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Comparing
Distributions

Learn to compare two data sets by analyzing their center and spread, then draw meaningful conclusions in context.

Pre-Algebra

What Is a Distribution?

A distribution describes how data values are arranged: where they cluster, how they spread out, and what shape they form. Think of it as the "personality" of a data set.

Center
The "typical" value: mean or median.
Spread
How much the data varies: range, IQR, or MAD.
Shape
Symmetric, skewed left, or skewed right.

Measuring the Center

Mean
Add all values and divide by how many there are: \(\bar{x} = \frac{\sum x_i}{n}\). Sensitive to outliers.
Median
The middle value when data is ordered. Resistant to outliers: better for skewed data.

Measuring the Spread

Range
\(\text{max} - \text{min}\). Simple but affected by outliers.
IQR
\(Q_3 - Q_1\). Measures the middle 50%. Resistant to outliers.
MAD
Mean Absolute Deviation: average distance from the mean. Shows typical variability.

Why Compare Distributions?

Comparing distributions lets you make evidence-based decisions. Instead of guessing, you can see which group scored higher, which treatment worked better, or which team was more consistent.

1
Compare centers to see which group has a higher or lower typical value.
2
Compare spreads to see which group is more consistent or more variable.
3
State a conclusion in context using the actual topic (not just numbers).

The Comparison Framework

Center Spread Conclusion Mean or Median Range, IQR, or MAD In Context

Center vs Spread

Center Spread
Question it answers What is the typical value? How much do values vary?
Common measures Mean, Median Range, IQR, MAD
Higher value means Data is typically larger Data is more spread out
Lower value means Data is typically smaller Data is more consistent
Real-world example "Team A scores more on average" "Team B is more consistent"

How to Compare Step-by-Step

1
Calculate center for both groups (mean or median).
2
Calculate spread for both groups (range, IQR, or MAD).
3
Compare the centers: Which group has a higher/lower typical value? By how much?
4
Compare the spreads: Which group is more consistent or more variable?
5
Write a conclusion in context: Use the real-world topic to explain what the numbers mean.

Visualizing Two Distributions

Below are dot plots showing quiz scores for two classes. Each dot represents one student's score.

Class A 5 6 7 8 9 10 Class B 5 6 7 8 9 10

Example 1: Calculating Center

Using the quiz data from the dot plots:
Class A: 6, 7, 7, 7, 8, 8, 8, 8, 9, 9
Class B: 5, 5, 6, 7, 8, 9, 9, 10, 10, 10

Class A Mean
\(\bar{x} = \frac{6+7+7+7+8+8+8+8+9+9}{10} = \frac{77}{10} = 7.7\)
Class B Mean
\(\bar{x} = \frac{5+5+6+7+8+9+9+10+10+10}{10} = \frac{79}{10} = 7.9\)

Example 2: Calculating Spread

Class A
Range: \(9 - 6 = 3\)
\(Q_1 = 7,\; Q_3 = 8\)
IQR: \(8 - 7 = 1\)
Class B
Range: \(10 - 5 = 5\)
\(Q_1 = 6,\; Q_3 = 10\)
IQR: \(10 - 6 = 4\)

Example 3: Writing a Contextual Conclusion

Now we put it all together with a real-world conclusion:

✓
Compares centers with actual values.
✓
Compares spreads with actual values.
✓
Uses context: "quiz scores," "students," "performance."

Example 4: Reading Box Plots

Two basketball players' points per game over a season:

5 10 15 20 25 30 Player A Player B Points Per Game

Using MAD to Compare Spread

The Mean Absolute Deviation tells you the average distance each data point is from the mean. Here is how to compute it:

1
Find the mean \(\bar{x}\).
2
Find each distance from the mean: \(|x_i - \bar{x}|\).
3
Average those distances: \(\text{MAD} = \frac{\sum |x_i - \bar{x}|}{n}\).

Common Mistakes

✕ Only comparing center
Saying "Group A scored higher" without mentioning spread. You need both center and spread for a complete comparison.
✕ No context
Writing "the mean is 7.7 and the IQR is 1" without saying what the data represents. Always tie numbers to the real-world situation.
✕ Confusing spread with center
Saying "Group B has a higher range so they did better." A higher spread means more variability, not better performance.

Summary Comparison Checklist

Step What to Do Example Phrase
1. Center Compare means or medians "Group A's median is higher..."
2. Spread Compare range, IQR, or MAD "Group B is more consistent..."
3. Context Use the real topic "...in terms of quiz scores"
4. Direction Say higher/lower, more/less "scored 2 points higher on average"
5. Values Include actual numbers "median of 18 vs 16"

Question 1 of 5

Team A has a mean score of 82 and an IQR of 6. Team B has a mean score of 78 and an IQR of 14. Which statement is the best comparison?

Question 2 of 5

If Distribution X has a range of 20 and Distribution Y has a range of 8, what can you conclude about spread?

Question 3 of 5

Which measure of center is most appropriate when a distribution is skewed or contains outliers?

Question 4 of 5

Store A's daily sales have a median of $450 and a MAD of $30. Store B's daily sales have a median of $450 and a MAD of $120. What is the best conclusion?

Question 5 of 5

A student writes: "The mean of Group A is 15 and the mean of Group B is 12." What is missing from this comparison?