The Bonferroni correction is a statistical method used to reduce the risk of Type I errors (false positives) when you run multiple hypothesis tests. Every time you test a hypothesis, there’s a chance you’ll incorrectly…
Type I Error (False Positive) You reject a true null hypothesis — you conclude something is happening when it actually isn’t. Example:A medical test says a patient has a disease, but they actually don’t. Type…
🧩 What “Independent Samples” Means Two samples are independent when the individuals in one group have no relationship to the individuals in the other group. This is the setup for the independent‑samples t‑test, also called…
A one‑sample t‑test checks whether the mean of a single sample is significantly different from a known or hypothesized population mean. It answers the question: “Is my sample mean different enough from the population mean…
A t‑test is a hypothesis test used when you want to compare means but you don’t know the population standard deviation and your sample size is small. It’s used for: One‑sample t‑test → compare one…
The significance level, usually written as , is the threshold for how much evidence you require before rejecting the null hypothesis. It is the probability of making a Type I error: So choosing α is…
The p‑value is the probability of getting a result as extreme as (or more extreme than) your sample result if the null hypothesis were true. In other words: The p‑value tells you how surprising your…
Instead of using a p‑value, you compare your test statistic (like a z‑score or t‑score) to a critical value that marks the boundary of the rejection region. If your test statistic falls beyond the critical…
🧪 Ronald Fisher: The p‑value Rebel Philosophy: Evidence, not decisions Fisher believed statistics should help scientists measure evidence against a null hypothesis. Key ideas Fisher’s vibe: The scientist as a detective, gathering clues and weighing…
At the heart of every hypothesis test are two competing statements about a population: They must be:Mutually exclusive (can’t both be true)Exhaustive (cover all possibilities)About population parameters, not sample statistics Let’s break down how to…
Hypothesis testing is a structured way to use sample data to make decisions or draw conclusions about a population. It answers questions like: It’s the backbone of inferential statistics. 🎯 The Core Idea You start…
⭐ Central Limit Theorem (CLT) The Central Limit Theorem says something surprisingly powerful: If you take many random samples and compute their means,the distribution of those sample means will be approximately normal,even if the original…
⭐ Z‑Score A z‑score tells you how many standard deviations an observation is from the mean. What it does Example Population mean , standard deviation .What is the z‑score of ? Interpretation:The value is 1.5…
⭐ Histogram vs. Density Plot Both visualize distributions, but they answer slightly different questions and behave differently. 📊 Histogram A histogram groups data into bins and shows counts (or proportions) in each bin. Key features…
A random variable that follows a geometric distribution satisfies: This means: The probability you still have to wait more trials does NOT depend on how long you’ve already been waiting. Your past failures don’t change…
The geometric distribution models the number of trials needed until the first success occurs in a sequence of independent Bernoulli trials (like repeated coin flips). Think of it as the math of “How long until…
⭐ Binomial Distribution The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has the same probability of success. Think of it as the math of “How…
The expected value (mean) of random variables adds even if the variables are dependent. This is the magic part: Expectation is always linear — no independence required. Formally, for any random variables and : And…
A histogram is a graph that shows how data are distributed by grouping values into bins (intervals) and showing how many observations fall into each bin. It’s perfect for visualizing: Think of it as stacking…
A probability density function describes the distribution of a continuous random variable. If is continuous, its PDF is a function such that: The key idea For continuous variables: The PDF is not a probability. Probability…
A probability mass function is a function that gives the probability of each individual value of a discrete random variable. If is a discrete random variable, then its PMF is: It tells you: A PMF…
Most random variables fall into two big categories: Everything else is a refinement of these two. 🎯 1. Discrete Random Variables A discrete random variable takes countable values — usually integers. Key features Examples Common…
