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Compare Two
Populations Using Data

Learn to use measures of center and measures of variability to draw informal comparative inferences about two different populations.

7th Grade Math

What Does It Mean to Compare Populations?

A population is the entire group you want to study. When we compare two populations, we look at numerical data from each group and ask: is there a meaningful difference between them?

Population A
Word lengths in a 7th-grade science textbook
Population B
Word lengths in a 4th-grade science textbook

Measures of Center

Measures of center tell us where the "middle" of a data set is. The two key measures are:

Mean
Add all values and divide by the count: \(\bar{x} = \frac{\sum x_i}{n}\)
Median
The middle value when data is ordered from least to greatest

Measures of Variability

Variability tells us how spread out the data values are. Two data sets can have the same mean but very different spreads.

Range
Maximum value minus minimum value: \(\text{Range} = x_{\max} - x_{\min}\)
IQR
Interquartile range: \(\text{IQR} = Q_3 - Q_1\). Measures spread of the middle 50%.
MAD
Mean Absolute Deviation: average distance of each value from the mean.

Visualizing Two Populations

Dot plots help us see the shape, center, and spread of each data set side by side.

Population A: 7th-Grade Words 2 3 4 5 6 7 8 9 mean ≈ 5.3 Population B: 4th-Grade Words 2 3 4 5 6 7 8 9 mean ≈ 3.9 Number of Letters per Word

The Comparative Inference Rule

Mean A Mean B Difference in Centers

When the two distributions barely overlap, the difference is meaningful. When they overlap a lot, the populations may not be truly different.

Using MAD to Gauge the Difference

The Mean Absolute Deviation (MAD) measures how spread out values are from the mean. We can express the difference between two means in terms of MAD:

> 2 MADs apart
Strong evidence of a real difference between populations
1 to 2 MADs
Some difference, but not conclusive
< 1 MAD
Populations likely overlap too much to draw conclusions

Center vs. Variability

Both are essential for comparison. One without the other can be misleading.

Measures of Center Measures of Variability
What it tells Where data clusters How spread out data is
Examples Mean, Median Range, IQR, MAD
Question it answers "What is typical?" "How much do values vary?"
For comparing Which group is higher or lower? Which group is more consistent?

How to Compare Two Populations

1
Collect random samples from each population and organize the data.
2
Calculate measures of center (mean or median) for each sample.
3
Calculate measures of variability (MAD, IQR, or range) for each sample.
4
Compare the difference in centers to the variability. Is the gap large relative to the spread?
5
Draw an inference: state whether the populations appear to be different and support with evidence.

Example 1: Test Scores

Two classes took the same math test. Here are their scores:

Class A
72, 75, 78, 80, 82, 85, 88
Class B
60, 65, 70, 72, 75, 78, 80

Example 1: Finding the MAD

Example 2: Plant Heights

A student grew plants under two conditions and measured their heights (cm) after 3 weeks:

Sunlight
12, 14, 15, 15, 16, 18
Shade
6, 7, 8, 9, 9, 11
Sunlight Group Shade Group 4 6 8 10 12 14 16 18 20 4 6 8 10 12 14 16 18 20 mean ≈ 15 mean ≈ 8.3

Example 2: Drawing the Inference

Common Mistakes to Avoid

✕ Comparing Only Centers
Saying "Group A scored higher" without checking if the difference is large relative to the spread. A 2-point mean difference with MAD of 10 is not meaningful.
✕ Ignoring Sample Size
Drawing a strong conclusion from just 3 or 4 data points. Larger samples give more reliable inferences.
✕ Confusing MAD with Range
Range uses only the max and min. MAD uses every data point, making it a more reliable measure of variability.

Summary: Key Measures

Measure What It Tells You Formula / Method
Mean Average value \(\bar{x} = \frac{\sum x_i}{n}\)
Median Middle value Order data, find the center
Range Total spread \(x_{\max} - x_{\min}\)
IQR Middle 50% spread \(Q_3 - Q_1\)
MAD Typical distance from mean \(\frac{\sum |x_i - \bar{x}|}{n}\)

Question 1 of 5

Two groups of students measured their jump distances. Group A has a mean of 48 inches and Group B has a mean of 52 inches. Both groups have a MAD of 6 inches. What is the difference in means expressed as a multiple of MAD?

Question 2 of 5

Which measure of center is most resistant to outliers?

Question 3 of 5

Team X has a mean score of 85 with a MAD of 3. Team Y has a mean score of 76 with a MAD of 4. What is the ratio of the difference in means to the average MAD?

Question 4 of 5

A data set has values: 10, 12, 14, 16, 18. The mean is \(14\). What is the MAD?

Question 5 of 5

Two dot plots show data from School A and School B. The distributions overlap almost completely. What can you conclude?