▭ · Box Plots
Five numbers hold a distribution on one line — the box plot.
Two classes both average 70 — are their score distributions the same?
Compare data with medians and quartiles.
The chart invented by the man who coined the word 'software'.
Open the immersive version →A reading path containing this lesson: What a collection of numbers tells us · 3/3
Why does this question matter?
A histogram shows the full face of a distribution but is heavy for side-by-side comparison. Comparing ten classes on one screen needed a lighter summary.
The solution: quarter the data. Sort it, mark the middle (median), the middle of the lower half (Q1), and of the upper half (Q3) — with min and max, five numbers form the skeleton.
Draw those five as a box with whiskers and you get the box plot: position (median), spread (box length), skew (where the median sits in the box), and extremes — groups compared in one strip.
Pause and try it yourself
Make a prediction first. Change one input at a time and compare what happens with what you expected.
Predict first — two classes both averaging 70. Are their score distributions the same?
Find the five numbers from Class A's scores
Press the buttons in order and watch the box plot assemble.
Read the idea from another angle
A box plot is a distribution's X-ray.
The flesh (individual values) is invisible but the skeleton (five numbers) is sharp. For many groups, boxes side by side replace stacks of histograms — comparison finishes at a glance.
Quartiles are the language of rank.
Q1 bounds the bottom 25%, Q3 the top 25%. If your score sits above the box's middle, you're in the upper half — rank sense and distribution sense meet at the quartiles.
Now read the definitions with their conditions
What the two-class comparison confirmed, in mathematical language.
The five-number summary
Five points that quarter the data — 25% of it lives in each stretch.
Interquartile range
The width of the middle 50% — the box's length. A spread ruler that extremes can't shake.
Flagging outliers (convention)
Points beyond the whiskers plot separately as outliers — the net that catches data errors and oddities.
Make one more distinction
A longer box means more students in that range.
Nearly the opposite. The box always holds the middle 50% — a longer box means that same 50% is spread wider. Box length is spread (IQR), never headcount.
Equal means make similar groups.
One class averaging 70 may sit tightly in 60–82 while another scatters across 40–100. The mean gives one position — only spread reveals the group, and the box plot shows it in five numbers.
Where else can this idea help?
Reading salary distributions
'Average salary $50k' may be dragged up by a few high earners. The box plot's median and upper whisker expose the felt distribution — the picture that catches the mean's trap.
Comparing classes and schools
Line up ten classes' boxes and you instantly see who performs evenly and where the gaps yawn — the standard tool of education statistics.
Sports analytics
A player's per-game box plot shows average level and volatility at once. The short-boxed player is the consistent one.
Quality control
One dot beyond the whiskers in a factory's measurements flags a bad lot — the 1.5×IQR rule is industry's anomaly detector.
Explain it in your own words
Try another question
Q1Five-number summary of 3, 5, 7, 9, 11?Show answer ▾
min 3, Q1 4 (between 3 and 5), median 7, Q3 10 (between 9 and 11), max 11. Mark the median first, then the middles of each half.
Q2The median line sits left of the box's center — what does that say?Show answer ▾
Within the middle 50%, the lower stretch (Q1–median) is tight and the upper (median–Q3) wide — a right-tailed (positively skewed) distribution. Salaries are the classic case.
Q3With Q1=64 and Q3=76, is a score of 95 an outlier?Show answer ▾
IQR = 12; upper fence = 76 + 1.5×12 = 94. Since 95 > 94, convention flags it — a signal to check whether it's brilliance or a grading error.
Try answering yourself before revealing it — getting it wrong is where learning starts.
What question comes next?
A reading path containing this lesson: What a collection of numbers tells us · 3/3
Previous concept
Frequency & Histograms
See the shape first, and the five-number summary earns its keep.
← Frequency & Histograms labLeads to next
Variance
You measured spread by box length; now measure it precisely by every point's distance — variance.
Go to the Variance lab →Related lab
Frequency & Histograms
The box plot compresses the distribution's face into a five-number skeleton.
Go to the Frequency & Histograms lab →Related lab
Variance
Another spread ruler — IQR and standard deviation are two answers to one question.
Go to the Variance lab →