# Histograms Questions, Worksheets and Revision

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## What you need to know

When displaying grouped data, especially continuous data, a histogram is often the best way to do it – specifically in cases where not all the groups/classes are the same width, and by “width” we mean the upper bound of the class takeaway the lower bound. Histograms are like bar charts with 2 key differences:

• There are no gaps between the bars, and
• It’s the area (as opposed to the height) of each bar that tells you the frequency of that class.

In order to make this work, when drawing a histogram, we plot frequency density on the y-axis rather than frequency. The frequency density for each group is found using the formula:

$\text{frequency density} = \dfrac{\text{frequency}}{\text{class width}}$

In this topic we will at both drawing and interpreting histograms.

Example: Below is a grouped frequency table of the lengths of 71 pieces of string. Construct a histogram of the data.

To construct a histogram, we will need the frequency density for each class. Dividing the frequency of the first class by its width, we get

$\text{frequency density } =\dfrac{8}{20-0} = 0.4$

Then, for the second class we get

$\text{frequency density }=\dfrac{13}{30-20}=1.3$

Once we have calculated the frequency density with the remaining groups, then it is good to add a third column to the table containing the frequency density values, see: below.

Once this new column is completed, all that remains is to plot the histogram.

With lengths on the $x$-axis frequency density on the $y$-axis, each bar that we draw will have width equal to its class width, and height equal to the relevant frequency density.

The resulting histogram is shown below

The other part of this topic is interpreting histograms when they’re given to you pre-drawn. You might see them without the frequency density axis labelled, like the example below.

Example: Below is a histogram times taken to complete a quiz. 44 people took between 0 and 1.5 minutes. Work out how many people took between 3 and 4 minutes.

It’s imperative here to remember that it’s the area that matters. So, to answer this question, we’re going to use the information to work out how much 1 small square of area is worth.

Between 0 and 1.5 minutes includes all of the first bar and some of second. From 0 to 1 minutes there are $10\times 12 =120$ small squares, and from 1 to 1.5 there are $5\times 20=100$ small squares (marked on the graph below for clarity).

So, in total there are $100+120=220$ small squares between 0 and 1.5 minutes, and the question tells us that this accounts for 44 people. Therefore, 1 person is equal to

$220 \div 44=5\text{ small squares}$.

Now, reading from the graph we get that there are $11 \times 10 = 110$ small squares between 3 and 4 minutes, so given that 5 small squares is one person, there must be

$110 \div 5 = 22$

who took between 3 and 4 minutes to do the quiz. Note: as 110 is half of 220, the answer must be half of 44 – this is a nice time-saver answer, but it won’t always be that nice unfortunately.

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### Example Questions

We must calculate the frequency density for each class by dividing each frequency by its associated class width. Writing the frequency density in its own column, we get

Now we need to plot the histogram with height on the $x$-axis and frequency density on the $y$-axis. The result should look like

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We want to see how many people one small square of area is worth.

The question tells us that there were 54 people who could hold their breath for 1 minute or more, so that means 54 people account for the whole of the last bar and part of the second-to-last bar. The number of small squares after 60 seconds is

$(5 \times 15) + (15 \times 4) = 75 + 60 = 135$

As seen on the graph below if you’re unsure.

Then, if this area constitutes 54 people, we must have that

$\text{1 person } = \dfrac{135}{54} = 2.5\text{ small squares}$

The number of small squares between 20 and 40 is

$(5 \times 32) + (5 \times 20) = 160 + 100 = 260$

Also shown on the graph below. Therefore, the number of people who held their breath for between 20 and 40 seconds is

$\dfrac{260}{2.5} = 104\text{ people}$.

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## Histograms Worksheets and Revision

Histograms 3
Level 6-7

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