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Mathematics

Percentiles and Quartiles

Five-Number Summaries, Box Plots, and Reading Test Scores Right — A TLDR Primer

Your test score says you're in the 73rd percentile — but what does that actually mean? Your stats homework asks for Q1 and Q3, and the textbook buries the method under pages of theory before getting to a single worked example. This guide cuts straight to what you need.

**TLDR: Percentiles and Quartiles** covers the core ideas of rank-based statistics from the ground up. You'll learn exactly what a percentile is and why it tells you something a class average can't. You'll compute medians, quartiles, and the interquartile range by hand, step by step, with real numbers. You'll see both standard methods for finding any percentile — and you'll understand why your graphing calculator sometimes gives a slightly different answer than your teacher's formula.

The guide then shows you how to turn a five-number summary into a box plot, apply the 1.5 IQR rule to spot outliers, and compare distributions side by side. The final sections connect the math to the real world: how to understand SAT percentile scores, interpret a pediatric growth chart, and make sense of income statistics you'll encounter in the news.

Written for high school and early college students, this guide is short by design — no filler, no padding, just the concepts, the procedures, and the worked examples you need to walk into class or an ap statistics descriptive statistics review session with confidence.

If quartiles and box plots are on your next exam, grab this and get to work.

What you'll learn
  • Define percentile and quartile and explain what a 'rank' actually measures
  • Compute the median, Q1, Q3, and the interquartile range from a data set
  • Find any percentile using the linear interpolation method and check it against software
  • Build and interpret a box plot, including identifying outliers with the 1.5 IQR rule
  • Read SAT, ACT, and growth-chart percentile reports without common misinterpretations
What's inside
  1. 1. What Percentiles and Quartiles Actually Mean
    Introduces rank-based statistics, defines percentile and quartile in plain language, and contrasts them with the mean.
  2. 2. Finding the Median and the Quartiles
    Step-by-step procedure for computing Q1, Q2 (median), and Q3 from a data set, including the odd/even count split and the IQR.
  3. 3. Computing Any Percentile
    Two standard methods (nearest-rank and linear interpolation) for finding the kth percentile, with worked examples and a note on why different calculators give different answers.
  4. 4. Box Plots and Outliers
    How to draw a box plot from a five-number summary, the 1.5 IQR outlier rule, and how to compare distributions using side-by-side box plots.
  5. 5. Reading Percentiles in the Real World
    How to interpret SAT/ACT percentile rankings, pediatric growth charts, and income percentiles — including the misconceptions students bring in.
  6. 6. Percentiles vs. Other Summaries: When to Use What
    Compares percentile-based summaries to mean and standard deviation, explains robustness to outliers, and previews where these ideas reappear in later statistics courses.
Published by Solid State Press
Percentiles and Quartiles cover
TLDR STUDY GUIDES

Percentiles and Quartiles

Five-Number Summaries, Box Plots, and Reading Test Scores Right — A TLDR Primer
Solid State Press

Percentiles and Quartiles

Five-Number Summaries, Box Plots, and Reading Test Scores Right — 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 What Percentiles and Quartiles Actually Mean
  2. 2 Finding the Median and the Quartiles
  3. 3 Computing Any Percentile
  4. 4 Box Plots and Outliers
  5. 5 Reading Percentiles in the Real World
  6. 6 Percentiles vs. Other Summaries: When to Use What
Chapter 1

What Percentiles and Quartiles Actually Mean

Suppose your friend scored 84 out of 100 on a standardized test. Is that good? You cannot answer that question from the score alone — you need to know how everyone else did. That is exactly the problem that percentiles and quartiles are designed to solve.

The Core Idea: Rank, Not Value

When statisticians talk about percentiles and quartiles, they are thinking about rank — where a data point sits relative to every other value in a dataset, not what the value itself is. Rank is an ordinal measurement: it tells you order (first, second, third…) rather than the actual number.

Think of a race. The runner who finishes first has rank 1. You do not need to know their exact time to know they beat everyone else. Percentiles and quartiles work the same way — they organize data from smallest to largest and then ask: what position does a given value occupy?

A percentile expresses rank as a percentage of the group. The $k$th percentile is the value at or below which $k$ percent of the data falls. So if your test score is at the 70th percentile, roughly 70% of test-takers scored at or below you, and about 30% scored above you.

A quartile is a special case of a percentile. Quartiles split a sorted dataset into four equal-sized pieces — four "quarters." There are three dividing lines:

  • Q1 (first quartile) = 25th percentile. One quarter of the data falls at or below this point.
  • Q2 (second quartile) = 50th percentile. Half the data falls at or below this point. Q2 has another name you already know: the median.
  • Q3 (third quartile) = 75th percentile. Three quarters of the data falls at or below this point.

About This Book

If you need statistics help for high school students made plain and fast, this book is for you. Maybe you're prepping for an AP Statistics descriptive statistics review, studying for the SAT, or trying to decode a standardized test score report that lists your child at "the 73rd percentile." Maybe you're a tutor looking for a tight, reliable refresher before a session.

This primer covers how to find quartiles and percentiles by hand, walks through the five-number summary, and explains box plot and five-number summary construction step by step. It also covers the interquartile range and outliers — including how to flag them with the standard fence rule — and shows you how to read growth chart percentiles the same way a pediatrician does. Concise by design, with no filler.

Read straight through in order; the sections build on each other. Work every example yourself before reading the solution, then use the problem set at the end to confirm you've got it.

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