Compare Two Populations Using Data

SECTION 1

Why Do We Compare Groups Using Data?

People have been comparing groups for a long time. A farmer might wonder, "Do my tomato plants grow taller with fertilizer A or fertilizer B?" A coach might ask, "Which team scores more points per game?" These are great questions, but you need more than a guess to answer them. You need data — actual numbers that describe what is happening.

Over many centuries, mathematicians and scientists built tools to summarize and compare data. Let's look at some key moments in the history of statistics (the study of collecting and understanding data).

1700s

Early Averages

Astronomers began using the mean (average) to combine many measurements of star positions. This helped reduce errors.

1800s

Measuring Spread

Scientists realized that knowing just the average wasn't enough. They developed measures of variability (how spread out the data is) to understand differences between groups.

1900s

Random Sampling Takes Off

Researchers discovered that picking a random sample (a group chosen by chance) from a larger population gives trustworthy results without studying everyone.

2000s

Data Everywhere

With computers and the internet, comparing populations became easier than ever. Sports stats, medical studies, and even social media all rely on comparing groups of data.

Today's big question is the same one people have asked for hundreds of years: How can we tell if two groups are truly different, or if they're basically the same? In this lesson, you will learn to use measures of center and variability to answer that question.

SECTION 2

Core Ideas: Center, Variability, and Inference

Before you compare two groups, you need to understand three big ideas. These are the building blocks of every comparison you will make.

Measures of Center

A measure of center tells you where the "middle" of a data set is. The two most common are the mean (add all values and divide by how many) and the median (the middle value when data is in order).

Measures of Variability

A measure of variability tells you how spread out the data is. Key examples are the range (max − min), the interquartile range (IQR), and the mean absolute deviation (MAD).

Random Samples

A random sample is a smaller group chosen by chance from a larger population. Because each member had an equal chance of being picked, the sample's statistics are fair representatives of the whole group.

Informal Comparative Inference

An informal comparative inference is a conclusion you draw about two populations based on their sample data. It's "informal" because you use reasoning and visual evidence rather than advanced formulas.

✦ KEY TAKEAWAY

Think of comparing two populations like comparing two basketball teams. The mean is like each team's average points per game — it tells you who typically scores more. The variability is like how consistent each team is. A team that scores between 50 and 100 points is unpredictable, while a team that always scores around 75 is very consistent. You need both pieces — the center AND the spread — to make a fair comparison.

SECTION 3

Seeing the Difference: Dot Plots Side by Side

One of the best ways to compare two populations is to line up their data displays. The diagram below shows dot plots for the number of letters in randomly sampled words from a 4th-grade science book and a 7th-grade science book. Notice how the dots cluster in different places.

Each dot is one word from a random sample. The cyan dots show 4th-grade words (mean ≈ 4.1 letters). The violet dots show 7th-grade words (mean ≈ 5.6 letters). The dashed lines mark each group's mean.

Look at where most of the dots sit. The 4th-grade words bunch up around 3 to 5 letters. The 7th-grade words bunch up around 5 to 7 letters. The means are separated by about 1.5 letters. This visual gap is our first clue that the 7th-grade book uses longer words. But we also need to check the spread — if the data were very spread out, the difference in means might not mean much.

SECTION 4

The Math Behind Comparing Populations

To compare two populations, you calculate the same statistics for each sample and then see how those numbers differ. Here are the key formulas you'll use.

MEAN (AVERAGE)

Mean = (Sum of all values) ÷ (Number of values)

Add up every data value, then divide by how many values there are. The mean tells you the typical value in the data set.

MEDIAN

Median = Middle value when data is listed from least to greatest

If there is an even number of values, the median is the mean of the two middle values. The median is useful when there are outliers (extreme values) that could pull the mean away from the center.

MEAN ABSOLUTE DEVIATION (MAD)

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

For each value, find how far it is from the mean (ignoring negative signs). Add those distances and divide by the number of values. A small MAD means data is close together. A large MAD means data is spread out.

COMPARING TWO POPULATIONS

Difference in means ÷ MAD = How many MADs apart the means are

A helpful rule of thumb: if the difference between two means is 2 or more MADs, the two populations are likely meaningfully different. If it's less than 1 MAD, they may be similar.

💡 Why MADs Matter

Imagine two students both have a mean test score of 80. Student A's scores are 78, 80, 82 (MAD = 1.3). Student B's scores are 60, 80, 100 (MAD = 13.3). Even though their means are the same, Student B's grades are way more spread out. When comparing populations, the spread matters just as much as the center!

SECTION 5

Box Plots: Another Way to Compare

Dot plots are great, but when you have a lot of data, box plots (also called box-and-whisker plots) give you a cleaner picture. A box plot shows five key numbers: the minimum, the first quartile (Q1), the median, the third quartile (Q3), and the maximum. The box in the middle holds the middle 50% of the data. The diagram below compares daily steps walked by students at two different schools.

School A (amber) has a median of 6,000 steps and an IQR of 3,000. School B (emerald) has a median of 9,000 steps and an IQR of 3,000. Since the medians are 3,000 steps apart and the IQRs are the same, School B students clearly walk more.

Notice that School B's box doesn't overlap much with School A's box. When two box plots have little or no overlap, that's strong evidence the populations are different. If the boxes overlap a lot, the populations may be similar. Also compare the IQR (the width of each box). Here, both IQRs are 3,000, so the groups have similar spread. The key difference is the location of the center.

SECTION 6

Worked Example: Comparing Test Scores

A teacher gives the same math quiz to two classes. She randomly selects 10 scores from each class. Let's compare the two classes step by step.

Class A scores: 72, 75, 78, 80, 82, 84, 85, 88, 90, 96

Class B scores: 60, 65, 70, 74, 76, 78, 80, 82, 85, 90

Comparing Class A and Class B

Step 1 — Find the Mean of Each ClassClass A: (72 + 75 + 78 + 80 + 82 + 84 + 85 + 88 + 90 + 96) ÷ 10 = 830 ÷ 10 = 83.0. Class B: (60 + 65 + 70 + 74 + 76 + 78 + 80 + 82 + 85 + 90) ÷ 10 = 760 ÷ 10 = 76.0.

Mean A = 83.0 | Mean B = 76.0

Step 2 — Find the Median of Each ClassBoth classes have 10 values, so the median is the average of the 5th and 6th values. Class A: (82 + 84) ÷ 2 = 83.0. Class B: (76 + 78) ÷ 2 = 77.0.

Median A = 83.0 | Median B = 77.0

Step 3 — Find the MAD of Each ClassFor Class A, find each distance from the mean of 83: |72−83| + |75−83| + |78−83| + |80−83| + |82−83| + |84−83| + |85−83| + |88−83| + |90−83| + |96−83| = 11 + 8 + 5 + 3 + 1 + 1 + 2 + 5 + 7 + 13 = 56. MAD = 56 ÷ 10 = 5.6. For Class B, find each distance from the mean of 76: |60−76| + |65−76| + |70−76| + |74−76| + |76−76| + |78−76| + |80−76| + |82−76| + |85−76| + |90−76| = 16 + 11 + 6 + 2 + 0 + 2 + 4 + 6 + 9 + 14 = 70. MAD = 70 ÷ 10 = 7.0.

MAD A = 5.6 | MAD B = 7.0

Step 4 — Compare the Means Using MADThe difference in means = 83.0 − 76.0 = 7.0. We can use the average of the two MADs as a reference: (5.6 + 7.0) ÷ 2 = 6.3. Now divide: 7.0 ÷ 6.3 ≈ 1.1 MADs apart.

The means are about 1.1 MADs apart

Step 5 — Draw a ConclusionSince the difference is about 1.1 MADs, there is some evidence that Class A tends to score higher than Class B. However, because 1.1 is only slightly above 1 MAD, the difference is moderate — not overwhelming. If the classes were 2 or more MADs apart, we'd be more confident.

Class A likely scores somewhat higher than Class B.

SECTION 7

When Does This Method Work Best?

Comparing populations with measures of center and variability is powerful, but like any tool, it has strengths and limitations. Knowing these helps you avoid mistakes.

Strengths and Limitations of Informal Comparative Inference

Strengths	Limitations
Uses real numbers, not just opinions, to compare groups.	Results depend on the sample. A bad sample can lead to wrong conclusions.
Works with any numerical data — scores, heights, weights, word lengths, etc.	Outliers (extreme values) can pull the mean and MAD, making comparisons misleading.
Quick to calculate with a small data set.	With very small samples (fewer than 10), the results may not represent the whole population well.
Gives you both center and spread, painting a fuller picture.	It's "informal" — it can't prove a difference with 100% certainty like advanced tests can.

✦ KEY TAKEAWAY

Think of a random sample like a taste test of soup. If you stir the pot well (random selection), one spoonful gives you a good idea of the flavor. But if you only taste from the top, you might get a misleading impression. The bigger and more random your sample, the more trustworthy your comparison will be.

SECTION 8

From Informal to Formal: What Comes Next?

Right now, you are learning to make informal inferences — reasonable conclusions based on center and spread. In high school and college, you'll learn formal statistical tests that use probability to determine how confident you can be in your comparison. The table below shows the connection.

Informal vs. Formal Statistical Inference

Feature	Informal Inference (This Lesson)	Formal Inference (Future Courses)
Tools Used	Mean, median, MAD, IQR, dot plots, box plots	Standard deviation, t-tests, p-values, confidence intervals
Conclusion Type	"Group A is likely higher than Group B"	"There is a statistically significant difference (p < 0.05)"
Math Level	Arithmetic and basic reasoning	Algebra 2 and beyond
Certainty	Good estimate, but not exact	Includes a specific confidence percentage

Everything you're learning now builds the foundation for those advanced tools. The idea is the same: compare centers, account for spread, and decide if the difference is meaningful. You're already doing real statistics!

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

Why isn't it enough to compare only the means of two data sets? Why do you also need to look at a measure of variability like the MAD?

PROBLEM 2 — BASIC CALCULATION

A sample of 5 plant heights (in cm) from Garden X is: 12, 15, 14, 13, 16. A sample from Garden Y is: 18, 20, 17, 19, 21. Find the mean of each sample.

PROBLEM 3 — INTERMEDIATE

Using the same Garden X data {12, 15, 14, 13, 16} and Garden Y data {18, 20, 17, 19, 21}, calculate the MAD of each sample and express the difference in means as a number of MADs.

PROBLEM 4 — APPLIED

A school nurse measures the number of hours of sleep for random samples of 6th graders and 8th graders. 6th graders: 9, 10, 8, 9, 10, 9, 8, 10. 8th graders: 7, 8, 6, 7, 8, 7, 6, 7. The school is deciding whether to change the start time for 8th graders. Based on these samples, do 8th graders get less sleep? Use both the mean and MAD in your answer.

PROBLEM 5 — CRITICAL THINKING

Two data sets have means that are only 0.5 MADs apart. A classmate says, "The groups are definitely the same because the difference is small." Do you agree? Explain what other factors you might consider.

SUMMARY

Pulling It All Together

To compare two populations, start by collecting random samples from each group. Calculate a measure of center (like the mean or median) and a measure of variability (like the MAD, range, or IQR) for each sample. Then compare: how far apart are the centers, and how does that gap relate to the spread of the data?

Use visual displays like dot plots and box plots to see differences at a glance. If the difference in means is 2 or more MADs, you have strong evidence the populations are different. If it's less than 1 MAD, the populations may be similar. This skill — drawing informal comparative inferences — is the foundation of all statistical comparison and will serve you well in every subject.