Histograms and Box Plots

A histogram and a box plot are two ways of picturing how a set of numbers is spread out. Both work with one numerical variable (like test scores or heights), but each one shows the data in a different style. Once you can read both, you can compare two groups quickly and describe their shape, center, and spread.

Histograms

• A histogram groups numerical data into equal-width intervals called bins (for example 0-20, 20-40, 40-60).
• The height of each bar is the frequency — how many data values fall into that bin.
• Bars touch each other because the data is continuous, not separate categories.
• The tallest bar marks the most common interval; the shortest bar marks the rarest.

The Five-Number Summary

1. Min — the smallest value in the data set.
2. Q1 (lower quartile) — 25% of the values are at or below Q1.
3. Median (Q2) — the middle value; half the data is at or below it.
4. Q3 (upper quartile) — 75% of the values are at or below Q3.
5. Max — the largest value in the data set.

Box Plots

• A box plot (box-and-whisker plot) draws the five-number summary on a number line.
• The box stretches from Q1 to Q3 and a line inside it marks the median.
• Whiskers extend from the box out to the min on the left and the max on the right.
• The middle 50% of the data lives inside the box.

Range and IQR

• Range = max - min — how far the data stretches end to end.
• Interquartile range (IQR) = Q3 - Q1 — the width of the middle box.
• A small IQR means the middle of the data is tightly packed; a large IQR means it is spread out.

Worked Example: A Box Plot

1. Five-number summary: min = 12, Q1 = 20, median = 27, Q3 = 35, max = 48.
2. Range = 48 - 12 = 36.
3. IQR = 35 - 20 = 15.
4. Half of the values are at or below 27 (the median); a quarter are between 27 and 35.