Lesson 3: Make Comparative Inferences About Populations

Property

In a box plot, the length of whiskers and the height of the box indicate data variability: longer whiskers show greater spread in the extreme values, while a taller box indicates greater spread in the middle $50\%$ of the data (interquartile range).

Examples

Property

To draw valid statistical conclusions when comparing two data sets, you must systematically analyze and state the differences in BOTH the Center and the Spread, and support your claims with specific numerical evidence.

1. Compare Centers: Use the Median (or Mean) to determine which group generally has a higher or lower typical value.
2. Compare Spreads: Use the IQR (or Standard Deviation/MAD) to determine which group is more consistent/reliable (smaller spread) or more variable/unpredictable (larger spread).

Examples

• Comparing Test Scores: "Class A performed better on average with a higher median score (85 vs 78). However, Class A also has a larger IQR (15 vs 8), indicating that Class B was much more consistent in their performance."
• Comparing Reaction Times: "Group 1 shows a faster average reaction time (0.45 seconds vs 0.52 seconds) and a lower Standard Deviation (0.08 vs 0.15). This suggests Group 1 is not only faster, but their performance is far more reliable."
• Evaluating Overlap: "Store X has the exact same median sales as Store Y ($2,500), but Store X's massive Range suggests its daily performance is highly unpredictable compared to Store Y."

Explanation

A statistical conclusion is not a guessing game; it is a structured argument. If you only compare the centers, you are only telling half the story! A student who scores 50 and 100 has the exact same average (75) as a student who scores 75 and 75, but they are completely different types of students. Always pair your center statistic with your spread statistic. Use words like "typical" or "higher average" for the center, and words like "consistent", "reliable", or "variable" to describe the spread.