# Draw Histograms

In this worksheet, students practice drawing histograms.

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, Pearson Edexcel, OCR

Curriculum topic:   Statistics

Curriculum subtopic:   Statistics, Interpreting and Representing Data

Difficulty level:

### QUESTION 1 of 10

At the higher level in GCSE maths, you have to be able to make the leap from using a bar chart to represent data to a histogram.

When do I use a histogram?

Histograms are used when you have continuous data that is grouped.

What is the difference between histograms and bar charts.

Both Histograms and Bar Charts are drawn with rectangular bars bu there is one main difference.

In a bar chart, the height is used to represent frequency and in a histogram area is used to represent the frequency.

Example: The table gives the heights of a group of boys.

 Height, h (cm) 151 < h ≤ 153 153 < h ≤ 154 154 < h ≤ 155 155 < h ≤ 159 159 < h ≤ 160 Frequency 120 90 150 240 40

Step 1: Find the widths of the bars you need to draw.

This is also called the class width. This can easily be found by subtracting the lowest value of the class from the largest value.

 Height, h (cm) 151 < h ≤ 153 153 < h ≤ 154 154 < h ≤ 155 155 < h ≤ 159 159 < h ≤ 160 Frequency 120 90 150 240 40 Class Width 2 1 1 4 1

Step 2: Find out how high we need to draw the bars.

We have already said that in a histogram, the area of the bar represents the frquency.

We can say that Frequency = Class Width x Height of the bar

And this can be rearranged into the equation.

Height =
 Frequency Class Width

The correct name of the hight of the bar in a histogram is the Frequency Density, this makes our formula...

Frequency Density =
 Frequency Class Width
 Height, h (cm) 151 < h ≤ 153 153 < h ≤ 154 154 < h ≤ 155 155 < h ≤ 159 159 < h ≤ 160 Frequency 120 90 150 240 40 Class Width 2 1 1 4 1 Frequency Density 60 90 150 60 40

Step 3: Draw your histogram.

To draw the histogram, we plot the class on the bottom (don't forget to use a continuous scale) and the frequency density up the side.

Dividing frequency by class width will give...

The heights of 400 tulips are shown in a grouped frequency table.

 Height, h (cm) 24 < h ≤ 26 26 < h ≤ 27 27 < h ≤ 28 28 < h ≤ 31 31 < h ≤ 37 Frequency 50 50 60 120 120

Match the groups with the correct class width.

## Column B

24 < h ≤ 26
3
26 < h ≤ 27
1
27 < h ≤ 28
6
28 < h ≤ 31
2
31 < h ≤ 37
1

The heights of 400 tulips are shown in a grouped frequency table.

 Height, h (cm) 24 < h ≤ 26 26 < h ≤ 27 27 < h ≤ 28 28 < h ≤ 31 31 < h ≤ 37 Frequency 50 50 60 120 120 Class Width 2 1 1 3 6

What are the frequency densities for each group?

 Frequency Density 24 < h ≤ 26 26 < h ≤ 27 27 < h ≤ 28 28 < h ≤ 31 31 < h ≤ 37

The heights of 400 tulips are shown in a grouped frequency table.

 Height, h (cm) 24 < h ≤ 26 26 < h ≤ 27 27 < h ≤ 28 28 < h ≤ 31 31 < h ≤ 37 Frequency 50 50 60 120 120 Class Width 2 1 1 3 6 Freq Density 25 50 60 40 20

Which of the two histograms have been drawn correctly?

A

B

The ages of a groups pf children are shown in the grouped frequency table.

 Age (Years) 11 < x ≤ 14 14 < x ≤ 16 16 < x ≤ 17 17 < x ≤ 21 Frequency 21 24 10 20

Match the groups with the correct class width.

## Column B

11 < x ≤ 14
1
14 < x ≤ 16
4
16 < x ≤ 17
2
17 < x ≤ 21
3

The ages of a groups pf children are shown in the grouped frequency table.

 Age (Years) 11 < x ≤ 14 14 < x ≤ 16 16 < x ≤ 17 17 < x ≤ 21 Frequency 21 24 10 20 Class Width 3 2 1 4

What are the frequency densities for each group?

 Frequency Density 11 < x ≤ 14 14 < x ≤ 16 16 < x ≤ 17 17 < x ≤ 21

The ages of a groups pf children are shown in the grouped frequency table.

 Age (Years) 11 < x ≤ 14 14 < x ≤ 16 16 < x ≤ 17 17 < x ≤ 21 Frequency 21 24 10 20 Class Width 3 2 1 4 Frequency Density 7 12 10 5

Which of these two histograms have been drawn correctly?

A

B

The ages of a groups pf children are shown in the grouped frequency table.

 Time (minutes) 0 < x ≤ 8 8 < x ≤ 12 12 < x ≤ 16 16 < x ≤ 20 Frequency 72 48 54 36

Match the groups with the correct class width.

## Column B

0 < x ≤ 8
4
8 < x ≤ 12
4
12 < x ≤ 16
4
16 < x ≤ 20
8

The ages of a groups pf children are shown in the grouped frequency table.

 Time (minutes) 0 < x ≤ 8 8 < x ≤ 12 12 < x ≤ 16 16 < x ≤ 20 Frequency 72 48 54 36 Class Width 8 4 4 4

What are the frequency densities for each group?

 Frequency Density 0 < x ≤ 8 8 < x ≤ 12 12 < x ≤ 16 16 < x ≤ 20

The ages of a groups pf children are shown in the grouped frequency table.

 Time (minutes) 0 < x ≤ 8 8 < x ≤ 12 12 < x ≤ 16 16 < x ≤ 20 Frequency 72 48 54 36

Has this histogram been drawn completely correctly?

Yes

No

• Question 1

Dividing frequency by class width will give...

EDDIE SAYS
It's vital that you get the language here, there is a mark in the exam for labelling the histogram axes correctly
• Question 2

The heights of 400 tulips are shown in a grouped frequency table.

 Height, h (cm) 24 < h ≤ 26 26 < h ≤ 27 27 < h ≤ 28 28 < h ≤ 31 31 < h ≤ 37 Frequency 50 50 60 120 120

Match the groups with the correct class width.

## Column B

24 < h ≤ 26
2
26 < h ≤ 27
1
27 < h ≤ 28
1
28 < h ≤ 31
3
31 < h ≤ 37
6
EDDIE SAYS
All we have to do to find the class width is subract the lowest number from the higher number for the group
• Question 3

The heights of 400 tulips are shown in a grouped frequency table.

 Height, h (cm) 24 < h ≤ 26 26 < h ≤ 27 27 < h ≤ 28 28 < h ≤ 31 31 < h ≤ 37 Frequency 50 50 60 120 120 Class Width 2 1 1 3 6

What are the frequency densities for each group?

 Frequency Density 24 < h ≤ 26 26 < h ≤ 27 27 < h ≤ 28 28 < h ≤ 31 31 < h ≤ 37
EDDIE SAYS
Remember the formula for frequency density? fd = freq / class wirth
• Question 4

The heights of 400 tulips are shown in a grouped frequency table.

 Height, h (cm) 24 < h ≤ 26 26 < h ≤ 27 27 < h ≤ 28 28 < h ≤ 31 31 < h ≤ 37 Frequency 50 50 60 120 120 Class Width 2 1 1 3 6 Freq Density 25 50 60 40 20

Which of the two histograms have been drawn correctly?

B
EDDIE SAYS
One of the biggest mistakes people make when they draw histograms is that they draw they frequency instead of the frequency density
• Question 5

The ages of a groups pf children are shown in the grouped frequency table.

 Age (Years) 11 < x ≤ 14 14 < x ≤ 16 16 < x ≤ 17 17 < x ≤ 21 Frequency 21 24 10 20

Match the groups with the correct class width.

## Column B

11 < x ≤ 14
3
14 < x ≤ 16
2
16 < x ≤ 17
1
17 < x ≤ 21
4
EDDIE SAYS
All we have to do to find the class width is subract the lowest number from the higher number for the group
• Question 6

The ages of a groups pf children are shown in the grouped frequency table.

 Age (Years) 11 < x ≤ 14 14 < x ≤ 16 16 < x ≤ 17 17 < x ≤ 21 Frequency 21 24 10 20 Class Width 3 2 1 4

What are the frequency densities for each group?

 Frequency Density 11 < x ≤ 14 14 < x ≤ 16 16 < x ≤ 17 17 < x ≤ 21
EDDIE SAYS
Remember the formula for frequency density? fd = freq / class wirth
• Question 7

The ages of a groups pf children are shown in the grouped frequency table.

 Age (Years) 11 < x ≤ 14 14 < x ≤ 16 16 < x ≤ 17 17 < x ≤ 21 Frequency 21 24 10 20 Class Width 3 2 1 4 Frequency Density 7 12 10 5

Which of these two histograms have been drawn correctly?

A
EDDIE SAYS
We need to take care with is where the bars are drawn. Option A has the first bar starting at 11 Option B has the first bar starting at 10. Which one does the table tell us we should be using?
• Question 8

The ages of a groups pf children are shown in the grouped frequency table.

 Time (minutes) 0 < x ≤ 8 8 < x ≤ 12 12 < x ≤ 16 16 < x ≤ 20 Frequency 72 48 54 36

Match the groups with the correct class width.

## Column B

0 < x ≤ 8
8
8 < x ≤ 12
4
12 < x ≤ 16
4
16 < x ≤ 20
4
EDDIE SAYS
All we have to do to find the class width is subract the lowest number from the higher number for the group
• Question 9

The ages of a groups pf children are shown in the grouped frequency table.

 Time (minutes) 0 < x ≤ 8 8 < x ≤ 12 12 < x ≤ 16 16 < x ≤ 20 Frequency 72 48 54 36 Class Width 8 4 4 4

What are the frequency densities for each group?

 Frequency Density 0 < x ≤ 8 8 < x ≤ 12 12 < x ≤ 16 16 < x ≤ 20
EDDIE SAYS
Remember the formula for frequency density? fd = freq / class wirth
• Question 10

The ages of a groups pf children are shown in the grouped frequency table.

 Time (minutes) 0 < x ≤ 8 8 < x ≤ 12 12 < x ≤ 16 16 < x ≤ 20 Frequency 72 48 54 36

Has this histogram been drawn completely correctly?

No
EDDIE SAYS
This is very nearly correct. The only thing that is incorrect is that the top of the third bar has not been drawn fully across the bar. While this may seem like a small thing, it would be enough to lose a mark in the exam.
---- OR ----

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