Matching Statistics and Comparing Data

Lesson 2 of 2: Compare Center and Spread

In this lesson:

• Match the statistic pair to a distribution's shape
• Compare groups fairly using center and spread

Learning Objectives for This Unit

By the end of this unit, you should be able to:

1. Compute the mean, median, IQR, and SD
2. Judge shape and select fitting statistics
3. Explain why median and IQR resist outliers
4. Compare data sets with matched statistics
5. Write comparisons of center and spread

Which Average Tells the Truth?

A news report says the average home costs $1.2 million.

• But every listing you find is near $400,000

Both can be true — a few mansions pull the mean far above a typical home.

Which number is honest?

Resistance Tells You the Answer

From Lesson 1: the mean is dragged by extremes; the median holds.

• A few mansions are those extreme values
• They pull the mean to $1.2M, but the median stays near $400K

The median describes a typical home — the shape decided the statistic.

The Shape Decides the Statistic

• Symmetric, no outliers → mean and SD
• Skewed or outliers → median and IQR

Symmetric Data Earns the Mean

A histogram of test scores is roughly symmetric.

• Single peak, both sides taper evenly, no extreme values

The shape earns the mean and SD:

Mean , SD — a typical score is 75, give or take 8.

Skewed Data Earns the Median

• A long right tail of high earners pulls the mean up
• The median and IQR honestly describe a typical household

Outliers Also Earn the Median

A box plot of marathon times has one very slow finisher flagged as an outlier.

• That extreme value would inflate the mean and the SD

Report the median and IQR — the resistant pair:

Skew or an outlier → median and IQR, every time.

Why the Mean Sits Above the Median

In a right-skewed distribution, the long tail pulls the mean up.

• Mean reflects the few high earners
• Median reflects the many typical households

The bigger the skew, the larger the gap — and the more the mean misleads.

Quick Check: Name the Pair

For each shape, name the statistic pair:

1. Symmetric, bell-like histogram of heights
2. Strongly right-skewed home prices
3. Data with one flagged outlier

Decide all three, then advance.

Answers: 1) mean and SD (symmetric); 2) median and IQR (skewed); 3) median and IQR (outlier).

Comparing Two Groups Needs One Rule

To compare one distribution you choose a pair. To compare two, add one rule:

• Both groups must use the same pair
• Comparing one group's mean to another's median is meaningless

Read each shape, pick a pair that fits both, then compare center and spread.

Compare Two Quiz Classes Fairly

Both symmetric, so use mean and SD:

• Class A: mean 82, SD 6
• Class B: mean 76, SD 11

A Comparison Is Two Sentences

For Class A versus Class B:

• Center: Class A's mean is higher — 82 vs 76
• Spread: Class B is more variable — SD 11 vs 6

Both findings matter: Class A is uniformly solid; Class B has high and low extremes.

Predict: Same Center, Different Spread

Two groups have the same median of 50.

• Group 1: small IQR — scores packed near 50
• Group 2: large IQR — scores spread widely

Are these two groups the same?

Commit to an answer, then advance.

A Complete Comparison Needs Both

The groups are not the same — only the spread distinguishes them.

• Same center, but one is reliable and one is volatile
• Comparing center alone would call them equal — and be wrong

A complete comparison always states center and spread.

When One Set Is Skewed

Comparing a skewed set with a symmetric one? The same-pair rule holds.

• The skewed set demands median and IQR
• So both sets use median and IQR

If either set is skewed, the resistant pair wins.

The Method Scales to Three Groups

• Compare centers by lining up the medians
• Compare spreads by box widths and whiskers

Two groups or ten — same method.

Your Turn: Compare Two Teams

Both roughly symmetric, both mean 80:

• Team X: mean 80, SD 4
• Team Y: mean 80, SD 12

Write a center sentence and a spread sentence, then advance.

Answer: Centers equal (both 80). Team Y is far more variable (SD 12 vs 4); Team X is more consistent.

Full Task: Judge, Choose, Compare

Given two real data sets:

1. Judge each shape
2. Choose one matched pair for both
3. Compute it for each set
4. Write a two-sentence comparison

Do the whole task unaided, then advance.

Answer: Skewed → median/IQR for both; symmetric → mean/SD. Two sentences, in context.

Key Takeaways From This Lesson

✓ Shape first: symmetric → mean/SD; skewed → median/IQR

✓ Compare groups with the same matched pair

✓ Report center and spread

A skewed set forces median/IQR for all groups

Naming only the center is incomplete

Next: what these differences mean, and handling outliers.