A histogram groups quantitative data into equal-width intervals (bins) and draws a bar for each bin, with no gaps between bars because the number line has no gaps. You met it with dot plots in Grade 6 and learned to describe a distribution by its center, spread and shape. Here we make "shape" precise.
Key idea
Symmetric: the two halves are near mirror images. Skewed right: a long, thin tail stretches toward large values. Skewed left: the long tail stretches toward small values. Unimodal means one peak; bimodal means two clear peaks, often a sign of two different groups mixed together.
Why it works
Skew is named for the tail, because the tail is what pulls on the summaries. A few very large values stretch the right side of the picture out like a tail, even though the bulk of the data sits on the left. So "skewed right" means "stretched to the right," not "most of the data is on the right."
Worked example: Reading a histogram
Commute times: 0–10 min: 3 workers, 10–20: 9, 20–30: 12, 30–40: 6, 40–50: 3, 50–60: 1, 60–70: 1. Describe the shape.
- Find the peak: the 20–30 minute bin is tallest, and there is only one peak, so the distribution is unimodal.
- Compare the sides: on the left the bars drop off within 20 minutes; on the right they trail off slowly out to 70 minutes.
- The long tail is on the right.
Answer: Unimodal and skewed right.
Worked example: Predicting a shape from context
What shape would you expect for scores on a very easy quiz worth 10 points?
- Most students score near the top: 8, 9 or 10. Scores cannot go above 10, so the data pile up against that wall.
- A few students score low, which makes a thin tail stretching toward small values.
- The tail points left.
Answer: Skewed left.
Common mistake
Saying "skewed right" when the big pile of data is on the right. Skew is named for the long tail, not the pile. Check: point to where the bars get short and stretched out. That direction is the name of the skew.
A histogram's shape can change with the bin width. Very wide bins hide detail; very narrow bins make a jagged picture. Statisticians often try a few widths before describing the shape.