Statistical Methods: Graphical Representation of Data

Title of the Course: Statistical Methods
Class: Second Year, First Semester
Teacher:
Dr. Ramkrishna Singh Solanki
Assistant Professor: Mathematics and Statistics
Contact: +919826026464
email: rsolankisolanki_stat@jnkvv.org
College of Agriculture Balaghat
Murjhad Farm, Waraseoni, M.P. 481331

Topic
“Graphical Representation of Data”

A graphical representation is a visual display of data and statistical results. It is more
often and effective than presenting data in tabular form. There are different types of
graphical representation and which is used depends on the nature of the data and
the nature of the statistical result.
There are different types of graphical representation. Some of them are as follows
• Bar Graph/Diagram
• Histogram
• Pie chart/ diagram or circle graph
• Frequency Polygon
• Frequency curve

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
3 4 2 3 8 10 6 1 7 8 4 7

Bar graph/diagram/chart
A bar chart or bar graph is a chart or graph that presents categorical data with rectangular
bars with heights or lengths proportional to the values that they represent. The bars can
be plotted vertically or horizontally. A vertical bar chart is sometimes called a column
chart.

Histogram
It is similar to a Bar Chart, but a histogram groups numbers into ranges .
The height of each bar shows how many fall into each range.

What is the difference between a bar chart and a histogram?
Gaps between bars in a bar graph but in the histogram, the bars are adjacent to each
other.

Width of bars: In Histogram-Need not to be same but in Bar chart- same.

Pie chart/ diagram or circle graph
A Pie Chart is a type of graph that displays data in a circular graph. It is a circle in
which different components are represented through the sections or portions of a
circle.

Frequency Polygon
A Frequency Polygon is a graph that displays the data by using lines that connect points
plotted for frequencies at the midpoint of classes. The heights of the midpoints represent
frequencies.

Frequency curve
It is a limiting form of a histogram or frequency polygon. It is formed by smoothly
joining the consecutive points on the graph with a specific pattern.

Statistical Methods: Graphical Representation of Data

An Ogive graph is a plot used in statistics to show cumulative frequencies.
Ogive graph

Importance or merits of graphical representation
• Attractive and Impressive: Graphs are always more attractive and impressive
than tables or figures.
• Simple and understandable presentation of data: Graphs help to present
complex data in a simple and understandable way. It saves time and energy for
both the statistician and the observer.
• Useful in comparison: Graphs provide an easy comparison of two or more
phenomena.
• Location of positional averages: Graphs provides a method of locating certain
positional averages like median, mode, quartiles, etc.
• Universal utility: Graphs can be used in all fields such as trade, economics,
government departments, advertisements, etc.
• Helpful in predictions: Through graphs, tendencies that could occur in the
near future can be predicted in a better way.

Variable
Quantitative
(Numerical)
Discrete
Continuous
Qualitative
(Categorical)
(attributive,)
Quantitative: Which can be a count or measured numerically. Ex. age, income, production
Discrete: Can take only specified number of values in a given range. Ex. Tiller, plants, days
Continuous: Can take infinitely many values in a given range. Ex. yield, weight, height
Qualitative: A variable whose value varies by attributes. Ex. Colour, Gender, Education

Frequency: The number of times the observation occurred/recorded in an experiment or
study.

Below are the no. of tillers of 30 plants in a field of a paddy crop:
2 5 1 2 3 6 1 0 7 3 1 6 2 4 3
6 2 5 7 1 3 6 2 4 7 0 5 2 6 4
Numbers of tillers (discrete variable) (x) Plants (frequency) (f)
0 2
1 4
2 6
3 4
4 3
5 3
6 5
7 3
Total 30

(x) (f) c.f. (less than type) c.f. (more than type)
0 2 2 (2+4+6+4+3+3+5+3) = 30
1 4 (2+4) = 6 (4+6+4+3+3+5+3) = 28
2 6 (2+4+6) = 12 (6+4+3+3+5+3) = 24
3 4 (2+4+6+4) =16 (4+3+3+5+3) = 18
4 3 (2+4+6+4+3) =19 (3+3+5+3) = 14
5 3 (2+4+6+4+3+3) =22 (3+5+3) = 11
6 5 (2+4+6+4+3+3+5) =27 (5+3) = 8
7 3 (2+4+6+4+3+3+5+3) = 30 3
Total 30
Cumulative frequency

Class (C.I.) (x) f c.f.
0-2 12 12
3-5 10 22
6-8 8 30
Total 30
(x) (f)
0 2
1 4
2 6
3 4
4 3
5 3
6 5
7 3
Total 30
Class (C.I.) (x) f c.f.
0-2 6 6
2-4 10 16
4-6 6 22
6-8 8 30
Total 30
Inclusive class
Exclusive class
0 - 2
Lower class
Limit
Upper class
Limit
Class width = U.L.–L.L.
= 2 - 0
= 2
Mid Point = (U.L.+L.L.)/2
= (2 + 0)/2
= 1

Class (C.I.) (x) f c.f. Mid point
0-2 12 12 1
3-5 10 22 4
6-8 8 30 7
Total 30
(x) (f)
0 2
1 4
2 6
3 4
4 3
5 3
6 5
7 3
Total 30
Class (C.I.) (x) f c.f. Mid point
0-2 6 6 1
2-4 10 16 3
4-6 6 22 5
6-8 8 30 7
Total 30
Inclusive class
Exclusive class

How to convert an inclusive series into an Exclusive series.
Pocket
Expenses
No of
Students
(x) (f)
20-29 10
30-39 8
40-49 6
50-59 4
60-69 2
First, we find the difference between the
upper limit of class interval and the lower limit
of the next class interval.
For example the upper limit of the class
interval 20-29 is 29.The lower limit of the next
class interval 30-39 is 30. The difference is 30
minus 29 = 1.
Secondly, half of that difference is
added to the upper limit of a class interval and
half is subtracted from the lower limit of the
class interval.
Half of the difference found in first step
will be 0.5.Add 0.5 to the upper limit and
subtract 0.5 from the lower limit for each class
interval.
Pocket
Expenses
No of
Students
(x) (f)
19.5-29.5 10
29.5-39.5 8
39.5-49.5 6
49.5-59.5 4
59.5-69.5 2

Frequency distribution?
An arrangement of statistical data that exhibits the
frequency of the occurrence of the values of a variable

Grouped data and Ungrouped data

How to construct a bar graph on graph paper?
Now we will discuss about the construction of bar graphs or column graph. In
brief let us recall about, what is bar graph?
Bar graph is the simplest way to represent a data.
● In consists of rectangular bars of equal width.
● The space between the two consecutive bars must be the same.
● Bars can be marked both vertically and horizontally.
● The height of bar represents the frequency of the corresponding observation.

Graph Paper: special paper with small squares on it.

The following data gives the information of the no. of children
involved in different activities.
Activities Dance Music Art Cricket Football
No. of Children 30 40 25 20 35
Steps in construction of bar graphs/column graph:

Step 1:
On a graph, draw two lines
perpendicular to each
other, intersecting at 0. The
horizontal line is x-axis and
vertical line is y-axis.
0
Y axis
X axis
1 cm.

Step 2:
Along the horizontal
axis, choose the uniform
width of bars and gap
between the bars and
write the names of the
data items whose values
are to be marked.
0
Y axis
X axis
Dance Art
Music F. ball
Cricket
Activities

Step 3:
Along the vertical
axis, choose a
suitable scale in
order to
determine the
heights of the
bars for the given
values (frequency
is taken along Y
axis).
0
Y axis
X axis
Dance Art
Music F. ball
Cricket
Activities
No.
Of
Children
10
40
20
30
50
Scale: Y axis
1 cm.= 10 children
60

Step 4:
Calculate the
heights of the
bars according to
the scale chosen
and draw the bars
Dance: 30
Children
0
Y axis
X axis
Dance Art
Music F. ball
Cricket
Activities
No.
Of
Children
10
40
20
30
50
Scale: Y axis
1 cm.= 10 children

Step 4:
Calculate the
heights of the
bars according to
the scale chosen
and draw the bars
Music: 40
Children
0
Y axis
X axis
Dance Art
Music F. ball
Cricket
Activities
No.
Of
Children
10
40
20
30
50
Scale: Y axis
1 cm.= 10 children

Step 4:
Calculate the
heights of the
bars according to
the scale chosen
and draw the bars
Art: 25 Children
0
Y axis
X axis
Dance Art
Music F. ball
Cricket
Activities
No.
Of
Children
10
40
20
30
50
Scale: Y axis
1 cm.= 10 children

Step 4:
Calculate the
heights of the
bars according to
the scale chosen
and draw the bars
Cricket: 20
Children
0
Y axis
X axis
Dance Art
Music F. ball
Cricket
Activities
No.
Of
Children
10
40
20
30
50
Scale: Y axis
1 cm.= 10 children

Step 4:
Calculate the
heights of the
bars according to
the scale chosen
and draw the bars
Football: 35
Children
0
Y axis
X axis
Dance Art
Music F. ball
Cricket
Activities
No.
Of
Children
10
40
20
30
50
Scale: Y axis
1 cm.= 10 children

How to construct a Histogram ?
Grouped data are often represented graphically by histograms. A histogram
consists of rectangles, each of which has width equal or proportional to the size
of the concerned call interval, and height equal or proportional to the
corresponding frequency. In a histogram, consecutive rectangles have a common
side. For this, the class intervals are made overlapping in all cases.

Construct a histogram for the following frequency distribution.
Height (in cm) 101-110 111-120 121-130 131-140 141-150
Step I: Observe the class intervals of the distribution. If they are
nonoverlapping (inclusive series) (discontinuous), Change them into
overlapping (exclusive series) (continuous) classes.
Height (in cm) 100.5-110.5 110.5-120.5 120.5-130.5 130.5-140.5 140.5-150.5

Step 2:
On a graph, draw two lines
perpendicular to each
other, intersecting at 0. The
horizontal line is x-axis and
vertical line is y-axis.
0
Y axis
X axis

Step 3: Along the horizontal axis, choose a suitable scale in order to determine the width of
the bars for the given values (class boundaries on the x-axis).
0
Y axis
X axis
100.5
Height (in cm)
110.5 120.5 130.5 140.5 150.5
Scale: X axis 1 cm.= 10 cm. height

Step 4: Along the vertical axis, choose a suitable scale in order to determine
the heights of the bars for the given values (frequency is taken along Y axis).
0
Y axis
X axis
100.5
Height (in cm)
110.5 120.5 130.5 140.5 150.5
Scale: X axis 1 cm.= 10 cm. Height
Y axis 1 cm. = 3 children
No.
Of
Children
3
6
9
12
15
18

Step 5: Draw a bar extending from the lower value of each interval to the lower value of the next interval.
The height of each bar should be equal to the frequency of its corresponding interval.
0
Y axis
X axis
100.5
Height (in cm)
110.5 120.5 130.5 140.5 150.5
No.
Of
Children
3
6
9
12
15
18

Step 5: Draw a bar extending from the lower value of each interval to the
lower value of the next interval. The height of each bar should be equal to the
frequency of its corresponding interval.
0
Y axis
X axis
100.5
Height (in cm)
110.5 120.5 130.5 140.5 150.5
No.
Of
Children
3
6
9
12
15
18

Example for unequal class interval.
0
Y axis
X axis
100.5
Height (in cm)
110.5 130.5 140.5 150.5
No.
Of
Children
3
6
9
12
15
18

Frequency polygon
0
Y axis
X axis
100.5
Height (in cm)
110.5 120.5 130.5 140.5 150.5
No.
Of
Children
3
6
9
12
15
18

Frequency curve
0
Y axis
X axis
100.5
Height (in cm)
110.5 120.5 130.5 140.5 150.5
No.
Of
Children
3
6
9
12
15
18

Frequency curve
0
Y axis
X axis
100.5
Height (in cm)
110.5 120.5 130.5 140.5 150.5
No.
Of
Children
3
6
9
12
15
18
Frequency polygon
Histogram

What is Cumulative Frequency Curve or the Ogive in Statistics
First we prepare the cumulative frequency table, then the cumulative frequencies
are plotted against the upper or lower limits of the corresponding class intervals.
By joining the points the curve so obtained is called a cumulative frequency curve
or ogive.
There are two types of ogives :
Less than ogive : Plot the points with the upper limits of the class as abscissa and
the corresponding less than cumulative frequencies as ordinates. The points are
joined by free hand smooth curve to give less than cumulative frequency curve or
the less than Ogive. It is a rising curve.
Greater than ogive : Plot the points with the lower limits of the classes as
abscissa and the corresponding Greater than cumulative frequencies as
ordinates. Join the points by a free hand smooth curve to get the “More than
Ogive”. It is a falling curve.

Exercise: Using a graph paper, drawn the Ogives for the following distribution
which shows a record of the weight in kilograms of 200 students.
Weight (kg) No. of students
40-45 5
45-50 17
50-55 22
55-60 45
60-65 51
65-70 31
70-75 20
75-80 9
Total 200

Step 1: Observe the class intervals of the distribution. If
they are nonoverlapping (inclusive series) (discontinuous),
Change them into overlapping (exclusive series)
(continuous) classes.

Step 3: Draw cumulative frequency table (less than and more than type).
Weight frequency c.f. (less than) c.f. (more than)
40-45 5 5 200
45-50 17 22 195
50-55 22 44 178
55-60 45 89 156
60-65 51 140 111
65-70 31 171 60
70-75 20 191 29
75-80 9 200 9
Total 200

Step 4: Mark class intervals along X-axis and c.f. along Y-axis.
0
Y axis
X axis
40
Weight (in kg)
45 50 55 60 65
Scale: X axis 1 cm.= 5 kg
Y axis 1 cm.= 25 students
80
75
70
c.f.
25
50
75
100
125
150
175
200

Step 5:
(i) Less than ogive : Plot the points (x, y) where x is the U. L. of a class and y is
corresponding cumulative frequency.
(ii) More than ogive: Plot the points (x, y) where x is the L. L. of a class and y is
corresponding cumulative frequency.
x 45 50 55 60 65 70 75 80
Y 5 22 44 89 140 171 191 200
x 40 45 50 55 60 65 70 75
Y 200 195 178 156 111 60 29 9
Less than ogive points
More than ogive points

0
Y axis
X axis
40
Weight (in kg)
45 50 55 60 65
80
75
70
c.f.
25
50
75
100
125
150
175
200
Less than ogive points
More than ogive points

0
Y axis
X axis
40
Weight (in kg)
45 50 55 60 65
80
75
70
c.f.
25
50
75
100
125
150
175
200
Less than ogive
More than ogive
Step 6: Join the points obtained in step 5 by a free hand .to get the ogives

Statistical Methods: Graphical Representation of Data

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