1. The document describes histograms and how to construct them from frequency data by calculating frequency densities. 2. An example shows how to construct a histogram from a frequency table by dividing frequencies by interval widths to obtain frequency densities. 3. Another example constructs a frequency table from a histogram and calculates the total number of observations.

1. Histograms
BTEOTSSSBAT draw and
interpret histograms

2. Histograms are similar to bar charts but there
are two main differences:
There are no gaps between the bars in a
histogram
The area of each bar is proportional to the
frequency that it represents. Hence, total area
is proportional to the total frequency.
The vertical axis is
labelled frequency
density.
frequency density = frequency divided by interval width

3. f = 15
f =4
f=6
frequency density = frequency divided by interval width
So, frequency = frequency density x interval width (area of the bars)
Frequencies = 1 x 6 = 6
= 3 x 5 = 15
= 2 x 2 = 4

4. frequency density = frequency divided by interval width
So, frequency = frequency density x interval width

5. Example 1
Display the following data as a histogram
Mass (g) 0-20 20-40 40-60 60-80 80-100
No of letters 10 18 24 14 18

6. Example 1
Display the following data as a histogram
Mass (g) 0-20 20-40 40-60 60-80 80-100
No of letters 10 18 24 14 18
solution
First find the intervals/ widths: NB the interval
widths don’t
For each one the width/ interval is 20
have to be the
same!

7. Example 1
Display the following data as a histogram
Mass (g) 0-20 20-40 40-60 60-80 80-100
No of letters 10 18 24 14 18
solution
First find the intervals/ widths:
For each one the width/ interval is 20
Then find the frequency density using frequency divided by interval/width

8. Example 1
Display the following data as a histogram
Mass (g) 0-20 20-40 40-60 60-80 80-100
No of letters 10 18 24 14 18
solution
First find the intervals/ widths:
For each one the width/ interval is 20
Then find the frequency density using frequency divided by interval/width
Mass (g) Interval Frequency Frequency density
width = frequency / width
0-20 20 10 0.5
20-40 20 18 0.9
40-60 20 24 1.2
60-80 20 14 0.7
80-100 20 18 0.9

9. Mass (g) Interval Frequency Frequency density
width = frequency / width
0-20 20 10 0.5
20-40 20 18 0.9
40-60 20 24 1.2
60-80 20 14 0.7
80-100 20 18 0.9

10. Finding frequencies from a histogram
To find a frequency in each interval use
frequency = interval width × frequency density

11. Example 2
A passengers' association conducted survey on punctuality of trains
using a particular station. The histogram shows the results.
•Construct the frequency distribution
•How many trains were there in the survey?
Frequency density
Time late (mins)

12. Example 2
A passengers' association conducted survey on punctuality of trains
using a particular station. The histogram shows the results.
•Construct the frequency distribution
•How many trains were there in the survey?
Minutes late Frequency density Frequency
= width x freq density
Frequency density
0-5
5-10
10-20
20-30
30-50
50-80
Time late (mins)

13. Example 2
A passengers' association conducted survey on punctuality of trains
using a particular station. The histogram shows the results.
•Construct the frequency distribution
•How many trains were there in the survey?
Minutes late Frequency density Frequency
= width x freq density
Frequency density
0-5 6.4
5-10 8.8
10-20 3
20-30 1.2
30-50 0.6
50-80 0.2
Time late (mins)

14. Example 2
A passengers' association conducted survey on punctuality of trains
using a particular station. The histogram shows the results.
•Construct the frequency distribution
•How many trains were there in the survey?
Minutes late Frequency density Frequency
= width x freq density
Frequency density
0-5 6.4 5 x 6.4
5-10 8.8 5 x 6.4
10-20 3 10 x 3
20-30 1.2 10 x 1.2
30-50 0.6 20 x 0.6
50-80 0.2 30 x 0.2
Time late (mins)

15. Example 2
A passengers' association conducted survey on punctuality of trains
using a particular station. The histogram shows the results.
•Construct the frequency distribution
•How many trains were there in the survey?
Minutes late Frequency density Frequency
= width x freq density
Frequency density
0-5 6.4 5 x 6.4 = 32
5-10 8.8 5 x 6.4 =32
10-20 3 10 x 3 = 30
20-30 1.2 10 x 1.2 = 12
30-50 0.6 20 x 0.6 = 12
50-80 0.2 30 x 0.2 = 6
Time late (mins)

16. Example 2
A passengers' association conducted survey on punctuality of trains
using a particular station. The histogram shows the results.
•Construct the frequency distribution
•How many trains were there in the survey?
Minutes late Frequency density Frequency
= width x freq density
Frequency density
0-5 6.4 5 x 6.4 = 32
5-10 8.8 5 x 6.4 = 32
10-20 3 10 x 3 = 30
20-30 1.2 10 x 1.2 = 12
30-50 0.6 20 x 0.6 = 12
50-80 0.2 30 x 0.2 = 6
Time late (mins) Total trains =
32 + 32 + 30 + 12 + 12 + 6 =

17. 3. Abraham measured the time, in
seconds, that birds spent on each
individual visit to his bird table. The
birds made a total of 236 individual
visits.
The histogram shows some of the
results.
(a) Use the information in the histogram
to complete the frequency table
below.
(b) Use the table to complete the
histogram
Frequency Freq
density
0-10 50
10-20
20-25
25-30 20
30-50
Total 236

18. 3. Abraham measured the time, in
seconds, that birds spent on each
individual visit to his bird table. The
birds made a total of 236 individual
visits.
The histogram shows some of the
results.
(a) Use the information in the histogram
to complete the frequency table
below.
(b) Use the table to complete the
histogram
Frequency Freq
density
0-10 50 50/ 10 = 5
10-20
5 20-25
25-30 20
30-50
Total 236

19. 3. Abraham measured the time, in
seconds, that birds spent on each
individual visit to his bird table. The
birds made a total of 236 individual
visits.
The histogram shows some of the
15 results.
(a) Use the information in the histogram
to complete the frequency table
below.
(b) Use the table to complete the
histogram
10
Frequency Freq
density
0-10 50 50/ 10 = 5
10-20 70 7 x 10 = 70
5 20-25
25-30 20
30-50
Total 236

20. 3. Abraham measured the time, in
seconds, that birds spent on each
individual visit to his bird table. The
birds made a total of 236 individual
visits.
The histogram shows some of the
15 results.
(a) Use the information in the histogram
to complete the frequency table
below.
(b) Use the table to complete the
histogram
10
Frequency Freq
density
0-10 50 50/ 10 = 5
10-20 70 7 x 10 = 70
5 20-25 76 15.2 x 5
25-30 20
30-50
Total 236

21. 3. Abraham measured the time, in
seconds, that birds spent on each
individual visit to his bird table. The
birds made a total of 236 individual
visits.
The histogram shows some of the
15 results.
(a) Use the information in the histogram
to complete the frequency table
below.
(b) Use the table to complete the
histogram
10
Frequency Freq
density
0-10 50 50/ 10 = 5
10-20 70 7 x 10 = 70
5 20-25 76 15.2 x 5
25-30 20 20/5= 10
30-50
Total 236

22. 3. Abraham measured the time, in
seconds, that birds spent on each
individual visit to his bird table. The
birds made a total of 236 individual
visits.
The histogram shows some of the
15 results.
(a) Use the information in the histogram
to complete the frequency table
below.
(b) Use the table to complete the
histogram
10
Frequency Freq
density
0-10 50 50/ 10 = 5
10-20 70 7 x 10 = 70
5 20-25 76 15.2 x 5
25-30 20 20/5= 10
30-50
Total 236

23. 3. Abraham measured the time, in
seconds, that birds spent on each
individual visit to his bird table. The
birds made a total of 236 individual
visits.
The histogram shows some of the
15 results.
(a) Use the information in the histogram
to complete the frequency table
below.
(b) Use the table to complete the
histogram
10
Frequency Freq
density
0-10 50 50/ 10 = 5
10-20 70 7 x 10 = 70
5 20-25 76 15.2 x 5
25-30 20 20/5= 10
30-50 20 20/20 = 1
Total 236

24. 3. Abraham measured the time, in
seconds, that birds spent on each
individual visit to his bird table. The
birds made a total of 237 individual
visits.
The histogram shows some of the
15 results.
(a) Use the information in the histogram
to complete the frequency table
below.
(b) Use the table to complete the
histogram
10
Frequency Freq
density
0-10 50 50/ 10 = 5
10-20 70 7 x 10 = 70
5 20-25 76 15.2 x 5
25-30 20 20/5= 10
30-50 20 20/20 = 1
Total 236