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Mathematics

Statistical Power and Sample Size

Type II Error, Effect Size, and How Big Your Study Needs to Be — A TLDR Primer

Statistical power is one of the most tested and most misunderstood ideas in introductory statistics — and most textbooks bury it under pages of theory before getting to the point. This concise primer cuts straight to what you actually need to know.

Whether you're preparing for an AP Statistics exam, working through an intro stats course, or trying to make sense of why a study's results don't replicate, this guide covers the full picture: null and alternative hypotheses, Type I and Type II error, and exactly what it means when a test has "80% power." From there, it walks through the four levers that push power up or down — effect size, sample size, alpha, and variability — with concrete numbers, not hand-waving.

You'll learn how to measure effect size using Cohen's d and its equivalents for proportions, and you'll work through complete sample-size calculations for two-sample mean tests and proportion tests, step by step. The final section connects all of it to real research: what underpowered studies look like, why the replication crisis happened, and how to spot the post hoc power fallacy when you read a paper.

This guide is short by design. Every section leads with the one sentence you need to take away, follows with worked examples, and calls out the misconceptions students get wrong most often. No filler, no detours — just the concepts, the formulas, and the intuition to use them.

If statistical power has felt slippery until now, start here.

What you'll learn
  • Define statistical power and explain its relationship to Type I and Type II error
  • Identify the four levers that determine power: effect size, sample size, significance level, and variability
  • Compute and interpret Cohen's d and other standardized effect sizes
  • Calculate the required sample size for a two-sample test of means or proportions
  • Critique studies as 'underpowered' and recognize the consequences for published research
What's inside
  1. 1. Hypothesis Testing in 90 Seconds: The Setup You Need
    Refreshes null/alternative hypotheses, p-values, and Type I vs Type II errors so the rest of the book has solid footing.
  2. 2. What Statistical Power Actually Is
    Defines power as 1 minus beta, gives the intuition with overlapping distributions, and explains why 0.80 became the convention.
  3. 3. The Four Levers: What Power Depends On
    Walks through how effect size, sample size, alpha, and variability each push power up or down, with worked numerical intuition.
  4. 4. Measuring Effect Size: Cohen's d and Friends
    Introduces standardized effect sizes for mean differences and proportions, with rules of thumb and worked examples.
  5. 5. Calculating the Sample Size You Need
    Derives and applies the sample-size formula for a two-sample test of means and a test of proportions, with full worked examples.
  6. 6. Underpowered Studies and Why Power Matters in the Real World
    Explains the replication crisis, the danger of small-n studies, post hoc power fallacy, and what to look for when reading research.
Published by Solid State Press
Statistical Power and Sample Size cover
TLDR STUDY GUIDES

Statistical Power and Sample Size

Type II Error, Effect Size, and How Big Your Study Needs to Be — A TLDR Primer
Solid State Press

Statistical Power and Sample Size

Type II Error, Effect Size, and How Big Your Study Needs to Be — A TLDR Primer

Copyright © 2026 Solid State Press.

Published by Solid State Press.

All rights reserved. No part of this book may be reproduced or transmitted in any form without prior written permission, except for brief quotations in critical reviews and educational use.

Part of the TLDR Study Guides series.

Contents

  1. 1 Hypothesis Testing in 90 Seconds: The Setup You Need
  2. 2 What Statistical Power Actually Is
  3. 3 The Four Levers: What Power Depends On
  4. 4 Measuring Effect Size: Cohen's d and Friends
  5. 5 Calculating the Sample Size You Need
  6. 6 Underpowered Studies and Why Power Matters in the Real World
Chapter 1

Hypothesis Testing in 90 Seconds: The Setup You Need

Every statistical test begins with a decision: is this data surprising enough to change my mind? To make that decision formally, statisticians frame it as a contest between two competing claims.

The null hypothesis ($H_0$) is the default, skeptical position — usually the claim that nothing interesting is happening. A drug has no effect. Two groups have the same mean. A coin is fair. The alternative hypothesis ($H_a$ or $H_1$) is the claim you are trying to find evidence for: the drug works, the groups differ, the coin is biased.

You never prove either hypothesis. Instead, you ask: if the null hypothesis were true, how unlikely would this data be? That probability is the p-value. A small p-value means the data would be rare under $H_0$, which is evidence against $H_0$. A large p-value means the data is unremarkable under $H_0$, so you have no strong reason to doubt it.

To make a binary decision — reject or do not reject — you set a threshold in advance called alpha ($\alpha$), the significance level. The most common choice is $\alpha = 0.05$. If $p < \alpha$, you reject $H_0$. If $p \geq \alpha$, you fail to reject it.

A common mistake is to read "fail to reject" as "prove $H_0$ is true." It isn't. Failing to reject just means the data didn't give you enough evidence to overturn the default. The null hypothesis could still be false — you just didn't catch it.

Two Ways to Be Wrong

Any binary decision can go wrong in exactly two ways, and both have names.

A Type I error is a false positive: you reject $H_0$ when it is actually true. You concluded the drug works, but it doesn't. By design, the probability of making a Type I error equals $\alpha$. When you set $\alpha = 0.05$, you are accepting a 5% chance of a false positive every time you run a test under a true null. Alpha is sometimes called the false-positive rate for this reason.

About This Book

If you are taking AP Statistics, an intro college stats course, or any research-methods class where your professor just said "power" and watched half the room go blank, this book is for you. It is also for anyone who has tanked a hypothesis testing question involving beta error and had no idea why.

This is a focused Type II error statistics study guide that covers what statistical power and sample size explained together actually looks like in practice. You will work through effect size from the ground up — including Cohen's d effect size for beginners — and follow complete sample size calculation statistics primer problems for both means and proportions. The book also touches on the replication crisis statistics explained simply, so the stakes of underpowered research feel real. Short by design, with no filler.

Read it straight through once to build the framework, then work every hypothesis testing power beta error guide example alongside the text, and finish with the practice problems to confirm what you know.

Keep reading

You've read the first half of Chapter 1. The complete book covers 6 chapters in roughly fifteen pages — readable in one sitting.

Coming soon to Amazon