Unit 1: Exploring One-Variable Data

Graphical Displays of Data

Constructing and comparing dotplots, histograms, and boxplots.

Advanced25 min lesson4 min readUpdated August 12, 2026Author not yet attributed

Seeing a Distribution's Shape Before Naming It

Two sections of the same exam produced these scores. Before reading on, just look: Section A — 62, 65, 68, 70, 72, 74, 75, 77, 79, 82, 85, 90. Section B — 55, 58, 60, 63, 65, 68, 70, 72, 75, 78, 88, 95. Can you tell, just from the list of numbers, which section's scores are more spread out, or which has a longer stretch on one side?

A list of numbers hides its own shape. Sorting them into a table of intervals — a histogram's underlying structure — makes the shape visible. Binning Section A's 12 scores into width-10 intervals: 60–69 has 3 scores (62, 65, 68), 70–79 has 6 (70, 72, 74, 75, 77, 79), 80–89 has 2 (82, 85), and 90–99 has 1 (90). The distribution is unimodal, mounded in the 70s, with a single higher score stretching it slightly to the right.

Definition — Distribution Shape Vocabulary

A distribution is symmetric if its left and right halves roughly mirror each other, skewed right if it has a longer tail stretching toward higher values (with most data bunched on the left), and skewed left if the longer tail stretches toward lower values. Unimodal means one clear peak; bimodal means two. An outlier is a value unusually far from the rest of the data — not just any value that looks a little unusual by eye.

Definition — Five-Number Summary and the Boxplot

The five-number summary — minimum, first quartile (Q1), median, third quartile (Q3), maximum — divides a sorted dataset into four roughly equal groups. IQR = Q3 − Q1 measures the spread of the middle 50% of the data. A value more than 1.5×IQR below Q1 or above Q3 is flagged as an outlier — a specific rule, not a visual judgment call.

Worked Example — Building a Five-Number Summary and Checking for Outliers

For Section A's 12 sorted scores, the median is the average of the 6th and 7th values: (74+75)/2 = 74.5. Q1 is the median of the lower six (62,65,68,70,72,74): (68+70)/2 = 69. Q3 is the median of the upper six (75,77,79,82,85,90): (79+82)/2 = 80.5. IQR = 80.5 − 69 = 11.5. Outlier fences: Q1 − 1.5(IQR) = 69 − 17.25 = 51.75, and Q3 + 1.5(IQR) = 80.5 + 17.25 = 97.75. Since 62 (the minimum) and 90 (the maximum) both fall inside those fences, Section A has no outliers by this rule.

Worked Example — Comparing Two Distributions by Their Five-Number Summaries

For Section B (55,58,60,63,65,68,70,72,75,78,88,95): median = (68+70)/2 = 69, Q1 = median of the lower six (55,58,60,63,65,68) = (60+63)/2 = 61.5, Q3 = median of the upper six (70,72,75,78,88,95) = (75+78)/2 = 76.5, IQR = 76.5 − 61.5 = 15. Fences: 61.5 − 22.5 = 39 and 76.5 + 22.5 = 99 — no formal outliers here either, since 95 < 99. But Section B's IQR (15) exceeds Section A's (11.5), and the gap between 78 and 88 is far larger than any other gap in either dataset — Section B's shape is stretched out toward higher values (right-skewed), even without a formal outlier. Skewness and outliers are related but genuinely separate ideas: a distribution can be clearly skewed with no single value extreme enough to flag.

Tip

Median and IQR describe center and spread using only the data's order, which is exactly why they stay reliable for a skewed distribution — mean and standard deviation, covered next, are pulled around by exactly the kind of stretched tail Section B has.

Common Mistakes

Reading 'right-skewed' as 'most of the data is on the right.'
Right-skewed means the tail stretches right — the bulk of the data actually sits on the left, with a smaller number of higher values pulling the distribution's shape out to the right, as Section B's data shows.
Calling any visually unusual value an outlier without applying the 1.5×IQR rule.
'Outlier' has a specific, computable definition — a value has to fall outside Q1 − 1.5(IQR) or Q3 + 1.5(IQR) to be flagged, not just look far from the pack.

Key Takeaways

A histogram's shape — symmetric, skewed left/right, unimodal/bimodal — reveals structure a raw list of numbers hides.
The five-number summary (min, Q1, median, Q3, max) and IQR describe center and spread using only the data's order.
Skewness and outliers are distinct ideas — a distribution can be clearly skewed without any single value meeting the formal 1.5×IQR outlier rule.

Summary

Shape, the five-number summary, and IQR describe a distribution without needing every individual value. The next lesson introduces mean and standard deviation — numerical summaries that use every value directly, and behave differently when a distribution is skewed.

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