Graphical Representation of
Frequency Distribution
Frequency Distribution
Frequency Distribution is a classification of data showing the number of occurrences
of each interval or category of a variable. In the case of a quantitative variable the
class intervals are arranged in order of magnitude. A frequency distribution provides a
basic ordering of data, and is usually the first step in statistical analysis.
We know that the frequency distribution can be presented in a tabular form but if the
same data are representation graphically. The graphical representation makes visual
comparison of data easier and gives a more lasting impression than is possible by any
other means.
Objective
1. The histogram is particularly appropriate to cases in which classes are not of equal
width.
2. The main purpose of the polygon is to find the mode of the given series. Mode can
be ascertained fairly accurately by the apex of the polygon.
3. Ogive is useful in determining the partition values graphically.
4. While the pie chart is perhaps the most ubiquitous statistical chart in the business
world and the mass media, it is rarely used in scientific or technical publications
Histogram
The histogram is composed of a set of rectangles one over each class-interval on the
horizontal to the frequencies of the classes. Thus in case of equal class interval the
heights of the rectangles will be proportional to the frequencies while for classes of
unequal width, the heights of the rectangles will be proportional to the ratios of the
frequencies to the width of the corresponding classes.
Example 01: Draw a histogram for the following data
Frequency Polygon and Frequency curve
The frequency polygon of a grouped frequency distribution is constructed by joining
by means of straight lines the points whose abscissas are the mid-points of the classes
and the ordinates are the corresponding frequencies. Thus a frequency polygon can
also be obtained from a histogram by joining the mid-points of the upper sides of the
adjacent rectangles by means of straight lines.
To draw the frequency curve it is necessary first to draw the polygon. The polygon is
then smoothened out keeping in view the fact that the area of the curve should be
equal to that of the histogram.
A frequency polygon for 642 psychology test scores is shown in Figure 1. The first
label on the X-axis is 35. This represents an interval extending from 29.5 to 39.5.
Since the lowest test score is 46, this interval has a frequency of 0. The point labeled
45 represents the interval from 39.5 to 49.5. There are three scores in this interval.
There are 150 scores in the interval that surrounds 85.
Cumulative Frequency Curve (Ogives)
To construct cumulative frequency curve or ogive it is necessary first to form the
frequency table. Then the upper limits of the classes are taken as the x-coordinates and
the cumulative frequencies as the y-coordinates and the points are plotted. The points
are joined by a free hand smooth curve to give the ogive.
Example 02: Draw a 'less than' ogive curve for the following data:
To Plot an Ogive:
 We plot the points with coordinates having abscissae as actual limits and ordinates
as the cumulative frequencies, (10, 2), (20, 10), (30, 22), (40, 40), (50, 68), (60,
90), (70, 96) and (80, 100) are the coordinates of the points.
 Join the points plotted by a smooth curve.
 An Ogive is connected to a point on the X-axis representing the actual lower limit
of the first class.
Scale: X -axis 1 cm = 10 marks, Y -axis 1cm = 10 c.f.
Pie Chart
A pie chart (or a circle graph) is a circular chart divided into sectors, illustrating
relative magnitudes or frequences or percents. In a pie chart, the arc length of each
sector (and consequently its central angle and area), is proportional to the quantity it
represents. Together, the sectors create a full disk. It is named for its resemblance to a
pie which has been sliced
Example: The following example chart is based on preliminary results of the election
for the European Parliament in 2004. The following table lists the number of seats
allocated to each party group, along with the derived percentage of the total that they
each make up. The values in the last column, the derived central angle of each sector,
is found by multiplying the percentage by 360°.
Group
Seats
Percent (%)
Central angle (°)
EUL
39
5.3
19.2
PES
200
27.3
98.4
EFA
5.7
42
20.7
EDD
15
2.0
7.4
ELDR
67
9.2
33.0
EPP
276
37.7
135.7
UEN
27
3.7
13.3
Other
66
9.0
32.5
Total
732
99.9*
360.2*
Assignment:
1. Write short notes on the following.
(i) Histogram (ii) Ogive
(iii) Pie chart
2. What is frequency distribution? Draw histogram, frequency polygon for the
following data.
Marks
No. of students
0-4
04
4-8
06
8-12
10
12-16
08