Histograms (Equal Class Intervals)

Draw a histogram when all the classes are the same width, and read the modal class, totals and how many are above a value from it.

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Histogram or Bar Chart?

A histogram shows grouped data that is measured, like time or height. It looks like a bar chart, with three differences.

Bar chart	Histogram
gaps between bars	bars touch
a label under each bar	a number scale along the bottom
any categories	classes of measured data

Key idea

When every class has the same width and the side is labelled Frequency, the height of each bar is simply the frequency.

Drawing the Bars

The 2 or 3-mark question: "Draw a histogram for this information." 40 students timed their homework, in minutes.

Time (minutes)	Frequency
$0 < t \leq 10$	5
$10 < t \leq 20$	9
$20 < t \leq 30$	12
$30 < t \leq 40$	10
$40 < t \leq 50$	4

Steps

1
Along the bottom: an even scale, 0, 10, 20, 30, 40, 50.
2
Up the side: frequency, from 0, in equal steps up past 12.
3
Each bar goes from the start of its class to the end, up to its frequency.
4
The bars touch. No gaps.

Worked example

1
$20 < t \leq 30$ — Take one row of the table. This class starts at 20 minutes and ends at 30.
2
$bar from 20 to 30$ — The bar covers the whole class, so its left edge is at 20 and its right edge is at 30 on the bottom scale.
3
$height 12$ — All the widths are equal and the side says Frequency, so the bar goes up to its frequency, 12.
4
$next bar: 30 to 40$ — The next class begins where this one ends, at 30. So the bars touch, with no gap.

Common mistake

Drawing thin bars with gaps, or one bar at the middle of each class. The bars cover the whole class.

Key idea

You get a mark for each correct bar, so one slip costs only a little. Use a ruler, and check every height against the table.

Reading the Histogram

The 1 or 2-mark questions: "Write down the modal class", "How many took more than 30 minutes?" and "What percentage is that?"

Worked example

1
$5 + 9 + 12 + 10 + 4 = 40$ — Add the height of every bar. Each height is a frequency, so this gives the total: 40 students.
2
$tallest bar: 20 < t \leq 30$ — The modal class is the class with the highest frequency. With equal widths, that is the tallest bar. Write the class, not the number 12.
3
$30 < t \leq 40 and 40 < t \leq 50$ — “More than 30” means the bars that start at 30 or later. The bar from 20 to 30 ends at 30, so it does not count.
4
$10 + 4 = 14$ — Add the heights of those two bars: 14 students took more than 30 minutes.
5
$\frac{14}{40} = 0.35$ — A percentage compares a part with the whole, so divide by the total.
6
$0.35 \times 100 = 35%$ — Multiply by 100 to turn the decimal into a percentage.

Common mistake

Taking the bar from 20 to 30 as "more than 30". More than 30 starts at 30, so only the bars to the right count.

Fill the Gaps: Table and Histogram

A common 3-mark question gives half a table and half a histogram, and you fill in the rest. 50 seedlings were measured. The histogram already shows the first three bars, with heights 6, 11 and 15.

Height (cm)	Frequency
$0 < h \leq 4$	6
$4 < h \leq 8$	?
$8 < h \leq 12$	15
$12 < h \leq 16$	10
$16 < h \leq 20$	?

Worked example

1
$4 < h \leq 8 : height 11$ — Read the bar for 4 to 8 against the side scale. Its height is the frequency, so 11 goes in the table.
2
$6 + 11 + 15 + 10 = 42$ — Add the four frequencies you now know.
3
$50 - 42 = 8$ — There are 50 seedlings in total, so the last class holds what is left: 8.
4
$12 < h \leq 16 : bar to 10$ — Now draw the bars the graph is missing. This one goes from 12 to 16, up to its frequency, 10.
5
$16 < h \leq 20 : bar to 8$ — The last bar goes from 16 to 20, up to 8. It touches the bar before it.
6
$6 + 11 + 15 + 10 + 8 = 50$ — Check: the table now adds up to the total, so nothing was misread.

Key idea

Before you read any bar, find what one small square is worth. If 5 small squares make 2, each one is $2 \div 5 = 0.4$ .

Common mistake

Reading a bar by eye. Line a ruler up from the top of the bar to the side scale.

When the Side Says Frequency Density

Some histograms label the side Frequency density, even when every class is the same width. Then the height is not the frequency, and you need this:

Formula

frequency = frequency density \times class width

Frequency is how many are in the class. Frequency density is the height of the bar. Class width is the top of the class minus the bottom.

Race times, in classes 5 minutes wide. The bar for $15 < t \leq 20$ has height 2.4.

Worked example

1
$width = 20 - 15 = 5$ — Top minus bottom gives the class width, 5. Every class here is 5 wide.
2
$frequency = 2.4 \times 5$ — The side says frequency density, so multiply the height by the width.
3
$= 12$ — So 12 runners are in this class. A count must be a whole number, which is a good check.
4
$9 \div 5 = 1.8$ — Going the other way, to draw a class with frequency 9: divide by the width. The bar goes up to 1.8.

Key idea

The modal class is still the tallest bar, because every bar was divided by the same width.

Common mistake

Saying 2.4 runners. A frequency density is not a count. Multiply by the width first.

Which Method? Read the Question

Check the widths of the classes first.

You see	You do
Equal widths, draw it	frequency up the side, bars touch
Modal class	the tallest bar
How many more than	add the bars to the right
What percentage	divide by the total, times 100
A frequency is missing	the total minus all the others
The side says frequency density	height × width = frequency
Widths are different	Histograms (Unequal Class Intervals)
Estimate the mean	Estimating the Mean for Grouped Data

Key idea

The tallest bar is the modal class only because the widths are equal. With different widths, use frequency density.

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