Boxplots

A boxplot represents a batch of numbers using five landmark values, computed after sorting the data from smallest to largest. The median (Q2) splits the data into a lower and upper half. The first quartile (Q1) is the median of the lower half; the third quartile (Q3) is the median of the upper half. The interquartile range, IQR = Q3 − Q1, measures the spread of the middle 50% of observations and is the plot's core spread statistic, exactly as introduced in The Interquartile Range and Boxplots. The box itself is drawn from Q1 to Q3, with a line inside it marking the median. From each end of the box, a whisker extends outward — not to the true minimum and maximum, but to the most extreme data point that still lies within a fence set at 1.5 × IQR beyond the box edge. Any observation further out than that fence is plotted individually as a point (or asterisk), flagged as a candidate outlier. This convention, formalized by John Tukey in the 1970s, is what separates a boxplot from a simple "min-to-max" range bar: it builds in an automatic, distribution-free outlier detector. Reading a boxplot is reading shape without a curve. If the median line sits near the center of the box and whiskers are roughly equal, the batch is close to symmetric. If the median sits near one edge of the box, or one whisker is much longer than the other, the batch is skewed in the direction of the longer whisker/more distant edge. Multiple boxplots placed on a common axis let you compare medians (typical value), box heights (spread of the central mass), whisker lengths (spread of the extremes), and the number and position of outlier points — across as many groups as fit on the page.

Step 1 — Sort the data. All quartile calculations require the observations in ascending order. For n observations, this is the foundation for every later step.

Step 2 — Find the median (Q2). If n is odd, it is the middle value. If n is even, it is the average of the two middle values.

Step 3 — Find Q1 and Q3. Split the sorted data at the median into a lower half and upper half. (When n is odd, some conventions include the median in both halves — the "inclusive" method — others exclude it; be consistent and state which you used.) Q1 is the median of the lower half; Q3 is the median of the upper half.

Step 4 — Compute the IQR. IQR = Q3 − Q1. This is the height of the box.

Step 5 — Compute the fences. Lower fence = Q1 − 1.5 × IQR. Upper fence = Q3 + 1.5 × IQR. These fences are not drawn on the plot; they are working boundaries used only to decide whisker length and outlier status.

Step 6 — Draw whiskers. The lower whisker extends to the smallest data value that is still ≥ the lower fence. The upper whisker extends to the largest data value that is still ≤ the upper fence. Whiskers therefore always end on an actual data point, never on the fence itself.

Step 7 — Plot outliers. Any data point beyond either fence is drawn as an individual dot (sometimes with open circles for "mild" outliers beyond 1.5 × IQR and asterisks for "extreme" outliers beyond 3 × IQR — a refinement some software applies automatically). The constant "1.5" is a convention, not a law of nature: for data that are exactly normally distributed, it flags roughly 0.7% of observations per tail as outliers, a calibration Tukey chose because it worked well across many real datasets without being applied through any formal probability model — the boxplot's whole appeal is that it needs no distributional assumption to compute.

Boxplots were not "discovered" through a single crucial experiment the way, say, Meselson-Stahl confirmed DNA replication — they are a graphical convention, and their justification is methodological and statistical rather than a single decisive laboratory result. Still, the design was tested and refined empirically, and understanding that process is the "evidence" a biostatistics student needs. John W. Tukey introduced the boxplot in his 1977 book Exploratory Data Analysis, building on earlier "range bar" and "schematic plot" ideas used informally by statisticians for decades. Tukey's key innovation was the 1.5 × IQR fence rule for separating "ordinary" tail values from suspected outliers, chosen after examining how the rule behaved across many empirical and simulated datasets, including approximately normal distributions. The evidence for the 1.5 × IQR convention comes from probability calculations combined with simulation, not a wet-lab experiment: For data drawn from a true normal distribution, the theoretical quartiles are Q1 = μ − 0.6745σ and Q3 = μ + 0.6745σ, so IQR = 1.349σ. The fences sit at Q1 − 1.5×IQR ≈ μ − 2.698σ and Q3 + 1.5×IQR ≈ μ + 2.698σ, roughly ±2.7 standard deviations. Under the normal curve, only about 0.35% of observations fall beyond each fence — about 0.7% total, or roughly 1 observation in 143. That means for a perfectly normal sample of 200 points, you'd expect around 1–2 "outliers" flagged by pure chance — a manageable false-positive rate Tukey judged acceptable, since the goal of exploratory analysis is to flag candidates for scrutiny, not to make a formal hypothesis-test decision. Subsequent statisticians tested the rule's behavior on skewed and heavy-tailed distributions — exactly the shapes common in biological data (parasite counts, income, gene expression) — and found that the fixed 1.5 multiplier tends to over-flag outliers in right-skewed data, because the upper tail is genuinely longer than the lower tail even without any anomalous individual. This finding motivated adjusted variants such as the adjusted boxplot (Hubert & Vandervieren, 2008), which modifies the fence multiplier based on a robust skewness statistic (the medcouple) rather than using 1.5 uniformly. The existence and adoption of these adjustments is itself evidence: it confirms empirically that the classic rule's performance depends on distribution shape, and that biologists working with skewed data (very common in ecology and physiology) should interpret flagged "outliers" as candidates for inspection, not automatic exclusions. A separate, genuinely experimental line of evidence concerns how well people read boxplots. Perceptual studies in statistical graphics research (e.g., work descending from Cleveland & McGill's 1984 graphical-perception experiments) tested subjects' accuracy at judging quantities from different chart types. These controlled experiments — showing participants charts and measuring estimation error — established that people compare positions along a common scale (like quartile lines) far more accurately than they compare angles or areas, which is part of why the boxplot's design (aligned boxes and lines on a shared numeric axis) is graphically efficient for the comparative task biologists use it for: judging medians and spreads across many groups at a glance.

Example 1 — Parasite load in two lizard populations (asymmetric, skewed data). Ectoparasite mite counts on 11 lizards from Population X (disturbed habitat): 2, 3, 3, 4, 5, 6, 7, 8, 9, 22, 31 Sorted already. n = 11 (odd). Median (Q2): the 6th value = 6. Lower half (positions 1–5, excluding median): 2, 3, 3, 4, 5 → median of these = 3 = Q1. Upper half (positions 7–11): 7, 8, 9, 22, 31 → median of these = 9 = Q3. IQR = 9 − 3 = 6. Fences: Lower fence = 3 − 1.5(6) = 3 − 9 = −6 (no data below this, so lower whisker goes to the true minimum, 2). Upper fence = 9 + 1.5(6) = 9 + 9 = 18. Largest value ≤ 18 is 9 itself is inside the box... check the data beyond Q3: 22 and 31 both exceed the upper fence of 18. Upper whisker extends only to the largest value ≤ 18, which is 9 — wait, 9 is Q3 itself, so we need the largest upper-half value at or below 18 that is not itself an outlier. Scanning upper-half values (7, 8, 9, 22, 31): 7, 8, 9 are ≤ 18; 22 and 31 exceed it. So the upper whisker extends to 9, and 22 and 31 are flagged as outliers. This boxplot instantly shows: a tight, low cluster of mite counts (2–9) with two heavily infested outlier lizards (22, 31) — a pattern a mean and SD alone would badly obscure, since those two points would inflate the mean and SD without revealing that the bulk of the population is lightly infested. This is exactly the situation where a boxplot outperforms a bar-and-error-bar plot.

Example 2 — Comparing enzyme activity across three genotypes (reading, not building). Suppose a boxplot figure reports superoxide dismutase (SOD) activity (units/mg protein) in leaf tissue for wild-type, heterozygous, and homozygous knockout Arabidopsis lines, n = 15 plants per genotype:

Reading this table/plot:

• Median SOD activity drops roughly 73% from wild-type (52) to homozygous knockout (14), consistent with a dose-dependent gene effect.
• The heterozygote median (38) sits between the two homozygous classes but closer to the knockout, suggesting the wild-type allele is not fully dominant for this biochemical trait (incomplete dominance at the enzymatic level) — a pattern worth follow-up, though a boxplot alone cannot establish dominance or statistical significance; that requires a formal test such as ANOVA or a Kruskal-Wallis test.
• IQR shrinks from 20 (wild-type) to 12 (knockout), meaning the knockout plants are more uniformly low — plausible if residual SOD activity in knockouts comes from a single alternate pathway with little plant-to-plant variability, whereas wild-type activity may vary more due to additional regulatory inputs.
• The single outlier in the heterozygote group (65 units) is a candidate for follow-up: possibly a genotyping error, a contaminant, or genuine biological variation (e.g., an unusually active alternate SOD isoform) — the boxplot flags it, but does not explain it.

Boxplots appear throughout modern biology wherever many groups need honest, assumption-light comparison. In genomics, RNA-seq quality-control pipelines routinely display a boxplot of log-expression values per sample across dozens of samples in a single figure, letting analysts spot a failed library prep instantly by an outlier-shifted box before running any downstream statistics. In clinical trials, boxplots compare biomarker distributions (e.g., blood pressure, viral load, tumor size) across treatment arms, flagging patients with extreme responses for individual chart review — often the first step toward finding adverse-event patterns or non-responders. In conservation biology, boxplots of body condition indices, contaminant loads, or genetic diversity metrics across populations or protected areas support management decisions, since they communicate variability and outlier individuals to non-specialist stakeholders more effectively than a table of statistics. In agriculture and plant breeding, boxplots compare yield, disease resistance scores, or nutrient content across cultivar trials with many treatment levels, a direct descendant of the enzyme-activity comparison worked through above. And in quality control and diagnostics, clinical laboratories use boxplot-style control charts (closely related to Shewhart charts) to monitor assay performance and flag values outside expected ranges, connecting directly to the coming discussion of Outliers in Biological Data and to the broader ethical and design questions covered in Displaying Data Honestly — since a poorly chosen or manipulated axis, bin, or omitted outlier can turn an honest boxplot into a misleading one.

Ask the tutor: How do I decide whether to use a boxplot, a violin plot, or a strip plot for a given biological dataset?