Dot Plots and Histograms: Showing the Shape of Data

A statistical question expects varied answers

Before any display comes the question. “How tall am I?” has one answer, so there is no data set and nothing to plot.

“How tall are the students in my class?” expects many different answers, and that variety is what statistics studies. A question like that is called a statistical question.

Question	Statistical?
How old am I?	no — one answer
How old are the people in my family?	yes
What was yesterday’s temperature here?	no
What were the daily temperatures last month?	yes

A dot plot keeps every observation

Fourteen students recorded how many minutes of homework they did.

One dot per student, stacked above its value. Three facts come straight off the picture:

• The most common value is $30$ — the tallest stack.
• Most students fall between $20$ and $40$ — where the dots bunch.
• One student took $60$ minutes — a lone dot away from the rest.

A dot plot works well up to roughly $30$ or $40$ observations. Past that the stacks get too tall and the values too crowded to read.

A histogram groups values into intervals

For larger data sets, group the values into equal-width intervals called bins and draw a bar for each.

Here are $60$ test scores.

Each bar’s height is how many scores landed in that interval. Nineteen students scored between $70$ and $80$.

What a histogram gives up is the individual values — you can no longer tell whether those nineteen scored $71$ or $79$. What it gains is a readable shape from data far too large to dot.

Why the bars touch

This is the difference between a histogram and a bar chart, and it is not decoration.

A bar chart shows categories — colours, countries, sports. There is nothing between “red” and “blue”, so the bars are drawn apart to show they are separate things.

A histogram shows a number line. The interval $60$–$70$ runs straight into $70$–$80$ with nothing between them, so touching bars are telling the truth about the scale underneath. A gap in a histogram would mean an interval with no data, which is real information.

	Histogram	Bar chart
horizontal axis	numbers	categories
bars	touch	separated
order	fixed by the numbers	can be rearranged

Why bin width is a decision

The same data binned differently can look like a different data set.

Both histograms are correct, and they answer different questions. The wide bins say most students passed. The narrow bins also say where the peak sits.

Too wide hides structure — a gap or a second peak disappears inside a bar. Too narrow invents structure, turning ordinary bumpiness into what looks like a pattern.

That is the honest position: a histogram is a summary, and whoever draws it chooses how much detail to keep. Being able to say which bin width was used, and what it might be hiding, is part of reading one properly.

Describing the shape

Three words cover most distributions.

Symmetric — the two sides roughly mirror each other. The test scores above are close to this.

Skewed — one tail stretches further. Data skewed right has its long tail towards the high values, which is what salaries and house prices look like.

Outlier — a value far from the rest, like the $60$-minute student.

Shape decides which measure of center to report. A symmetric set is well described by its mean; a skewed one or one with an outlier is better described by its median.

Worked examples

Common mistakes

Practice problems