Normal Distribution
The bell curve and the 68-95-99.7 empirical rule, explained with worked examples โ from resting heart rate to manufacturing tolerances.
Read the guide โLoading...
Free articles ยท free calculators ยท 1-to-1 tutoring
Master Statistics Fundamentals From First Principles
From descriptive statistics and hypothesis testing to regression and correlation โ a free, GCSE-to-PhD statistics education platform built to make every concept click.
Free forever โ no account, no paywall, no login required
Core Concepts
The ideas every statistics course builds on
Statistics fundamentals are the precise tools for summarizing data, quantifying uncertainty, and drawing honest conclusions from a sample. Every field of research and data-driven decision-making builds on these foundations.
Pearson Correlation
What r actually measures, a full worked example, and exactly why correlation never proves causation โ with real mechanisms for false correlations.
Read the guide โBasic Probability Rules
The addition rule, multiplication rule, and conditional probability โ with worked examples including sampling without replacement.
Read the guide โHypothesis Testing Framework
Null and alternative hypotheses, p-values, Type I/II errors, and the full six-step framework โ worked through a real example start to finish.
Read the guide โCentral Limit Theorem
Why sample means become normally distributed regardless of the population's shape โ the result that makes most of inferential statistics possible.
Read the guide โConfidence Intervals
What "95% confidence" actually means, how to build an interval for a mean or proportion, and how to choose a sample size in advance.
Read the guide โWhy Statistics Fundamentals
The statistics education platform
you've been waiting for
More than a website โ a complete learning environment where statistics fundamentals become intuitive through worked examples and shown reasoning, not memorization.
๐
Complete Statistics Curriculum
From descriptive statistics through hypothesis testing to regression โ every core concept explained with depth and clarity by statistics educators.
๐งฎ
Free Calculators With Full Working
Every calculator shows the formula, the worked substitution, and the final answer โ not just a number. Understanding, not just answers.
๐ฏ
Conceptual Understanding First
We don't just hand you formulas. Every topic starts with why the method exists before introducing the mathematics.
๐ฉโ๐ซ
1-to-1 Statistics Tutoring
GCSE to PhD โ get direct, personalized help from a real statistics tutor, not just self-study articles.
๐
Real Worked Examples
Every article includes genuine worked problems with real numbers, not abstract formulas alone โ so the method actually sticks.
๐
Free & Accessible
Quality statistics education should never sit behind a paywall. Every article and calculator we build is completely free.
Comprehensive Curriculum
Every branch of statistics,
explained from the ground up
Our curriculum covers statistics fundamentals from descriptive statistics through hypothesis testing and regression. Each topic is built for genuine understanding, not exam prep alone.
Descriptive Statistics
Mean, Median, Mode, Standard Deviation, Variance, Quartiles, IQR
Probability
Basic Probability Rules, Conditional Probability, Bayes' Theorem, Expected Value
Distributions
Normal Distribution, Z-Scores, Binomial Distribution, the Empirical Rule
Sampling & Inference
Central Limit Theorem, Standard Error, Confidence Intervals, Sample Size
Hypothesis Testing
Null & Alternative Hypotheses, P-Values, T-Tests, Chi-Square, ANOVA
Regression & Correlation
Pearson Correlation, Simple Linear Regression, Rยฒ, Residual Analysis
Available now
Everything you need to master
statistics fundamentals
In-depth lessons across all core statistics topics
Free calculators with step-by-step worked solutions
Direct, personalized 1-to-1 statistics tutoring
Clear diagrams and visual explanations throughout
Organised study paths by topic and category
Step-by-step solutions for every worked example
100% free content โ no account or paywall, ever
Latest Articles
Start learning now
8 min read
Where Statistics Actually Shows Up in Machine Learning
Statistics Fundamentals Editorial Team
7 min read
Chebyshev's Inequality: A Distribution-Free Bound on Spread
Statistics Fundamentals Editorial Team
7 min read
Autocorrelation Explained: When Data Points Aren't Independent
Statistics Fundamentals Editorial Team
8 min read
The Friedman Test: Comparing 3+ Related Groups by Rank
Statistics Fundamentals Editorial Team
8 min read
The Durbin-Watson Test: Detecting Autocorrelation in Regression Residuals
Statistics Fundamentals Editorial Team
8 min read
Probability Trees: How to Build and Read Them, With Worked Examples
Statistics Fundamentals Editorial TeamWhat Are Statistics Fundamentals?
Statistics fundamentals are the core methods for summarizing data and drawing honest conclusions under uncertainty. Mastering them is one of the most broadly useful investments any student or working professional can make.
Descriptive Statistics
Descriptive statistics starts with mean, median, and mode โ three different answers to "what's typical," each with its own strengths. Pair these with standard deviation and the interquartile range to describe not just where data is centered, but how spread out it is.
All Descriptive Statistics Articles โProbability
Basic probability rules โ addition, multiplication, and conditional probability โ underpin everything else in statistics. Bayes' theorem extends this into formally updating a belief as new evidence arrives, from medical testing to spam filtering.
All Probability Articles โSampling & Inference
The Central Limit Theorem is the single result that makes inference possible โ it guarantees sample means become normally distributed regardless of the population's shape. This is what lets a confidence interval from a limited sample say something rigorous about an entire population.
All Sampling & Inference Articles โHypothesis Testing & Regression
The hypothesis testing framework is how you formally decide whether a difference is real or just noise. Correlation and regression model the relationship between variables โ and make clear exactly why correlation never proves causation.
All Articles โStatistics that actually
clicks.
Concise, insight-driven updates when we publish something worth reading โ a new article, calculator, or tool. No filler. No noise. Just the good stuff.
Free forever ยท No spam ยท Unsubscribe in one click
Explore
Everything on Statistics Fundamentals
Every article, topic, and resource โ all in one place for easy discovery.
All Articles
- A/B Testing: The Statistics Behind Product Decisions
- Autocorrelation Explained: When Data Points Aren't Independent
- Basic Probability Rules: Addition, Multiplication, and Conditional Probability
- Bayes' Theorem: The Formula, the Medical Testing Example, and Why Base Rates Matter
- Bayesian Credible Intervals: A Worked Example, Computed From Scratch
- Bayesian vs. Frequentist Statistics: Two Ways of Reasoning About Uncertainty
- Binomial Distribution: Formula, Worked Examples, and the Normal Approximation
- Bootstrap Sampling: Confidence Intervals Without a Formula
- Chebyshev's Inequality: A Distribution-Free Bound on Spread
- Chi-Square Test: Goodness-of-Fit and Independence, With Worked Examples
- Coefficient of Variation: Comparing Variability Across Different Scales
- Cohen's Kappa: Measuring Agreement Beyond Chance
- Confidence Intervals vs. Prediction Intervals in Regression
- Confidence Intervals: What They Mean and How to Build One
- Control Charts: Telling Real Process Shifts From Ordinary Noise
- Covariance: Measuring How Two Variables Move Together
- Cronbach's Alpha: Measuring the Reliability of a Scale
- Data Visualization Principles: Choosing Honest, Effective Charts
- Dummy Variables: Using Categorical Predictors in Regression
- Effect Size and Cohen's d: Why Significance Isn't Enough
- Expected Value: The Long-Run Average, Explained With Worked Examples
- Exploratory Data Analysis: The Critical First Step in Any Analysis
- False Discovery Rate and the BenjaminiโHochberg Procedure
- Frequency Distributions and Histograms: Choosing Classes and Reading the Shape
- Hypothesis Testing: The Complete Framework, Step by Step
- Interaction Effects in Regression: When One Predictor's Effect Depends on Another
- Joint, Marginal, and Conditional Probability: Reading a Two-Way Table
- K-Fold Cross-Validation: Estimating How a Model Performs on New Data
- Leverage, Outliers, and Cook's Distance: Finding Influential Points in Regression
- Linearity of Expectation: The Shortcut for Adding Averages
- Logistic Regression: Predicting Probabilities for a Binary Outcome
- Maximum Likelihood Estimation: The Method Behind the Formulas
- McNemar's Test: Comparing Paired Yes/No Outcomes
- Mean, Median, and Mode: When to Use Each One
- Median Absolute Deviation: A Spread Measure That Ignores Outliers
- Multinomial Logistic Regression: Beyond Two Outcome Categories
- Multiple Linear Regression: Modelling Several Predictors at Once
- Non-Parametric Tests: What to Use When a T-Test's Assumptions Don't Hold
- Normal Distribution: The Bell Curve, Explained
- Odds vs. Probability: What's the Actual Difference?
- One-Tailed vs. Two-Tailed Tests: How to Choose, With a Worked Example
- One-Way ANOVA: Comparing Three or More Group Means
- Outlier Detection: The IQR Rule, Z-Scores, and What to Do Next
- Pareto Charts: Finding the Vital Few Causes, With a Worked Example
- Partial Correlation: Controlling for a Third Variable
- PDF vs. CDF: Two Ways to Describe the Same Distribution
- Pearson Correlation: Formula, Worked Example, and Why It Isn't Causation
- Percentiles and Quartiles: What They Mean and How to Calculate Them
- Permutation Tests: A p-Value Built by Shuffling the Labels
- Permutations and Combinations: When Order Matters, and When It Doesn't
- Poisson Distribution: Formula, Worked Examples, and Real-World Uses
- Poisson Regression: Modeling Count Data From Predictors
- Polynomial Regression: Fitting a Curve Instead of a Straight Line
- Principal Component Analysis (PCA): A Step-by-Step Two-Variable Example
- Probability Trees: How to Build and Read Them, With Worked Examples
- Q-Q Plots: Checking Normality by Eye, With a Worked Example
- Regression to the Mean: Why Extreme Results Tend to Moderate
- Ridge and Lasso Regression: Shrinking Coefficients on Purpose
- Sampling Methods: Random, Stratified, Cluster, and Systematic Sampling
- Sensitivity, Specificity, and ROC Curves: Measuring How Good a Test Really Is
- Simple Linear Regression: The Least-Squares Line, Explained
- Simpson's Paradox: When Aggregated Data Reverses the Real Trend
- Skewness and Kurtosis: Measuring a Distribution's Shape
- Spearman's Rank Correlation: When the Relationship Isn't a Straight Line
- Standard Error: What It Measures and Why It Matters
- Statistical Power: Why 'Not Significant' Doesn't Mean 'No Effect'
- Statistics in the Real World: Three Full Worked Case Studies
- Stem-and-Leaf Plots: How to Draw, Read, and Compare Them
- Survivorship Bias: What the Missing Data Doesn't Tell You
- T-Tests: One-Sample, Two-Sample, and Paired โ With Full Worked Examples
- The Birthday Problem: Why 23 People Is Enough for a 50/50 Shared Birthday
- The Central Limit Theorem, Explained With Worked Examples
- The Durbin-Watson Test: Detecting Autocorrelation in Regression Residuals
- The Exponential Distribution: Modeling Wait Times and Lifetimes
- The F-Distribution: The Shape Behind ANOVA and the F-Test
- The F-Test for Overall Regression Significance
- The Finite Population Correction: Sampling Without Replacement From a Small Population
- The Friedman Test: Comparing 3+ Related Groups by Rank
- The Geometric Distribution: Trials Until the First Success
- The Hypergeometric Distribution: When Sampling Without Replacement Really Matters
- The InclusionโExclusion Principle: Counting and Probability for Overlapping Events
- The Jackknife: Estimating Bias and Standard Error by Leaving One Out
- The KaplanโMeier Survival Curve: Handling Censored Data
- The Kolmogorov-Smirnov Test: Comparing a Distribution's Whole Shape
- The Kruskal-Wallis Test: ANOVA Without the Normality Assumption
- The Law of Large Numbers: Why Sample Averages Converge on the Truth
- The Likelihood Ratio Test: Comparing Two Models Directly
- The Log-Normal Distribution: When Logarithms Are Normal
- The Monty Hall Problem: Why Switching Doors Really Does Double Your Odds
- The Multinomial Distribution: When There Are More Than Two Outcomes
- The Multiple Comparisons Problem: Why Testing Many Things Inflates False Positives
- The Negative Binomial Distribution: Waiting for the R-th Success
- The Runs Test: Checking Whether a Sequence Is Actually Random
- The Shapiro-Wilk Test: A Formal Verdict on Normality
- The T-Distribution vs. the Normal Distribution: What Actually Differs
- The Uniform Distribution: Formula, Worked Examples, and When to Use It
- The Weibull Distribution: Modeling Wear-Out and Reliability
- The Wilson Score Interval: A Better Confidence Interval for Proportions
- Time Series Basics: Trend, Seasonality, and Moving Averages
- Tukey's HSD: Finding Which Groups Actually Differ After ANOVA
- Types of Bias in Research: Selection, Response, and Measurement Bias
- Types of Data: Nominal, Ordinal, Interval, and Ratio Explained
- Unbiased Estimators and Mean Squared Error: What Makes an Estimate Good?
- Weighted Mean and Geometric Mean: Beyond the Simple Average
- Where Statistics Actually Shows Up in Machine Learning
- Which Statistical Test Should I Use? A Decision Framework
- Z-Score: Formula, Worked Examples, and How to Read One
All Calculators
- A/B Test Calculator
- Basic Probability Calculator
- Bayes' Theorem Calculator
- Binomial Probability Calculator
- Chi-Square Test Calculator
- Confidence Interval Calculator
- Descriptive Statistics Calculator
- Effect Size Calculator (Cohen's d)
- Final Grade Calculator
- Five-Number Summary & IQR Calculator
- Geometric Mean Calculator
- GPA & CGPA Calculator
- Logistic Regression Calculator
- Margin of Error Calculator
- Mean, Median & Mode Calculator
- Multiple Regression Calculator
- Normal Distribution Calculator
- One-Way ANOVA Calculator
- Outlier Calculator
- Paired t-Test Calculator
- Pearson Correlation Calculator
- Percent Change Calculator
- Percentile Rank Calculator
- Permutations & Combinations Calculator
- Poisson Distribution Calculator
- Sample Size Calculator
- Simple Linear Regression Calculator
- Skewness & Kurtosis Calculator
- Standard Deviation & Variance Calculator
- Standard Error Calculator
- t-Test Calculator
- Weighted Mean Calculator
- Z-Score Calculator
Topics
Tools
- All Calculators
- Descriptive Statistics Calculator
- Standard Deviation Calculator
- Z-Score Calculator
- Confidence Interval Calculator
- t-Test Calculator
- Chi-Square Test Calculator
- Pearson Correlation Calculator
- Linear Regression Calculator
- Basic Probability Calculator
- Bayes' Theorem Calculator
- Normal Distribution Calculator
- Five-Number Summary Calculator
- Binomial Probability Calculator
- Sample Size Calculator
- One-Way ANOVA Calculator
Frequently Asked Questions
Common questions about statistics fundamentals
What are the fundamentals of statistics?
The fundamentals of statistics span five core areas: descriptive statistics (mean, median, standard deviation), probability (the rules governing uncertainty), distributions (the normal distribution, binomial distribution, and others that describe how data is shaped), inferential statistics (using a sample to say something about a wider population, via confidence intervals and hypothesis testing), and regression (modelling the relationship between variables). These form the foundation for everything from a GCSE statistics exam to graduate-level research methods.
Is Statistics Fundamentals really free?
Yes, Statistics Fundamentals is completely free and always will be. We believe quality statistics education should be accessible to every student regardless of financial circumstances. All articles, calculators, and study guides are available at no cost. We also offer paid 1-to-1 tutoring for anyone who wants direct, personalized help.
Who creates the content on Statistics Fundamentals?
Our content is developed by statistics educators with experience teaching at GCSE, A-Level, and undergraduate level. Every article and calculator undergoes review to ensure mathematical accuracy and genuine pedagogical usefulness โ not just correct formulas, but explanations that actually build understanding.
What topics does Statistics Fundamentals cover?
We cover the full spectrum from GCSE statistics through postgraduate research methods: descriptive statistics, probability, distributions, sampling and inference, hypothesis testing (t-tests, chi-square, ANOVA), and regression and correlation. New articles and calculators are added regularly.
How is Statistics Fundamentals different from a textbook?
Unlike a traditional textbook, Statistics Fundamentals pairs every concept with an interactive calculator that shows the full step-by-step working, not just a final answer. We prioritize building genuine conceptual understanding โ the "why" behind a formula โ before asking you to apply it, which research shows leads to deeper, more lasting comprehension.
What is the best way to learn statistics?
The most effective approach combines conceptual understanding with genuine practice on real numbers. Start by grasping why a method exists and what question it actually answers, before memorizing its formula. Work through solved examples, then attempt problems independently using our calculators to check your own working. Consistency matters more than intensity โ short, regular sessions beat cramming.