Using Statistics to Compare Two Populations

Lesson 1 of 1

In this lesson:

• Choose the right measure of center and variability
• Write informal comparative inference statements
• Connect sample size to inference reliability

Learning Objectives for This Lesson

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

1. Choose and justify an appropriate measure of center
2. Choose the matching measure of variability
3. Compute mean, median, MAD, and IQR
4. Write a complete informal comparative inference
5. Explain why larger samples improve reliability

What Questions Should We Ask First?

Two classes tracked daily study minutes for one week:

• Class A: 20, 25, 25, 30, 30, 30, 35, 40, 45, 50
• Class B: 15, 20, 20, 25, 30, 35, 40, 45, 50, 80

Before computing anything, what should you ask?

Mean vs. Median: Two Measures of Center

• Mean: balance point — sum ÷ count; sensitive to extreme values
• Median: middle value — resistant to outliers
• When data is skewed or has outliers: median gives a fairer picture
• When data is roughly symmetric, no outliers: mean is appropriate

Outliers Pull the Mean, Not the Median

Data Set B is identical to A except for the value 27

Decision Rule: Choosing Your Measure Pair

Both pairs describe center and spread — choose whichever gives the most honest picture of the distribution.

IQR vs. MAD: Two Measures of Variability

• IQR = spread of the middle 50% of values; based on position
• MAD = average distance of all values from the mean; based on arithmetic
• IQR pairs with median — both position-based, both outlier-resistant
• MAD pairs with mean — both arithmetic, both use every value

Check-In: Choose the Right Measure Pair

For each data set, which measure pair would you use — Mean + MAD or Median + IQR?

1. Test scores: 72, 75, 78, 80, 81, 82, 85, 88, 90, 91
2. Home prices: $180K, $190K, $195K, $200K, $210K, $850K

Think before the next slide

Check-In: Choosing the Right Measure Pair

1. Test scores → Mean + MAD ✓ — roughly symmetric, no outliers
2. Home prices → Median + IQR ✓ — one extreme outlier ($850K)

The $850K value would pull the mean far above most prices — median gives a fairer picture

Introducing the Word Length Comparison

We'll compare word lengths in two science chapters:

• 4th-grade: 3, 4, 4, 5, 5, 5, 6, 6, 7, 8 letters
• 7th-grade: 4, 5, 6, 6, 7, 7, 8, 9, 10, 13 letters

Goal: Compute all four statistics, then draw an inference.

Median and IQR for 4th-Grade Words

Ordered: 3, 4, 4, 5, 5, 5, 6, 6, 7, 8

• Median = letters
• Lower half → Q1 = 4; Upper half → Q3 = 6
• IQR = letters

Median and IQR for 7th-Grade Words

Ordered: 4, 5, 6, 6, 7, 7, 8, 9, 10, 13

• Median = letters
• Lower half → Q1 = 6; Upper half → Q3 = 9
• IQR = letters

Mean and MAD for 4th-Grade Words

Mean = letters

Absolute deviations from 5.3:
2.3, 1.3, 1.3, 0.3, 0.3, 0.3, 0.7, 0.7, 1.7, 2.7

Sum = 11.6 → MAD = letters

Mean and MAD for 7th-Grade Words

Mean = letters

Sum = 20.0 → MAD = letters

Comparison Table: All Four Statistics

7th-grade is consistently higher — longer words and more variable.

Interpreting the Comparison Table Statistics

• Median difference: 7.0 − 5.0 = 2 letters
• 4th-grade IQR = 2; 7th-grade IQR = 3
• A 2-letter gap equals one full 4th-grade IQR — a meaningful difference

What inference can we draw about these two science texts?

Five Components of a Complete Inference

A complete informal inference statement must include:

1. Name both populations being compared
2. State the direction of the difference
3. Give the actual center values and the difference
4. Reference variability to contextualize the difference
5. Use hedging language — "tends to be," "on average," "based on this sample"

Inference Steps 1 and 2: Populations and Choice

Step 1 — Name the populations:
"We are comparing word lengths in a typical 7th-grade science chapter and a typical 4th-grade science chapter."

Step 2 — Choose measures and justify:
The 7th-grade data is skewed right (value of 13), so we use median + IQR.

Inference Steps 3 and 4: Numbers Matter

Step 3 — State the difference:
"Median: 7.0 letters (7th grade) vs. 5.0 letters (4th grade) — a gap of 2 letters."

Step 4 — Reference variability:
"4th-grade IQR = 2 letters, so a 2-letter gap equals one full IQR — a meaningful difference."

Inference Step 5: Hedging and Full Statement

Step 5 — State the inference with hedging:
"Based on these samples, words in 7th-grade science books tend to be longer than words in 4th-grade science books. This pattern likely holds more broadly, though a larger sample would give us greater confidence."

Complete inference: steps 1 through 5 combined.

Calibrating Confidence in Inference Statements

Language	Verdict
"7th-grade always uses longer words."	Over-confident
"We can't say — only 10 words."	Under-confident
"7th-grade tends to use longer words."	✓ Appropriate

Hedging is accuracy, not weakness.

Your Turn: Comparing Daily Screen Time

Group A (jobs): 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 hrs
Group B (no jobs): 3, 4, 4, 5, 5, 5, 6, 7, 7, 9 hrs

1. Compute median + IQR for each group
2. Choose the measure pair; justify
3. Write a complete five-component inference

Screen Time Statistics: Answers Revealed

Symmetric distributions — either measure pair valid.

Check-In: Evaluate This Inference Statement

"Students with after-school jobs use screens less. Group A's mean is 3.5 and Group B's is 5.5. The data proves this is always true."

What is missing or wrong? Use the five-component checklist.

Identify problems before advancing

Rate This Inference: Problems Found

1. ✓ Populations partially named (Group B missing)
2. ✓ Direction stated
3. ✓ Values given
4. No variability — missing component 4
5. Over-confident — "proves," "always true"

Fix: add IQR comparison; replace "proves" with "tends to"

How Reliable Is a 10-Word Inference?

What if we sampled only 3 words from each chapter?

• 7th-grade: 3, 5, 12 → Median 5, Mean 6.7
• 4th-grade: 4, 4, 11 → Median 4, Mean 6.3

One long word equalized the means — small samples are noisy.

Larger Samples Produce More Stable Estimates

As increases, sample medians cluster more tightly — estimates become more reliable.

Three Phrases: Shows, Suggests, Proves

"Informal" means no probability calculations, not careless reasoning.

Check-In: Which Language Fits Here?

Choose: "shows," "suggests," or "proves"

1. You sampled 4 words; 7th-grade median was higher.
2. You sampled 200 words from 50 chapters; results consistent.
3. You are reporting the mean of your 10 measured words.

Decide before advancing

Which Language Fits: Answers Revealed

1. 4 words sampled → "suggests" (tentative; very small sample)
2. 200 words, 50 chapters → "suggests" (stronger, but still a sample)
3. Reporting measured mean → "shows" (describing your data, not inferring)

"Proves" requires every word from every book — that's a census, not a sample.

Key Takeaways from Today's Lesson

• ✓ Choose measures first: symmetric → mean + MAD; skewed → median + IQR
• ✓ Complete inferences need: populations, direction, values, variability, hedging
• ✓ Gap compared to spread shows whether difference is meaningful
• ✓ Larger samples are more reliable — always hedge

Watch Out: Six Common Inference Errors

• No variability — inference without spread is incomplete
• Sample = population — "proves" overstates a sample
• Default to mean — check shape first
• High IQR = worse — variability describes the data
• Too certain — "always" and "proves" rarely fit
• Unfair comparison — use random comparable samples

Next: Applying Inference to Real Populations

In the next lesson, we'll apply informal comparative inference to real-world population data — income distributions, age distributions across countries, and survey results.

Coming up:

• Larger data sets with more complex distributions
• Choosing measures when data has multiple features to consider
• Connecting statistical inference to decision-making