 # Histogram (Non-Uniform Widths)

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What is a Histogram?

How to interpret and draw a histogram?

When constructing a histogram with non-uniform (unequal) class widths, we must ensure that the areas of the rectangles are proportional to the class frequencies.

Remember that the histogram differs from a bar chart in that it is the area of the bar that denotes the value, not the height. This means that we would need to consider the widths in order to determine the height of each rectangle.

Example

The following frequency distribution gives the masses of 48 objects measured to the nearest gram. Draw a histogram to illustrate the data.

 Mass (g) Frequency 10 – 19 20 – 24 25 – 34 35 – 50 51 – 55 6 4 12 18 8

Solution:

Evaluate each class widths.

 Mass (g) Frequency Class width 10 – 19 20 – 24 25 – 34 35 – 50 51 – 55 6 4 12 18 8 10 5 10 15 5

Since the class widths are not equal, we choose a convenient width as a standard and adjust the heights of the rectangles accordingly.

We notice that the smallest width size is 5. We can choose 5 to be the standard width. The other widths are then multiples of the standard width.

The table below shows the calculations of the heights of the rectangles.

 Mass (g) Frequency Class widths Rectangle’s height in histogram 10 – 19 20 – 24 25 – 34 35 – 50 51 – 55 6 4 12 18 8 10 5 10 15 5 2 × standard standard 2 × standard 3 × standard standard 6 ÷ 2 = 3 4 12 ÷ 2 = 6 18 ÷ 3 = 6 8 Histogram with uneven class widths
Higher GCSE topic revision
Understand what is a histogram.
Draw a histogram given some information.
Interpret a histogram.

GCSE Maths - Histograms - Unequal Class Intervals - Frequency Density - Higher A/A* grade
Example 1: The table below shows the length in mm of some worms found in Steve's garden. Draw a histogram to represent the information.

Example 2: The histogram shows the range of ages of members of a spots centre. Complete the frequency table.

Example 3: The histogram gives information about the heights of 540 Christmas trees. Work out an estimate for the number of Christmas trees with a height greater than 3 metres.

How to find the width and height of a class interval?
Example:
The lifetime of a bulb in hours is given in the table. If the width of the 95-105 class is 2cm and the height is 9c. Find the width and height of the 105-130 class.
Histogram (unequal class intervals)
Example:
The police wants to know how many cars exceed the speed limit. An officer stands with a speed gun and records the speeds of 1000 consecutive cars.
(a) Identify one possible source of bias for this experiment.

(b) The grouped frequency table represents the speeds of the 1000 cars.
On the grid, show the data on a histogram.

(c) The speed limit for the road is 30 miles per hour.
Two cars are chosen at random from the 1000 cars.
Estimate the probability that both cars are at least 10% above the speed limit.

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