Histograms

El método, paso a paso

1. 1

   Calculate the class width

   class 10 < x ≤ 30 → width 20

   Upper bound minus lower bound. Getting this wrong makes every subsequent figure wrong, and classes are often written in ways that make the width easy to misread — particularly when bounds are decimals.

2. 2

   Divide frequency by class width

   frequency 30, width 20 → frequency density 1.5

   Frequency density is what goes on the vertical axis. It answers 'how crowded is this class', which is what makes classes of different widths comparable at a glance.

3. 3

   Draw bars with no gaps

   each bar spans its class, touching its neighbours

   The data is continuous, so the classes meet rather than sitting apart. A gap would imply values that cannot occur, which is why histograms and bar charts are drawn differently.

4. 4

   Area equals frequency

   width 20 × density 1.5 = frequency 30

   This is the defining property. To find a frequency from a histogram you compute the area of the bar, not read its height — reading heights as frequencies is the error the entire topic is designed to expose.

5. 5

   Work backwards for part of a class

   half a class of width 20 → 10 × 1.5 = 15

   Questions frequently ask for the frequency in a range that cuts across a class. Assuming the data is evenly spread within the class, take the proportion of the width you need and multiply by the density.

Why height alone would mislead

If one class spans 10 units and another spans 50, plotting raw frequency as height makes the wide class look the same as the narrow one even though it covers five times as much ground. The eye compares areas, so unequal widths with equal heights create a false impression.

Frequency density corrects this by measuring how concentrated the data is per unit, which is why area then represents frequency honestly. It is a fix for a genuine visual problem rather than an arbitrary complication.

• Frequency density = frequency ÷ class width
• Area of bar = frequency
• No gaps — the data is continuous
• Read frequencies as areas, never heights
• Part of a class → that proportion of the width × density

Not a bar chart

A bar chart shows discrete categories with gaps and heights that represent frequency directly. A histogram shows continuous data with no gaps and, when widths differ, areas that represent frequency. They look similar and behave differently.

The most reliable tell is the horizontal axis: categories mean bar chart, a continuous numerical scale means histogram. Once that is settled, whether to read heights or areas follows.

When all the classes are the same width

With equal class widths, frequency density is just frequency divided by a constant, so the bars have the same relative heights either way and the axis can be labelled frequency without misleading anyone.

This is why some early textbooks show histograms with frequency on the vertical axis. It is a special case rather than the general rule, and exam questions overwhelmingly use unequal widths precisely because that is where the understanding shows.

How we teach histograms

We build the frequency density column as an explicit step in every table, even when the widths are equal. It keeps the habit intact for the questions where widths differ, which are the ones that carry the marks.

We also give the backwards questions — frequency from an area, or a partial class — equal time with the drawing. Students are generally competent at producing a histogram and much weaker at extracting information from one, which is what examiners actually ask.