This document discusses histograms and bar charts. Histograms show grouped data values along the horizontal axis and can be used for both discrete and continuous data. Bar charts show categories along the horizontal axis. The key difference is that histograms have bars of equal width to show frequency density, while bar charts have bars of varying widths to show frequencies. The document provides examples of how to construct histograms from raw data by grouping it into a frequency table first. It also discusses how to calculate mean from a frequency polygon by multiplying the frequency by the midpoint and summing these values.

1. Histograms are frequency diagrams that show grouped data values along the
horizontal axis. Histograms can be used to show both discrete and continuous data.
Bar charts are frequency diagrams that show categories along the horizontal axis
BAR CHART or HISTOGRAM?

2. A bar chart shows the frequency of events occurring. The height
of the bar is the frequency.
BAR CHARTS

3. BAR CHARTS versus HISTOGRAMS
Histograms are frequency diagrams that show grouped data values along the
horizontal axis. Histograms can be used to show both discrete and continuous
data.
Bar charts are frequency diagrams that show categories along the
horizontal axis

5. Key terms:
A histogram looks like a bar chart, except the area of the bar, and not the
height, shows the frequency of the data. Histograms are typically used
when the data is in groups of unequal width.
Discrete data =
Continuous data
information that can only take certain values,
information that we collect that can be counted.
= data that can take any value.
the number of people in a class
height
temperature
test questions answered correctly
length

6. Histograms for Continuous Data
A histogram for continuous data is a frequency diagram that shows
part of the number line on the horizontal axis.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ……
5
Frequency
Length in cm
4 6 8 10 12 14
10
15
20
Length of Objects
Class Interval
4 to 6
This histogram has 5 Class Intervals each of class width 2
Class Interval
10 to 12
Modal Class

7. Drawing a Histogram from raw data.
The following data show the number of runs scored by an
opening batsman during matches last season.
42 34 39 49 62 99 18 24 37 38
59 112 47 72 85 68 53 101 45 12
0 33 42 61 80 30 21 27 56 90
5 35 94 63 75 33 20 44 33 71
Frequency
Tally
Runs
Histograms for Discrete Data
When a lot of data needs to be
sorted, one of the most efficient
ways is to use a frequency
table.

8. Drawing a Histogram from raw data.
The following data show the number of runs scored by an
opening batsman during matches last season.
42 34 39 49 62 99 18 24 37 38
59 112 47 72 85 68 53 101 45 12
0 33 42 61 80 30 21 27 56 90
5 35 94 63 75 33 20 44 33 71
Frequency
Tally
Runs
100 - 119
80 - 99
60 - 79
40 - 59
20 - 39
0 - 19
1. Choose a suitable
grouping and display
in a frequency table.
2. Complete the frequency table
then draw the histogram.
0-19 20-39 40-59 60-79 80-99 100-119
Runs Scored
2
5
7
9
13
4
f
5
10
15
0
Runs scored in matches

9. HISTOGRAMS
frequency
density
Class width/ class interval
Class width Frequency
0<v≤20 0.3
20<v≤40
40<v≤60
60<v≤80
80<v≤100
20 40 60
0 80 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Equal class Intervals
In a histogram the heights of the bars give the frequency.

10. HISTOGRAMS
frequency
density
Class width/ class interval
Class width Frequency
0<v≤40
40<v≤60
60<v≤80
80<v≤100
Frequency = frequency density X class width
20 40 60
0 80 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Unequal class Intervals

11. HISTOGRAMS
Class width Frequency
(Fd x class width)
0<v≤40
40<v≤60
60<v≤80
80<v≤100
Frequency = frequency density X class width

12. 20 40 60
0 80 100
10
20
30
40
50
60
80
70
90
100
120
Speed mph
frequency
density
Recorded Speeds of Cars on a Motorway
Use the histogram to work out the missing frequencies from the table:
Step 1:
Remember frequency
is not the height of
the bar, but the area
Area = L x w
Frequency = frequency density X class width (speed)
Speed
frequency
density
Step 2:
Add an extra column in
your frequency table, to
help

13. Class
width (Cw)
Frequency
density (Fd)
Frequency
(Fd x Cw)
0-30 8 8x30=240
30-40 32 32x10=320
40-50
50-60
60-70
70-80
80-120
Frequency = frequency density X class width

14. 20 40 60
0 80 100
10
20
30
40
50
60
80
70
90
100
120
Speed mph
frequency
density
Recorded Speeds of Cars on a Motorway
Speed f fd
0 - 30 240 8
30 - 40 320 32
40 - 50 500 50
50 - 60 780 78
60 - 70 960 96
70 - 80 820 82
80 - 120 640 16

15. Answer the questions first:
1) What is this graph about?
2) What type of graph is this? Bar
chart or histogram? What is the
difference?
3) What is on the x- axis?
4) What is on the y-axis?
5) What does 0≤w<20 means?
6) What does frequency means?
7) Is frequency the same as
frequency density?

16. Class
width
(Cw)
Frequency
density (Fd)
Frequency
(Fd x Cw)
0-20 a=
20-30 b=
30-40 c=
40-60 d=
60-80 e=

17. Any questions?

18. FREQUENCY POLYGONS Use the
frequency
polygon to work
out an estimate
for the mean
rounded to 1 dp

19. A frequency distribution can be shown using a frequency polygon.
5
Frequency Polygons
5-9
0
10
15
20
10-14 15-19 20-24 25-30
Frequency
Marks
Test Scores
A Frequency Polygon can be drawn onto an existing histogram.
1. Mark the mid - points of each bar at the top with a point.
2. Draw in straight lines connecting points.
3. Extend lines if necessary ½ a class interval beyond first and last bars

20. How do you find the mean of a frequency polygon?
Step 1: Find the midpoint of each interval. – we already
have the midpoints!
Step 2: Multiply the frequency of each interval by its mid-
point.
Step 3: Get the sum of all the frequencies (f) and the sum
of all the fx.
Divide 'sum of fx' by 'sum of f ' to get the mean.

21. How do you find the mean of a frequency polygon?
Step 1: Find the midpoint of each interval. – we already have the
midpoints!
Step 2: Multiply the frequency of each interval by its mid-point.
Step 3: Get the sum of all the frequencies (f) and the sum of all the fx.
Divide 'sum of fx' by 'sum of f ' to get the mean.
∑(f)= 2+4+.. ∑(fx)= 80+..
Class interval midpoint frequency Frequency x
mid-point
40 2 2x40=80
52.5 4 4x52.5 =
65
77.5
90
Calculate the mean:
∑(fx) ÷ ∑(f)=

22. Frequency Diagrams

23. a) Frequency Diagram
b) Frequency Poligon
Step 1: choose a scale for x-axis (Time, in mins)
and label the graph
Step 2: choose a scale for y-axis (Frequency)
and label the graph
Step 3:
!A frequency diagram shows the
frequencies for different groups. To plot a
frequency polygon of grouped data, plot
the frequency at the midpoint of each
group.

24. HISTOGRAMS
GROUP FREQUENCY TABLES
Frequency polygons