Measures of Center & Variability — I can summarize a distribution with measures of center and variability (mean/median, IQR/MAD). - Pre Algebra

SECTION 1

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.

~3000 BCE • Ancient Record-Keeping

Egyptians and Babylonians kept records of harvests and taxes. They added totals and divided to find typical amounts — an early form of the mean.

1669 • The Arithmetic Mean

Christiaan Huygens formally described the arithmetic mean (average) as the best single estimate for a group of measurements.

1882 • The Median Gains Popularity

Francis Galton promoted the median as a better measure when data has extreme values, like very high incomes.

1900s • Spread Matters Too

Statisticians realized that knowing the center is not enough. Measures like the IQR and MAD were developed to describe how spread out data is.

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?

SECTION 2

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)

Add up all the values, then divide by how many values there are. The mean is pulled toward extreme values (outliers).

Median (Middle Value)

Put the values in order from least to greatest. The median is the middle number. If there are two middle numbers, average them.

IQR (Interquartile Range)

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

MAD (Mean Absolute Deviation)

Find the mean, then find how far each value is from the mean. Average those distances. The MAD tells you the typical distance from the mean.

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KEY TAKEAWAY

Think of a basketball team. The center (mean or median) is like the team's typical height. The variability (IQR or MAD) tells you whether all the players are about the same height or if some are much taller or shorter than others. You need both pieces of information to really understand the team!

SECTION 3

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.

Both classes share a mean of 7, but Class A's dots cluster tightly while Class B's dots spread across a wide range. The variability is the key difference.

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.

SECTION 4

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!

MEAN (AVERAGE)

Mean = Sum of all values ÷ Number of values

Add every number in your data set. Then divide by how many numbers there are. For example, for the set {3, 5, 7}: Mean = (3 + 5 + 7) ÷ 3 = 5.

MEDIAN (MIDDLE VALUE)

Median = Middle value of the ordered data set

Step 1: Put the numbers in order from least to greatest. Step 2: If there is an odd number of values, the median is the one right in the middle. If there is an even number of values, average the two middle numbers.

IQR (INTERQUARTILE RANGE)

IQR = Q3 − Q1

Q1 (first quartile) is the median of the lower half. Q3 (third quartile) is the median of the upper half. The IQR tells you the range of the middle 50% of your data.

MAD (MEAN ABSOLUTE DEVIATION)

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

The vertical bars | | mean "absolute value" — just ignore any negative signs. First find the mean. Then find how far each value is from the mean. Finally, average all those distances.

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Quick Tip

Mean and MAD are a pair — they go together. Median and IQR are a pair — they go together. If your data has outliers (extreme values), use the median and IQR because they are not pulled by extreme numbers.

SECTION 5

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.

The data is ordered from least to greatest. The median splits the data in half. Then we find Q1 (median of the lower half) and Q3 (median of the upper half). The IQR = Q3 − Q1.

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.

SECTION 6

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.

Push-Up Data: Full Statistical Summary

1
Step 1 — Order the Data

Arrange the values from least to greatest: 8, 10, 11, 12, 14, 15, 18, 20.

Ordered: 8, 10, 11, 12, 14, 15, 18, 20
2
Step 2 — Find the Mean

Add all values: 8 + 10 + 11 + 12 + 14 + 15 + 18 + 20 = 108. Divide by the number of values: 108 ÷ 8 = 13.5.

Mean = 13.5
3
Step 3 — Find the Median

There are 8 values (an even number). The two middle values are the 4th and 5th: 12 and 14. Average them: (12 + 14) ÷ 2 = 13.

Median = 13
4
Step 4 — Find Q1, Q3, and IQR

Lower half: 8, 10, 11, 12. The median of the lower half = (10 + 11) ÷ 2 = 10.5, so Q1 = 10.5. Upper half: 14, 15, 18, 20. The median of the upper half = (15 + 18) ÷ 2 = 16.5, so Q3 = 16.5. IQR = 16.5 − 10.5 = 6.

IQR = 6
5
Step 5 — Find the MAD

The mean is 13.5. Find how far each value is from 13.5: |8 − 13.5| = 5.5, |10 − 13.5| = 3.5, |11 − 13.5| = 2.5, |12 − 13.5| = 1.5, |14 − 13.5| = 0.5, |15 − 13.5| = 1.5, |18 − 13.5| = 4.5, |20 − 13.5| = 6.5. Sum of distances: 5.5 + 3.5 + 2.5 + 1.5 + 0.5 + 1.5 + 4.5 + 6.5 = 26. MAD = 26 ÷ 8 = 3.25.

MAD = 3.25
6
Step 6 — Interpret the Results

The typical student did about 13–13.5 push-ups. On average, students were about 3.25 push-ups away from the mean. The middle half of students completed between 10.5 and 16.5 push-ups (a span of 6).

SECTION 7

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

Comparison of center and variability measures

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KEY TAKEAWAY

Imagine five friends pool their money: $10, $10, $12, $11, $200. The mean is $48.60 — but that doesn't describe anyone! The median is $11, which feels much more "typical." When one value is wildly different from the rest, the median and IQR give a more honest picture.

SECTION 8

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

From middle-school stats to high-school stats

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.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A data set has a mean of 50 and a median of 50. Another data set also has a mean of 50 and a median of 50, but its MAD is much larger. What does a larger MAD tell you about the second data set compared to the first?

Show answer

A larger MAD means the values in the second data set are more spread out from the mean. Even though both sets have the same center (50), the second set's individual values are farther away from 50 on average. For example, the first set might be {48, 50, 52} while the second might be {20, 50, 80}. The MAD measures the typical distance from the mean, so a bigger MAD always means more variability.

Question 2: BASIC CALCULATION

Find the mean and median of this data set: 4, 7, 9, 10, 15.

Show answer

To find the mean, add all values: 4 + 7 + 9 + 10 + 15 = 45. Then divide by the number of values: 45 ÷ 5 = 9. The mean is 9. To find the median, the data is already in order and there are 5 values (odd number), so the middle value is the 3rd one: 9. The median is 9. In this case, the mean and median are the same because the data is fairly symmetric.

Question 3: INTERMEDIATE

A group of students recorded how many books they read last month: 1, 2, 3, 3, 4, 5, 5, 6, 20. Find the mean, median, and MAD. Which measure of center better represents the typical student?

Show answer

First, find the mean: (1 + 2 + 3 + 3 + 4 + 5 + 5 + 6 + 20) ÷ 9 = 49 ÷ 9 ≈ 5.4. The median is the 5th value in the ordered list: 4. For the MAD, find each distance from the mean (5.4): |1 − 5.4| = 4.4, |2 − 5.4| = 3.4, |3 − 5.4| = 2.4, |3 − 5.4| = 2.4, |4 − 5.4| = 1.4, |5 − 5.4| = 0.4, |5 − 5.4| = 0.4, |6 − 5.4| = 0.6, |20 − 5.4| = 14.6. Sum = 30. MAD = 30 ÷ 9 ≈ 3.3. The median (4) is a better measure of center because the outlier (20) pulls the mean up to 5.4, which is higher than most students' actual reading count.

Question 4: APPLIED

Two pizza restaurants track their delivery times (in minutes). Restaurant A: 20, 22, 25, 23, 21, 24, 22. Restaurant B: 15, 18, 30, 22, 35, 19, 21. Find the IQR for each restaurant. Which restaurant is more consistent?

Show answer

Restaurant A ordered: 20, 21, 22, 22, 23, 24, 25. Median = 22 (4th value). Lower half: 20, 21, 22 → Q1 = 21. Upper half: 23, 24, 25 → Q3 = 24. IQR = 24 − 21 = 3 minutes. Restaurant B ordered: 15, 18, 19, 21, 22, 30, 35. Median = 21 (4th value). Lower half: 15, 18, 19 → Q1 = 18. Upper half: 22, 30, 35 → Q3 = 30. IQR = 30 − 18 = 12 minutes. Restaurant A has a much smaller IQR (3 vs. 12), so it is more consistent. You can count on Restaurant A to deliver in a narrow window, while Restaurant B is unpredictable.

Question 5: CRITICAL THINKING

Create two different data sets that each have exactly 5 values. Both sets must have a median of 10, but one set should have an IQR of 2 and the other should have an IQR of 10. Explain your reasoning.

Show answer

One possible answer: Set 1 (IQR of 2): {8, 9, 10, 11, 12}. Median = 10. Q1 = 9, Q3 = 11, IQR = 11 − 9 = 2. Set 2 (IQR of 10): {2, 5, 10, 15, 20}. Median = 10. Q1 = 5, Q3 = 15, IQR = 15 − 5 = 10. To build these, start by placing 10 in the middle position (3rd value). For a small IQR, keep Q1 and Q3 close to 10. For a large IQR, push Q1 lower and Q3 higher. The 1st and 5th values can be anything as long as they stay outside Q1 and Q3. Many correct answers are possible — the key is controlling the 2nd and 4th values to set Q1 and Q3.

SUMMARY

Lesson Summary

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.