-a-handy-guide-for-biomedical-researchers\">Regression tests<\/a> model the relationship between predictor (independent) variables and an outcome (dependent) variable, enabling prediction and causal inference.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><td><strong>Regression Type<\/strong><\/td><td><strong>Predictors<\/strong><\/td><td><strong>Outcome<\/strong><\/td><td><strong>Use Case<\/strong><\/td><\/tr><\/thead><tbody><tr><td>Simple linear regression<\/td><td>1 continuous variable<\/td><td>1 continuous variable<\/td><td>Predict one numeric outcome from one input<\/td><\/tr><tr><td>Multiple linear regression<\/td><td>2+ continuous variables<\/td><td>1 continuous variable<\/td><td>Predict from multiple inputs simultaneously<\/td><\/tr><tr><td>Logistic regression<\/td><td>1+ variables (any type)<\/td><td>1 binary variable (yes\/no)<\/td><td>Classification and probability estimation<\/td><\/tr><tr><td>Nominal regression<\/td><td>1+ variables (any type)<\/td><td>1 nominal variable<\/td><td>Outcome with unordered categories<\/td><\/tr><tr><td>Ordinal regression<\/td><td>1+ variables (any type)<\/td><td>1 ordinal variable<\/td><td>Outcome with ordered categories<\/td><\/tr><tr><td>F-test \/ ANOVA<\/td><td>Categorical grouping variable<\/td><td>1 continuous variable<\/td><td>Compare variance across groups<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2><a id=\"_Toc231811369\">Core Statistical Tests Explained<\/a><\/h2>\n\n\n\n<h3>Z-Test<\/h3>\n\n\n\n<p>The Z-test is used when the sample size is large (n \u2265 30) and the population standard deviation is known. It compares the sample mean to a hypothesised population mean.<\/p>\n\n\n\n<p><strong>Formula: Z = (x\u0304 \u2212 \u03bc\u2080) \/ (\u03c3 \/ \u221an)<\/strong><\/p>\n\n\n\n<p>Decision rule: Reject H\u2080 if the computed Z exceeds the critical Z-value from the standard normal distribution (e.g., 1.96 for a two-tailed test at \u03b1 = 0.05).<\/p>\n\n\n\n<h3>T-Test<\/h3>\n\n\n\n<p>The t-test is used when the sample size is small (n &lt; 30) or the population standard deviation is unknown. It relies on the Student&#8217;s t-distribution, which has heavier tails than the normal distribution, reflecting greater uncertainty with smaller samples.<\/p>\n\n\n\n<p><strong>Formula: t = (x\u0304 \u2212 \u03bc\u2080) \/ (s \/ \u221an)<\/strong><\/p>\n\n\n\n<p>Decision rule: Reject H\u2080 if the computed t exceeds the critical value from the t-distribution with (n \u2212 1) degrees of freedom.<\/p>\n\n\n\n<h3>F-Test and ANOVA<\/h3>\n\n\n\n<p>The F-test compares the variances of two or more populations, or the variability between groups versus within groups. ANOVA extends this to compare means across three or more groups simultaneously, avoiding the inflated error that would result from multiple pairwise t-tests.<\/p>\n\n\n\n<p><strong>Formula: F = \u03c3\u2081\u00b2 \/ \u03c3\u2082\u00b2 (for two variances)<\/strong><\/p>\n\n\n\n<h3>Chi-Square Test<\/h3>\n\n\n\n<p>The chi-square test is used for categorical data. The chi-square test of independence assesses whether two categorical variables are related; the chi-square goodness-of-fit test evaluates whether observed frequencies match a hypothesised distribution.<\/p>\n\n\n\n<h2><a id=\"_Toc231811370\">Regression Analysis in Inferential Statistics<\/a><\/h2>\n\n\n\n<p>Regression analysis quantifies how changes in one or more predictor variables are associated with changes in an outcome variable. It is one of the most powerful and widely applied tools in inferential statistics.<\/p>\n\n\n\n<p>In simple linear regression, the relationship is modelled as:<\/p>\n\n\n\n<p><strong>y = \u03b1 + \u03b2x<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><td><strong>Symbol<\/strong><\/td><td><strong>Name<\/strong><\/td><td><strong>Meaning<\/strong><\/td><\/tr><\/thead><tbody><tr><td>Y<\/td><td>Dependent variable<\/td><td>The outcome being predicted<\/td><\/tr><tr><td>X<\/td><td>Independent variable<\/td><td>The predictor or input<\/td><\/tr><tr><td>\u03b1 (alpha)<\/td><td>Intercept<\/td><td>The predicted value of y when x = 0<\/td><\/tr><tr><td>\u03b2 (beta)<\/td><td>Regression coefficient \/ slope<\/td><td>The expected change in y for a one-unit increase in x<\/td><\/tr><tr><td>r\u00b2<\/td><td>Coefficient of determination<\/td><td>The proportion of variance in y explained by x (0 to 1)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>The regression coefficient \u03b2 is calculated as:<\/p>\n\n\n\n<p><strong>\u03b2 = rxy \u00d7 (\u03c3y \/ \u03c3x)<\/strong><\/p>\n\n\n\n<p>where rxy is the Pearson correlation coefficient, \u03c3y is the standard deviation of y, and \u03c3x is the standard deviation of x.<\/p>\n\n\n\n<p>Regression analysis assumes linearity, independence of errors, homoscedasticity (constant variance of errors), and normality of residuals. When data do not follow a normal distribution, mathematical transformations (such as taking logarithms or square roots) can be applied to meet these assumptions.<\/p>\n\n\n\n<h2><a id=\"_Toc231811371\">Worked Example: Applying Inferential Statistics<\/a><\/h2>\n\n\n\n<p>A logistics company wants to determine whether a new delivery algorithm reduces average delivery times compared to the existing system. Here is how inferential statistics would be applied:<\/p>\n\n\n\n<ul><li><strong>Sample setup:<\/strong> 100 orders are split into two groups: 50 orders using the new algorithm, 50 using the current system. Delivery times are recorded for all.<\/li><li><strong>Hypotheses:<\/strong> H\u2080: The new algorithm does not reduce delivery time. H\u2081: The new algorithm reduces delivery time.<\/li><li><strong>Significance level:<\/strong> \u03b1 = 0.05. This means a 5% risk of falsely concluding the new algorithm is better when it is not (Type I error).<\/li><li><strong>Test selection:<\/strong> Because two independent group means are being compared with continuous data, an independent samples t-test is appropriate.<\/li><li><strong>Calculation:<\/strong> Compute the means and standard deviations of both groups, then calculate the t-statistic. If the p-value &lt; 0.05, reject H\u2080.<\/li><li><strong>Confidence interval:<\/strong> A 95% confidence interval of [\u22125, \u22122] minutes would mean that deliveries are estimated to be 2 to 5 minutes faster with the new algorithm, with 95% confidence.<\/li><li><strong>Conclusion:<\/strong> If the p-value falls below 0.05, the company can confidently roll out the new algorithm, knowing the improvement is statistically significant and unlikely due to chance.<\/li><\/ul>\n\n\n\n<h2><a id=\"_Toc231811372\">Assumptions of Inferential Statistics<\/a><\/h2>\n\n\n\n<p>The validity of inferential conclusions depends on several assumptions. Violating them can lead to misleading results.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><td><strong>Assumption<\/strong><\/td><td><strong>What It Means<\/strong><\/td><td><strong>What Happens If Violated<\/strong><\/td><\/tr><\/thead><tbody><tr><td>Random sampling<\/td><td>Sample is selected without systematic bias<\/td><td>Conclusions may not generalise to the population<\/td><\/tr><tr><td>Independence<\/td><td>Observations do not influence one another<\/td><td>Standard errors are underestimated; tests are unreliable<\/td><\/tr><tr><td>Normality (parametric tests)<\/td><td>Data or sample means follow a normal distribution<\/td><td>Use non-parametric tests or rely on CLT with larger samples<\/td><\/tr><tr><td>Homogeneity of variance<\/td><td>Groups being compared have similar variances<\/td><td>Welch&#8217;s t-test or non-parametric alternatives are preferred<\/td><\/tr><tr><td>Adequate sample size<\/td><td>Sample is large enough to represent the population<\/td><td>Estimates are imprecise; tests lack statistical power<\/td><\/tr><tr><td>Correct test selection<\/td><td>The chosen test matches the data type and research design<\/td><td>Invalid results; incorrect conclusions<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2><a id=\"_Toc231811373\">Real-World Applications of Inferential Statistics<\/a><\/h2>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><td><strong>Field<\/strong><\/td><td><strong>Example Application<\/strong><\/td><\/tr><\/thead><tbody><tr><td>Medicine &amp; Clinical Trials<\/td><td>Determining whether a new drug reduces blood pressure more effectively than a placebo based on a patient sample.<\/td><\/tr><tr><td>Public Policy &amp; Polling<\/td><td>Estimating voting intentions of an entire electorate from a sample of a few thousand respondents.<\/td><\/tr><tr><td>Business &amp; Marketing<\/td><td>A\/B testing to determine whether a new website design increases conversion rates.<\/td><\/tr><tr><td>Education Research<\/td><td>Assessing whether a new teaching method leads to higher standardised test scores compared to traditional instruction.<\/td><\/tr><tr><td>Quality Control<\/td><td>Testing whether the defect rate of a manufactured batch differs from the acceptable standard without inspecting every item.<\/td><\/tr><tr><td>Economics<\/td><td>Estimating the effect of a minimum wage increase on employment levels using regional employment data.<\/td><\/tr><tr><td>Data Science &amp; Machine Learning<\/td><td>Evaluating model performance, comparing algorithms, and detecting statistically significant differences in prediction accuracy.<\/td><\/tr><tr><td>Environmental Science<\/td><td>Estimating average pollution levels across a region from sensor readings at selected monitoring stations.<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2><a id=\"_Toc231811374\">Key Takeaways<\/a><\/h2>\n\n\n\n<ul><li>Inferential statistics enables conclusions about populations from sample data, using probability to account for uncertainty.<\/li><li>The two main branches are hypothesis testing (assessing claims about population parameters) and regression analysis (modelling relationships between variables).<\/li><li>Sampling error is the difference between a sample statistic and the true population parameter; it is inevitable but manageable.<\/li><li>The Central Limit Theorem justifies using normal-distribution-based methods even when raw data is not normally distributed, particularly with large samples.<\/li><li>Confidence intervals provide a range of plausible values for a population parameter; a 95% confidence interval means 95% of such intervals from repeated sampling would contain the true parameter.<\/li><li>Hypothesis testing uses the p-value and a pre-set significance level (\u03b1) to decide whether to reject the null hypothesis.<\/li><li>Type I error (false positive) occurs when a true H\u2080 is rejected; Type II error (false negative) occurs when a false H\u2080 is not rejected.<\/li><li>Parametric tests (t-test, Z-test, ANOVA) are more statistically powerful but require distributional assumptions; non-parametric tests are used when those assumptions fail.<\/li><li>Comparison tests assess differences between groups; correlation tests measure association between variables; regression tests model predictive relationships.<\/li><li>The validity of inferential conclusions depends on representative sampling, adequate sample size, and appropriate test selection.<\/li><\/ul>\n\n\n\n<h2><a id=\"_Toc231811375\">Frequently Asked Questions<\/a><\/h2>\n\n\n\n<h3>What is the difference between inferential and descriptive statistics?<\/h3>\n\n\n\n<p>Descriptive statistics summarise and describe the characteristics of a data set you have actually collected, using measures such as the mean, median, and standard deviation. Inferential statistics go further by using the collected sample data to make probabilistic conclusions about a larger population that was not fully observed. Descriptive statistics involve no uncertainty; inferential statistics always involve uncertainty, which is quantified using confidence intervals and p-values.<\/p>\n\n\n\n<h3>When should I use a t-test versus a Z-test?<\/h3>\n\n\n\n<p>Use a Z-test when the sample size is large (typically n \u2265 30) and the population standard deviation is known. Use a t-test when the sample size is small (n &lt; 30) or when the population standard deviation is unknown and must be estimated from the sample. In practice, the t-test is more commonly used because population standard deviations are rarely known.<\/p>\n\n\n\n<h3>What does a p-value actually mean?<\/h3>\n\n\n\n<p>A p-value is the probability of obtaining a result at least as extreme as the one observed, assuming the null hypothesis is true. A small p-value (typically &lt; 0.05) suggests the data are unlikely under the null hypothesis, providing evidence to reject it.<\/p>\n\n\n\n<p>It is important to note that the p-value is not the probability that the null hypothesis is true, nor the probability that the result occurred by chance alone; it is a conditional probability given H\u2080.<\/p>\n\n\n\n<h3>What is a confidence interval and how is it interpreted?<\/h3>\n\n\n\n<p>A confidence interval is a range of values calculated from sample data that is expected to contain the true population parameter with a specified probability. For a 95% confidence interval, if the same study were repeated many times using different random samples, approximately 95% of the resulting intervals would contain the true population parameter.<\/p>\n\n\n\n<p>The interval reflects both the estimate and the uncertainty around it; wider intervals indicate greater uncertainty, usually due to smaller sample sizes or higher variability.<\/p>\n\n\n\n<h3>What are Type I and Type II errors, and why do they matter?<\/h3>\n\n\n\n<p>A Type I error occurs when the null hypothesis is true but is incorrectly rejected: a false positive. Its probability equals the significance level \u03b1.<\/p>\n\n\n\n<p>A Type II error occurs when the null hypothesis is false but is not rejected: a false negative. Its probability is denoted \u03b2.<\/p>\n\n\n\n<p>These errors matter because they have real consequences: a Type I error in a drug trial might lead to approving an ineffective drug, while a Type II error might cause a beneficial treatment to be discarded. Sample size, significance level, and effect size all influence the balance between these errors.<\/p>\n\n\n\n<h3>What is statistical power and why is it important?<\/h3>\n\n\n\n<p>Statistical power is the probability that a hypothesis test will correctly detect a true effect when one exists. It equals 1 \u2212 \u03b2, where \u03b2 is the probability of a Type II error. A commonly targeted power level is 0.80 (80%), meaning the test has an 80% chance of detecting a real effect.<\/p>\n\n\n\n<p>Power increases with larger sample sizes, larger effect sizes, higher significance levels, and reduced measurement error. A study with low power may miss real effects, wasting resources and potentially leading to incorrect conclusions.<\/p>\n\n\n\n<h3>How do I choose between parametric and non-parametric tests?<\/h3>\n\n\n\n<p>Use parametric tests (such as t-tests and ANOVA) when<\/p>\n\n\n\n<ul><li>your data are continuous (interval or ratio scale),<\/li><li>the sample is sufficiently large (or the data are approximately normally distributed), and<\/li><li>the variances of groups being compared are roughly equal.<\/li><\/ul>\n\n\n\n<p>Use non-parametric tests (such as the Mann-Whitney U test or Kruskal-Wallis test) when<\/p>\n\n\n\n<ul><li>your data are ordinal or categorical,<\/li><li>the normality assumption is violated,<\/li><li>the sample size is very small, or<\/li><li>you have outliers that would unduly distort parametric results.<\/li><\/ul>\n\n\n\n<p>Non-parametric tests are sometimes called distribution-free tests because they make no assumptions about the shape of the population distribution.<\/p>\n\n\n\n<h3>Can inferential statistics prove causation?<\/h3>\n\n\n\n<p>Inferential statistics alone cannot establish causation. It can demonstrate that a statistically significant association or difference exists, but association is not the same as causation.<\/p>\n\n\n\n<p>Establishing causation requires a well-designed experiment with random assignment of participants to conditions (a randomised controlled trial), or the use of causal inference methods in observational studies.<\/p>\n\n\n\n<p>Regression analysis can identify predictive relationships, but even a significant regression result does not prove that the predictor causes the outcome. Confounding variables, reverse causation, and coincidence must all be ruled out through <a href=\"https:\/\/researcher.life\/blog\/article\/what-is-research-design-types-examples\/\">study design<\/a> and careful interpretation.<\/p>\n","protected":false},"excerpt":{"rendered":"Inferential statistics are a way of drawing inferences (i.e., conclusions) about a population based on a smaller subset, known as a sample. As mentioned above, these conclusions or inferences are typically about whether certain variables are related to each other or whether certain groups differ from each other. Inferential statistics enable us to determine whether these variables or relationships are due to chance (i.e., whether they are statistically significant).","protected":false},"author":2,"featured_media":485,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_ayudawp_aiss_exclude":false,"_ayudawp_aiss_summary":"","_ayudawp_aiss_summary_provider":"","_ayudawp_aiss_summary_hash":""},"categories":[14],"tags":[23,24],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.6 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>What are Inferential Statistics? 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