Statistics Reference

How to Make a Box Plot

A box plot compresses a whole dataset into five numbers and a fence rule — and the fence rule is where most hand-drawn plots go wrong. This guide builds one from raw data step by step, using the same median-split quartiles and Tukey whiskers as the site's box plot generator, then shows how to read the finished picture.

The Build, Step by Step

Data: eight support-ticket resolution times (hours): 4, 2, 5, 12, 4, 7, 5, 4.

  1. Sort: 2, 4, 4, 4, 5, 5, 7, 12.
  2. Median (Q2): middle pair (4, 5) → 4.5.
  3. Quartiles by median-split: lower half {2, 4, 4, 4} → Q1 = 4; upper half {5, 5, 7, 12} → Q3 = 6.
  4. IQR: 6 − 4 = 2.
  5. Tukey fences: Q1 − 1.5×IQR = 1 and Q3 + 1.5×IQR = 9.
  6. Flag outliers: 12 > 9 → 12 is an outlier, drawn as a dot. Everything else lies within the fences.
  7. Whisker ends — the step most people get wrong: whiskers run to the most extreme values inside the fences, not to the fences and not to the min/max. Lower whisker → 2; upper whisker → 7 (the largest non-outlier), not 9 and not 12.
  8. Draw: box from 4 to 6, median line at 4.5, whiskers to 2 and 7, a dot at 12.

Paste the eight values into the box plot generator to see exactly this figure (downloadable as SVG); the quartile calculator and outlier calculator confirm steps 3–6 individually.

Reading a Box Plot

  • Center: the median line — half the tickets resolved within 4.5 hours.
  • Spread: the box holds the middle 50% of values (here a tight 4–6 band); total whisker span shows the non-outlier range.
  • Skew: compare the half-boxes and whiskers. Here the median hugs Q1 and the upper whisker is longer — right skew, confirmed numerically by the skewness calculator.
  • Outliers: each dot is a specific observation worth investigating — a data-entry error, a special cause, or a genuinely long case. Flagged is not the same as wrong.

Common Mistakes

  • Whiskers drawn to the fences (1 and 9). Fences are invisible thresholds, not data; whiskers end at real observations (2 and 7 here).
  • Whiskers drawn to min/max while also plotting outlier dots. Pick one convention; Tukey's (dots beyond fences) is the default in statistical software.
  • Forgetting to sort first. Every downstream number depends on order.
  • Mixing quartile methods mid-analysis. Median-split and interpolated quartiles differ on small samples (the percentiles guide shows how much); use one method throughout.
  • Reading the box as “most of the data.” It is exactly half. The whiskers-plus-box covers everything except flagged outliers.

When the Box Plot Is the Right Chart

Box plots excel at comparison: put five teams' resolution times side by side and medians, spreads, and outliers line up on one axis — a compactness histograms cannot match. Their cost is detail: bimodal shapes and gaps vanish into the box. A practical workflow is box plots to compare and screen, then the full descriptive statistics (or a histogram via the class width calculator) on anything that looks interesting.

Try the Box Plot Generator

Draw a labeled box and whisker plot with Tukey whiskers and outlier dots from your data — the worked example, drawn for you.

Frequently Asked Questions

What are the five numbers in a box plot?

The five-number summary: minimum, first quartile (Q1), median, third quartile (Q3), and maximum. The box spans Q1 to Q3 with a line at the median; the whiskers extend toward the min and max - but in the standard Tukey convention, only as far as the most extreme values within the fences, with anything beyond drawn as individual outlier points.

How far do the whiskers extend?

In the Tukey convention (used by this site's generator and most software), each whisker runs to the most extreme data value within 1.5 x IQR of its box edge - not to the fence itself, and not necessarily to the min/max. Values beyond the fences are plotted as separate dots. Some tools draw whiskers to the raw min and max instead; check the convention before reading.

How do I find outliers in a box plot?

Compute the fences: Q1 - 1.5 x IQR and Q3 + 1.5 x IQR. Any observation strictly beyond a fence is flagged and drawn as an individual point. In the worked example (Q1 = 4, Q3 = 6, IQR = 2), the fences are 1 and 9, so the value 12 is an outlier while everything else stays inside.

What does a longer whisker or half-box mean?

Asymmetry. If the median sits closer to Q1 and the upper whisker is longer, the data is right-skewed - the upper half of values spreads farther. A box plot is essentially a skewness diagram: symmetric data gives a centered median and similar whiskers.

Why does my box plot differ between Excel, R, and this site?

Two legitimate sources of variation: the quartile method (median-split/Tukey versus interpolated variants - they differ on small samples) and the whisker convention (Tukey fences versus min/max). The five raw numbers are the same data; the disagreements are in Q1/Q3 definitions and whisker rules. State the convention when the sample is small.

When is a box plot better than a histogram?

Box plots shine for comparing several groups side by side - five distributions as five compact boxes - and for flagging outliers automatically. Histograms show the full shape (bimodality, gaps) that a box plot compresses away. For one dataset, look at both; for many groups on one axis, box plots win.