Historical Context & Motivation
Imagine you just scored 85 on a test. Is that good? You can't tell unless you know how everyone else did. For thousands of years, people have wanted ways to summarize large groups of numbers with just one or two values. That idea is at the heart of statistics.
Ancient astronomers noticed that their measurements of star positions varied slightly every night. They needed a single "best guess" to use in their maps. Over time, mathematicians developed tools to find the center of a data set and to measure how spread out the values are.
Today you'll learn the same tools that scientists, coaches, and data analysts use every day. The big question is: How can we describe a whole data set with just a couple of numbers?
Core Principles & Definitions
To summarize data, you need two kinds of numbers. The first kind tells you where the center is — a typical value. The second kind tells you the variability — how spread out the values are from that center.
Mean (Average)
Median (Middle Value)
IQR (Interquartile Range)
MAD (Mean Absolute Deviation)
Visualizing Center & Spread
The dot plot below shows quiz scores for two different classes. Both classes have the same mean (7 points), but look how different the shapes are! Class A has scores bunched close together, while Class B has scores spread far apart.
This is why we need measures of variability. If you only reported the mean, you'd think both classes performed the same way. But a student in Class B might have scored a 2 or a 12 — very different experiences from Class A, where most students scored between 6 and 8.
The Formulas You Need
Let's look at each formula one at a time. Don't worry — each one is just addition, subtraction, and division!
Finding the IQR Step by Step
The IQR can feel tricky at first, so let's break it down visually. Imagine you have 11 quiz scores. The diagram below shows how to split the data into quartiles.
Notice that the overall median (8) is not included in either the lower or upper half when you have an odd number of values. You only use the numbers on each side of it to find Q1 and Q3.
Worked Example: Full Analysis
A gym teacher records the number of push-ups 8 students completed: 10, 15, 12, 8, 20, 14, 11, 18. Let's find the mean, median, IQR, and MAD.
When to Use Which Measure
Mean and median both describe the center. IQR and MAD both describe the spread. So when should you pick one over the other? The answer depends on whether your data has outliers (values that are much larger or smaller than the rest).
| Feature | Mean & MAD | Median & IQR |
|---|---|---|
| Best when | Data is roughly symmetric with no outliers | Data is skewed or has outliers |
| Effect of outliers | Pulled strongly by extreme values | Barely affected by extreme values |
| Uses every value? | Yes — every number changes the result | No — only position matters, not exact values |
| Example data | Test scores: 78, 82, 85, 88, 90 | Home prices: $150K, $160K, $170K, $180K, $2M |
Connection to Future Topics
The ideas you're learning now are the building blocks for more advanced statistics. In high school and college, you'll encounter related tools that do similar jobs in more powerful ways.
| What You Know Now | What Comes Next |
|---|---|
| Mean (average) | Weighted mean — some values count more than others (like final exams counting more than homework) |
| MAD (mean absolute deviation) | Standard deviation — a similar idea but it squares the differences before averaging, making it more useful in algebra |
| IQR (interquartile range) | Box-and-whisker plots — a visual that shows Q1, median, Q3, and outliers all at once |
| Median | Percentiles — the median is actually the 50th percentile; you'll learn about any percentile you want |
Don't worry about mastering those advanced topics now. The important thing is that by understanding mean, median, IQR, and MAD, you already have a strong foundation. Everything else is just an extension of these same ideas.
Practice Problems
Lesson Summary
To describe a data set, you need two types of measures. Measures of center tell you the typical value: the mean (add all values and divide by how many) and the median (the middle value when data is ordered). Measures of variability tell you how spread out the data is: the MAD (average distance from the mean) and the IQR (range of the middle 50%, found by Q3 − Q1).
Use the mean and MAD when your data is symmetric and has no outliers. Use the median and IQR when your data is skewed or contains extreme values. Together, a measure of center and a measure of variability give you a complete snapshot of any data set.