Three Ways to Picture the Same Data
Fifteen students report how many books they read over the summer: 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 7, 8, 12. Before reading on: staring at that list, can you quickly tell what a 'typical' student read, or does the shape of the data need to be pictured first?
Definition — Dotplot
A dotplot places every data value on a number line, stacking a dot for each repeated value. It shows every individual value at once, making it a good choice for smaller datasets.
Worked Example — Building a Dotplot
For the books-read data, count how many times each value appears: 2 (once), 3 (twice), 4 (three times), 5 (four times), 6 (twice), 7 (once), 8 (once), 12 (once). Stacking dots at each value on a number line from 2 to 12 shows a cluster mounded around 4–5, tapering off toward both ends — with the value 12 sitting noticeably apart from the rest, worth a closer look.
Definition — Histogram
A histogram groups data into equal-width intervals (bins) and shows the count (or frequency) in each — trading the dotplot's individual-value detail for a clearer view of overall shape, especially useful for larger datasets.
Worked Example — Building a Histogram
Grouping the same 15 values into width-3 bins: 1–3 has 3 values (2, 3, 3), 4–6 has 9 values (4, 4, 4, 5, 5, 5, 5, 6, 6), 7–9 has 2 values (7, 8), and 10–12 has 1 value (12). The histogram shows the same basic story as the dotplot — a mound in the middle bin, thinning out toward both ends — just at a coarser resolution.
Definition — Boxplot and the Five-Number Summary
A boxplot displays the five-number summary — minimum, first quartile (Q1), median, third quartile (Q3), maximum — as a box (spanning Q1 to Q3) with whiskers extending to the most extreme values that aren't flagged as outliers. A value is an outlier if it falls more than 1.5 × IQR (where IQR = Q3 − Q1) below Q1 or above Q3.
Worked Example — Building a Boxplot and Checking for Outliers
For the sorted 15 values, the median (the 8th value) is 5. The lower half (the first 7 values: 2,3,3,4,4,4,5) has median (Q1) 4. The upper half (the last 7 values: 5,5,6,6,7,8,12) has median (Q3) 6. IQR = 6 − 4 = 2. Outlier fences: Q1 − 1.5(IQR) = 4 − 3 = 1, and Q3 + 1.5(IQR) = 6 + 3 = 9. Since 12 > 9, the value 12 is a genuine outlier by this rule — confirming what the dotplot's isolated dot already hinted at.
Tip
A dotplot and a histogram of the same data should always tell a consistent shape story — if they seem to disagree, double-check the histogram's bin width, since bins that are too wide or too narrow can distort how a shape reads.
Common Mistakes
✗ Treating a histogram's bar heights as individual data points, the way a dotplot's dots are.
✓ A histogram bar's height is a count of how many values fall in that entire interval — the individual values inside a bar are no longer distinguishable, unlike in a dotplot.
✗ Calling any far-looking 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), not just look unusual by eye.
Key Takeaways
- Dotplots show every individual value; histograms trade that detail for a clearer view of overall shape, especially with larger datasets.
- A boxplot's five-number summary (min, Q1, median, Q3, max) and IQR summarize a distribution using only its order.
- An outlier is defined by the 1.5×IQR rule, not by visual judgment alone.
Summary
These three displays reveal a distribution's shape from different angles. The next lesson turns that visual shape into precise, contextual language.