Where Did These Ideas Come From?
Imagine you're on a basketball team. After ten games, your coach says, "Our team scores about 48 points per game." That single number tells you something useful about all ten games at once. People have been looking for ways to summarize data like this for hundreds of years.
Long before calculators existed, scientists and mathematicians needed ways to combine lots of measurements into one or two numbers. Here are a few key moments in that story.
Here's the big question these tools answer: If I have a whole list of numbers, how can I describe that list with just one or two numbers so someone else understands the big picture?
Core Definitions
In statistics, we split our summary numbers into two groups: measures of center (which tell you what's typical) and measures of variability (which tell you how spread out the data is). Let's define the four key terms.
Mean
Median
Interquartile Range (IQR)
Mean Absolute Deviation (MAD)
Seeing Center & Spread on a Number Line
Let's look at a data set and see how the mean, median, and spread actually look. Imagine a class of 9 students scored these points on a quiz:
2, 4, 5, 5, 6, 7, 7, 8, 10
In the diagram above, every pink dot is one student's score. Notice that the median sits right in the middle of the ordered list — there are 4 scores below it and 4 scores above it. The mean also happens to be 6 for this data set. The golden shaded region shows the IQR, which captures where the middle half of students scored.
Here's something important: the mean and median don't always match. If one student had scored 20 instead of 10, the mean would jump up, but the median would stay at 6. That's why we sometimes prefer the median — it isn't pulled by extreme values.
The Formulas
Now let's look at the actual math behind each measure. Don't worry — the formulas are simpler than they look! We'll plug in real numbers in the next section.
Notice that the mean and MAD are partners: you need the mean first before you can calculate the MAD. The median and IQR are also partners: you need the median (and quartiles) to find the IQR.
Step-by-Step Breakdown: How Quartiles Work
Finding the IQR can feel tricky because you first need to find Q1 (the first quartile) and Q3 (the third quartile). Let's break this down carefully with a visual.
Quartiles split ordered data into four equal groups. Here's how the process works using our quiz data: 2, 4, 5, 5, 6, 7, 7, 8, 10.
When the lower half has an even number of values (like our 4 values: 2, 4, 5, 5), you find Q1 by averaging the two middle numbers of that half. The same goes for Q3 with the upper half. The IQR of 3 means the middle half of students scored within a 3-point range — that's a pretty tight cluster!
Worked Example: Full Calculation
A student recorded how many minutes she practiced piano each day for one week:
15, 20, 25, 20, 30, 45, 25
Let's find the mean, median, IQR, and MAD.
15, 20, 20, 25, 25, 30, 45Sum = 15 + 20 + 20 + 25 + 25 + 30 + 45 = 180
Mean = 180 ÷ 7 ≈ 25.715, 20, 20, [25], 25, 30, 45IQR = Q3 − Q1 = 30 − 20 = 10|15 − 25.7| = 10.7
|20 − 25.7| = 5.7
|20 − 25.7| = 5.7
|25 − 25.7| = 0.7
|25 − 25.7| = 0.7
|30 − 25.7| = 4.3
|45 − 25.7| = 19.3
Now add those distances and divide by 7:
Sum of distances = 10.7 + 5.7 + 5.7 + 0.7 + 0.7 + 4.3 + 19.3 = 47.1
MAD = 47.1 ÷ 7 ≈ 6.7Mean vs. Median & IQR vs. MAD
Which measure should you use? It depends on the data! Let's compare them side by side.
| Feature | Mean & MAD | Median & IQR |
|---|---|---|
| What it measures | Center (mean) & average distance from center (MAD) | Center (median) & spread of the middle 50% (IQR) |
| Uses every value? | Yes — every number affects the mean and MAD | No — the median and IQR focus on position, not every value |
| Affected by outliers? | Yes — one extreme value can pull the mean a lot | Not much — the median stays steady even with extreme values |
| Best used when… | Data is roughly symmetric with no extreme outliers | Data is skewed or has outliers |
| Example scenario | Test scores where most students scored similarly | Home prices in a neighborhood (a few mansions could skew the mean) |
What Comes Next?
The mean, median, IQR, and MAD are your first toolkit for describing data. As you move into 7th and 8th grade, you'll learn even more powerful tools. Here's a sneak peek at how these ideas grow.
| What You Know Now | What You'll Learn Later |
|---|---|
| Mean — the average | Weighted mean — some values count more than others (like final exams counting more than homework) |
| Median — the middle value | Percentiles — the median is the 50th percentile; you can find any percentile |
| MAD — average distance from mean | Standard deviation — a related measure of spread used in high school and beyond |
| IQR — spread of the middle 50% | Box-and-whisker plots — a visual that shows Q1, median, Q3, and outliers all at once |
| Describing one data set | Comparing two data sets using these measures (Which class did better? By how much?) |
You're building a foundation right now that every scientist, economist, sports analyst, and data scientist uses every single day. The cool part? You already have enough tools to answer real questions about real data!
Practice Problems
Try these on your own. Click "Show Answer" when you're ready to check your work.
3, 7, 8, 10, 122, 3, 5, 7, 8, 9, 12, 15. Find the IQR of this data set.0, 1, 1, 2, 2, 6. Find the mean and MAD. Then explain: does the mean or the median better describe this player's typical game?78, 80, 82, 84, 86 Student B: 60, 70, 82, 94, 104. Both students have the same mean. Calculate the MAD for each student. Who is more consistent? Explain how you can tell.Lesson Summary
In this lesson, you learned four essential tools for describing data. The mean (average) adds all values and divides by the count, while the median finds the middle value in an ordered list. Together, these are your measures of center — they tell you what's typical. For measures of variability, the interquartile range (IQR) measures the spread of the middle 50% of data by subtracting Q1 from Q3, and the mean absolute deviation (MAD) tells you the average distance each data point is from the mean.
You also learned that the mean and MAD work best when data is symmetric and has no outliers, while the median and IQR are more reliable when extreme values are present. To fully describe a data set, you always need both a measure of center and a measure of variability — because knowing the "middle" means very little if you don't also know how spread out the data is around that middle.