DAT Quantitative Reasoning: Reading Box Plots and the Five-Number Summary

DAT High-Yield Concepts

Five-Number Summary: The key structural landmarks on a box plot are Minimum, First Quartile ( $Q_1$ ), Median (Second Quartile, $Q_2$ ), Third Quartile ( $Q_3$ ), and Maximum.
The 25% Rule: A box plot is partitioned into four equal quarters of data values. Each quarter (lower whisker, lower box segment, upper box segment, upper whisker) holds approximately $25\%$ of the total observations.
Interquartile Range (IQR): Measured directly off the plot as the width of the box from $Q_1$ to $Q_3$ ( $IQR = Q_3 - Q_1$ ).
Comparing Box Plots: Compare center using the median ( $Q_2$ ) and spread using the IQR or overall range ( $\text{Maximum} - \text{Minimum}$ ). Sample size ( $n$ ) cannot be determined from box plot dimensions alone.

1. Anatomy of a Box Plot and the Five-Number Summary

A box plot (or box-and-whisker plot) visually summarizes a univariate dataset using five key statistics known as the Five-Number Summary:

Minimum: The smallest value in the dataset (endpoint of the lower whisker).
First Quartile ( $Q_1$ ): The 25th percentile; marks the left edge of the box.
Median ( $Q_2$ ): The 50th percentile; indicated by a vertical line inside the box.
Third Quartile ( $Q_3$ ): The 75th percentile; marks the right edge of the box.
Maximum: The largest value in the dataset (endpoint of the upper whisker).

The five marks on a box plot, and the four quarters between them — A box plot carries exactly five marks: the minimum, the first quartile, the median, the third quartile and the maximum. The box spans the first to the third quartile, so its width is the interquartile range, here 34 minus 20, or 14. The four sections each hold about 25 percent of the values, so a wide section means the data there is spread out rather than crowded. Comparing two plots means comparing center and spread separately: clinic A and clinic B share a median of 26, but A's interquartile range is 14 against B's 6, so B is far more consistent. A box plot shows no individual values and no sample size.

The central box spans the middle $50\%$ of the data between $Q_1$ and $Q_3$ , while the whiskers extend outward to the extremes (Minimum and Maximum).

A simple example of box-and-whisker plot. (Source: ARAKI Satoru · CC BY-SA 4.0 · Wikimedia Commons)

2. The Four Quarters and the 25% Data Rule

Because quartiles divide an ordered dataset into four equal parts, each of the four sections of a box plot contains approximately $25\%$ of the observations:

Minimum to $Q_1$ : Lowest $25\%$ of data.
$Q_1$ to $Q_2$ (Median): Lower-middle $25\%$ of data.
$Q_2$ (Median) to $Q_3$ : Upper-middle $25\%$ of data.
$Q_3$ to Maximum: Highest $25\%$ of data.

Key Insight: Segment Length vs. Data Count
A longer whisker or wider box section does not mean that segment contains more data points! It means the data values in that interval are more spread out (variable). Each segment always represents

\approx 25\%

of the total dataset count.

Worked Example 1: Percentile & Range Interpretation

Problem: A box plot of patient ages in a dental study shows Minimum = 12, $Q_1 = 20$ , Median = 28, $Q_3 = 35$ , and Maximum = 48. What percentage of patients are between ages 20 and 48?

Step-by-step Solution:

Identify the percentiles corresponding to age 20 and age 48: Age 20 is $Q_1$ (25th percentile). Age 48 is the Maximum (100th percentile).
Calculate the percentage between these boundaries: $100\% - 25\% = 75\%$

Answer: $75\%$ of patients are between 20 and 48 years old.

Common Exam Trap: mistaking segment length for frequency. Do not assume the section from 35 to 48 contains more patients than the section from 20 to 28 simply because 13 is wider than 8.

3. Interquartile Range (IQR) and Spread

The Interquartile Range (IQR) measures variability by determining the range of the middle $50\%$ of the data. On a box plot, the IQR is read directly as the horizontal length of the box:

IQR = Q_3 - Q_1

The total spread is given by the Range:

\text{Range} = \text{Maximum} - \text{Minimum}

4. Comparing Two Box Plots for Center and Spread

DAT questions frequently display two parallel box plots on a shared numerical axis and ask you to evaluate center and dispersion.

Feature	Statistic Used	How to Interpret on Plot
Center	Median ( $Q_2$ )	Position of the line inside the box along the number line. Higher line = larger typical value.
Spread (Variability)	IQR & Overall Range	Width of box (IQR) and total distance from min whisker to max whisker (Range). Wider box/whiskers = higher variability.
Sample Size ( $n$ )	Not visible	Box plots show relative percentages (25% chunks), never absolute sample sizes.

Worked Example 2: Comparing Dental Procedure Times

Problem: Two clinics record procedure duration (in minutes):

Clinic A: Min = 10 min, $Q_1 = 18\text{ min}$ , Median = 25 min, $Q_3 = 32\text{ min}$ , Max = 40 min.
Clinic B: Min = 15 min, $Q_1 = 22\text{ min}$ , Median = 30 min, $Q_3 = 42\text{ min}$ , Max = 55 min.

Compare their center and spread, and determine which clinic has a larger IQR.

Step-by-step Solution:

Compare Centers: Median A = 25 min, Median B = 30 min. Clinic B has a higher central procedure time.
Calculate IQR for both: $IQR_A = Q_3 - Q_1 = 32 - 18 = 14\text{ min}$ $IQR_B = Q_3 - Q_1 = 42 - 22 = 20\text{ min}$
Compare Spreads: Clinic B has a larger IQR (20 min vs. 14 min) and larger overall range ( $55 - 15 = 40\text{ min}$ vs. $40 - 10 = 30\text{ min}$ ), indicating greater variability in duration.

Answer: Clinic B has both a higher median procedure time and a larger interquartile range (20 min vs. 14 min).

Mnemonic for Five-Number Summary:
M-Q-M-Q-M — "Many Quick Math Questions Matter"
Minimum, Q1, Median, Q3, Maximum.

Critical Thinking Questions

If Box Plot A has a box twice as wide as Box Plot B, does Box Plot A necessarily contain twice as many data points? Explain.
A student states that $50\%$ of a dataset falls within the interquartile range (IQR). Is this statement always correct? Why?
Suppose two datasets have identical medians, but Dataset 1 has an IQR of 10 while Dataset 2 has an IQR of 25. Which dataset exhibits greater consistency around its center?

References

Sources returned by Google Search grounding during lesson verification.

Box Plots and the 5-Number Summary - Statistics — youtube.com
Comparing Two Box Plots - Dummies.com — dummies.com

Reading Box Plots and the Five-Number Summary

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