Measures of Center & Variability - 6th Grade Math

Section 1

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.

~1500s

Astronomers in Europe began averaging their observations of star positions. By adding measurements together and dividing, they got a single "best guess" — an early version of the mean.

1770s

French mathematician Pierre-Simon Laplace showed that averaging many measurements reduces errors. This made the mean a trusted tool in science.

1800s

Statisticians realized the mean can be fooled by extreme values. They started using the median (the middle value) as a more reliable summary when data contains outliers.

1900s

Researchers developed ways to measure variability — how spread out data is. The interquartile range (IQR) and mean absolute deviation (MAD) became popular tools in classrooms and labs.

Today

Every time your phone shows an "average screen time" or a weather app shows "typical temperatures," it uses measures of center and variability behind the scenes.

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?

Section 2

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

Add up all the values, then divide by how many values there are. This is the "average" you probably already know. It uses every single number in the data set.

Median

Put the values in order from least to greatest. The median is the middle value. If there are two middle values, find the mean of those two.

Interquartile Range (IQR)

Split the ordered data into four equal parts using quartiles. The IQR is the distance between the first quartile (Q1) and the third quartile (Q3). It tells you how spread out the middle 50% of data is.

Mean Absolute Deviation (MAD)

Find how far each value is from the mean, then average those distances. MAD tells you, on average, how much the data values differ from the mean.

✦

Key Takeaway

Think of center like the target on a dartboard — it shows where most data clusters. Think of variability like how scattered the darts are around that target. Two players might have the same average score, but one could be very consistent while the other is all over the place. You need both numbers to understand the full story.

Section 3

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

Dot plot showing quiz scores from 2 to 10 on a number line, with the mean at 6 and the median at 6 marked. Q1 is 4.5 and Q3 is 7.5, with the IQR shown as a shaded region.

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.

Section 4

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.

Mean (Average)

Mean = Sum of all values ÷ Number of values

Add every data value together, then divide by how many values you have.

Median (Middle Value)

1. Order the values from least to greatest 2. Find the middle position: (n + 1) ÷ 2

n = the number of data values. If n is even, average the two middle values.

Interquartile Range (IQR)

IQR = Q3 − Q1

Q1 = median of the lower half | Q3 = median of the upper half

Mean Absolute Deviation (MAD)

MAD = Sum of |each value − mean| ÷ Number of values

The vertical bars | | mean "absolute value" — always make the distance positive.

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.

✦

Key Takeaway

Think of absolute value like measuring distance on a map. Distance is always positive — you can't walk "negative 3 miles." When we figure out how far each data point is from the mean, we only care about the distance, not the direction. That's why we use absolute value in the MAD formula.

Section 5

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.

Diagram showing how 9 data values are split into a lower half and an upper half with the median in the middle. Q1 is the median of the lower half and Q3 is the median of the upper half.

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!

Section 6

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.

Full Calculation

1
Step 1 — Order the data from least to greatest

15, 20, 20, 25, 25, 30, 45

There are 7 values.
2
Step 2 — Find the Mean

Sum = 15 + 20 + 20 + 25 + 25 + 30 + 45 = 180 Mean = 180 ÷ 7 ≈ 25.7

On average, she practiced about 25.7 minutes per day.
3
Step 3 — Find the Median

Middle position = (7 + 1) ÷ 2 = the 4th value. 15, 20, 20, [25], 25, 30, 45

The median is 25 minutes.
4
Step 4 — Find Q1 and Q3

Lower half (below the median): 15, 20, 20. The middle of these three is 20. So Q1 = 20. Upper half (above the median): 25, 30, 45. The middle of these three is 30. So Q3 = 30. IQR = Q3 − Q1 = 30 − 20 = 10

The middle 50% of her practice times are spread across 10 minutes.
5
Step 5 — Find the MAD

First, find how far each value is from the mean (25.7). Remember to use absolute value! |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 ≈ 6.7

On average, each day's practice time differs from the mean by about 6.7 minutes.
6
Step 6 — Interpret the results

The student's typical practice time is around 25–26 minutes (mean ≈ 25.7, median = 25). Her practice times vary — some days she practiced much less (15 min) or much more (45 min). The MAD of 6.7 minutes and IQR of 10 minutes both confirm that there's a moderate amount of spread in her routine.

Section 7

Mean 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)

✦

Key Takeaway

Imagine you and four friends earned $10, $12, $11, $13, and $200 for doing chores this week. The mean would be $49.20 — but that doesn't describe most of you at all! The median ($12) is much more helpful here. When one value is way different from the rest (an outlier), the median and IQR give a more honest picture.

Section 8

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!

Section 9

Practice Problems

Try these on your own. Click "Show Answer" when you're ready to check your work.

Question 1: Problem 1 — Conceptual

What is the difference between a measure of center and a measure of variability? Give one example of each.

Show answer

A measure of center tells you what's typical or "in the middle" of a data set. Examples: mean or median. A measure of variability tells you how spread out the data is — whether the values are close together or far apart. Examples: IQR or MAD. You need both to fully describe a data set!

Question 2: Problem 2 — Basic Calculation

Find the mean and median of this data set: 3, 7, 8, 10, 12

Show answer

Mean: Add the values: 3 + 7 + 8 + 10 + 12 = 40. Divide by 5: 40 ÷ 5 = 8. Median: The data is already in order. There are 5 values, so the middle one is the 3rd value: 8. In this case, the mean and median are both 8. That happens when data is balanced (symmetric).

Question 3: Problem 3 — Intermediate

A group of friends tracked how many books they read over the summer: 2, 3, 5, 7, 8, 9, 12, 15. Find the IQR of this data set.

Show answer

There are 8 values. Split them into two halves of 4 each. Lower half: 2, 3, 5, 7 → Q1 = (3 + 5) ÷ 2 = 4. Upper half: 8, 9, 12, 15 → Q3 = (9 + 12) ÷ 2 = 10.5. IQR = Q3 − Q1 = 10.5 − 4 = 6.5. The middle 50% of readers read between about 4 and 10.5 books — a spread of 6.5 books.

Question 4: Problem 4 — Applied

A soccer player scored the following number of goals in 6 games: 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?

Show answer

Mean: (0 + 1 + 1 + 2 + 2 + 6) ÷ 6 = 12 ÷ 6 = 2. MAD: Distances from the mean (2): |0−2|=2, |1−2|=1, |1−2|=1, |2−2|=0, |2−2|=0, |6−2|=4. Sum = 8. MAD = 8 ÷ 6 ≈ 1.3. Median: (1 + 2) ÷ 2 = 1.5. The median (1.5) is a better description of a "typical" game because the player usually scores 0–2 goals. The score of 6 is an outlier that pulls the mean up. In 5 out of 6 games, the player scored 2 or fewer goals.

Question 5: Problem 5 — Challenge

Two students each took 5 quizzes. Their scores are shown below: Student A: 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.

Show answer

Both means are (78+80+82+84+86) ÷ 5 = 410 ÷ 5 = 82, and (60+70+82+94+104) ÷ 5 = 410 ÷ 5 = 82. ✓ Student A's MAD: Distances from 82: |78−82|=4, |80−82|=2, |82−82|=0, |84−82|=2, |86−82|=4. Sum = 12. MAD = 12 ÷ 5 = 2.4. Student B's MAD: Distances from 82: |60−82|=22, |70−82|=12, |82−82|=0, |94−82|=12, |104−82|=22. Sum = 68. MAD = 68 ÷ 5 = 13.6. Student A is much more consistent. Their MAD is only 2.4, meaning scores are usually within about 2–3 points of the average. Student B has a MAD of 13.6, meaning their scores jump around a lot. This is a perfect example of why the mean alone isn't enough — you need a measure of variability to see the whole picture!

Summary

Lesson Summary

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.