Graphical Summaries

Graphical summaries are visual tools that compress large datasets into clear, interpretable pictures. Rather than staring at hundreds of raw numbers, statisticians use graphs to reveal the shape, center, spread, and outliers of a distribution at a glance. The three most powerful graphical summaries at the high school level are the boxplot, the histogram, and the density plot — each highlighting different features of the same underlying data. A distribution describes how data values are spread across a range. It can be symmetric (like a bell curve), skewed left or right, unimodal (one peak), bimodal (two peaks), or uniform. Recognizing distributional shape is essential: it tells you which summary statistics are appropriate, whether the mean or median better represents the center, and whether standard deviation or IQR better captures spread. Mastering graphical summaries means being able to construct, read, and compare these displays critically. In this article you will learn to build a boxplot from scratch using the five-number summary, interpret density plots as smoothed histograms, and compare distributions side-by-side — skills that appear on standardized exams and underpin virtually every real-world data analysis.

Key takeaways

The Five-Number Summary
Constructing a Boxplot
Reading Skewness from a Boxplot
Histograms and Bin Width
Density Plots
Comparing Distributions

The Five-Number Summary

Every boxplot is built on five key statistics: the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum. Q1 is the median of the lower half of the data and Q3 is the median of the upper half. Together they define the interquartile range: IQR = Q3 − Q1, which contains the middle 50% of all observations.

Constructing a Boxplot

Draw a number line spanning the data range. Draw a rectangular box from Q1 to Q3 with a vertical line at the median. Then draw 'whiskers' extending from each side of the box to the farthest data point that is NOT an outlier. A point is flagged as an outlier if it falls below Q1 − 1.5·IQR or above Q3 + 1.5·IQR; such points are plotted individually as dots beyond the whiskers.

Reading Skewness from a Boxplot

If the median line sits closer to Q1 and the right whisker is longer, the distribution is right-skewed (positive skew) — a few very large values pull the tail right. If the median is closer to Q3 and the left whisker is longer, it is left-skewed. A centered median with equal whisker lengths signals symmetry. This asymmetry in a boxplot directly tells you whether the mean will be pulled above or below the median.

Histograms and Bin Width

A histogram divides the data range into equal-width bins and plots the frequency or relative frequency of observations falling in each bin. Bin width dramatically affects interpretation: too narrow and the histogram looks jagged and noisy; too wide and important shape features are hidden. A practical rule of thumb for bin count k is Sturges' formula: k = 1 + 3.322·log₁₀(n), where n is the sample size.

Density Plots

A density plot is a smoothed version of a histogram. It applies kernel density estimation (KDE) — placing a small smooth bump (kernel) over each data point and summing all bumps — to produce a continuous curve. The area under a density curve always equals 1, making it interpretable as a probability. The smoothness is controlled by a bandwidth parameter h: larger h gives a smoother curve, smaller h reveals more local bumps. Density plots are especially useful for comparing multiple groups without the arbitrary choice of bin width.

Comparing Distributions

Side-by-side boxplots are the standard tool for comparing several groups simultaneously: draw parallel boxplots on the same axis and compare medians, IQRs, and outlier patterns. Overlapping density plots (using transparency) let you compare shapes. When comparing, always comment on center (median or mean), spread (IQR or standard deviation), shape (skew, modality), and unusual features (outliers, gaps).

The Key Equation

\text{Outlier boundary} = \begin{cases} \text{Lower: } Q_1 - 1.5 \times \text{IQR} \\ \text{Upper: } Q_3 + 1.5 \times \text{IQR} \end{cases}

Q_1 — First Quartile (same as data): The median of the lower half of the dataset; 25% of data lie below this value.
Q_3 — Third Quartile (same as data): The median of the upper half of the dataset; 75% of data lie below this value.
\text{IQR} — Interquartile Range (same as data): IQR = Q3 − Q1; the range of the middle 50% of the data, a robust measure of spread.
1.5 — Tukey's multiplier (dimensionless): The conventional constant chosen by John Tukey; values beyond 3·IQR are sometimes called 'far outliers'.

Worked Example

Problem. The daily high temperatures (°F) for 11 consecutive days are: 58, 61, 63, 65, 66, 68, 70, 72, 75, 80, 95. Construct the five-number summary, calculate the IQR, identify any outliers using Tukey's fence, and describe what a boxplot would look like.

\text{Min}=58,\ Q_1=63,\ Q_2=68,\ Q_3=75,\ \text{Max}=95;\ \text{IQR}=12;\ \text{Outlier: }95

Answer. Five-number summary: Min = 58, Q1 = 63, Median = 68, Q3 = 75, Max = 95. IQR = 12°F. One outlier: 95°F. The distribution is mildly right-skewed.

Figure

Diagram

Origins · 1 minJohn Wilder Tukey and how this was discovered

John Wilder Tukey

1915–2000 · American

John Tukey invented the boxplot (which he called a 'box-and-whisker plot') in his landmark 1977 book Exploratory Data Analysis. He also introduced the concept of the stem-and-leaf plot and coined the word 'bit' (binary digit) in computing. His IQR-based outlier fences, still used today, provide a simple and robust way to flag unusual observations without assuming any particular distribution. Tukey also coined the word 'software' in 1958 — one of the earliest recorded uses of the term in print — while working at Bell Labs and Princeton University.

Measures of Center: Mean and Median — Boxplots display the median as their central reference point; understanding why the median is preferred over the mean for skewed distributions is essential for interpreting boxplot shape.
Measures of Spread: IQR and Standard Deviation — The IQR is the backbone of the boxplot construction and outlier detection; comparing it with standard deviation deepens understanding of robust vs. non-robust statistics.
The Normal Distribution — Density plots often reveal whether data approximates a normal (bell-curve) shape, which determines whether parametric statistical tests are appropriate.
Histograms and Frequency Tables — Histograms are the discrete precursor to density plots; understanding bin width and frequency helps students bridge from raw counts to smooth distributional curves.