Histogram Heights

Histogram Heights 📊

A **histogram** looks like a bar chart, but it plays by one crucial extra rule — used when the class intervals are **different widths**. On a histogram the **area** of each bar shows the frequency, not its height. Get that one idea and the grade-8/9 histogram questions become routine. (Higher tier.)

A quick word on sampling 🎯

Before representing data you have to **collect** it well. A good **sample** is representative of the whole population; a biased one leads to wrong conclusions. Grade-9 sample critiques are **specific**, not just "it is biased" — say *how*: "the survey was only taken in the morning, so people at work are underrepresented".

Spot the best critique

A café asks the first 10 customers on Monday morning whether the town needs more parking. Which is the strongest criticism of this sample?

• It only samples Monday-morning café customers, who may not represent the whole town
• It is biased
• Ten is an even number
• There is nothing wrong with it

Frequency density 📐

When classes have **unequal widths**, you cannot compare raw heights fairly. Instead a histogram plots **frequency density** up the vertical axis: **frequency density = frequency ÷ class width.** A class of 30 values spread over a width of 10 has frequency density 30 ÷ 10 = 3. Always label that vertical axis "frequency density", never "frequency".

Find the frequency density

An interactive activity.

Area is the frequency 🟪

Here is the golden rule: on a histogram, the **area** of a bar equals its **frequency**. Rearranging the density formula: **frequency = frequency density × class width.** So a short, wide bar can hold **more** values than a tall, narrow one. Never judge frequency by height alone — multiply density by width.

Work backwards

An interactive activity.

A histogram to read 📋

Here is the data behind the histogram coming up (times taken, in minutes): • 0 ≤ t < 10 — width 10, frequency 15, so FD = **1.5** • 10 ≤ t < 20 — width 10, frequency 30, so FD = **3.0** • 20 ≤ t < 40 — width 20, frequency 40, so FD = **2.0** • 40 ≤ t < 70 — width 30, frequency 30, so FD = **1.0** Notice the 10–20 bar is the **tallest** (FD 3), yet the 20–40 class actually has the **most** people (40).

Which class has the most people?

An interactive activity.

Match each class to its frequency density

Frequency from the bar

An interactive activity.

True or false?

"On a histogram, the tallest bar always represents the most people." Is this true or false?

• False — the largest AREA has the most people, and a shorter wider bar can beat a tall narrow one
• True — taller always means more people
• True, but only for equal-width classes

Order the method

An interactive activity.

The histogram rules

On a histogram the vertical axis shows frequency _____, worked out as frequency ÷ class _____. The frequency of a class is shown by the _____ of its bar, equal to frequency density × class width.