Statistics

submitted by
Sophiya prabin
To Choodamani sir
Mahendra multiple campus, dharan
BBMfaculty
Assignment of statistics
on Graphical representation of data

Introduction
 Graphical Representation of data means Presentation/Representation of data as
graph or chart
 It is a visual display of data and stastical result.
 It is considered to be more often and effective than presenting data in a tabular
form.
 This method of data presentation provides bases for comparison, forecasting
the trends, location of positional averages (such as Median, Mode and Quartiles
etc.) and establishing Correlation.

Functions/ Objectives
 To make complex data simple.
 To make Comparison easy.
 To make more detailed analysis.

Advantages
 Graphical Representation looks attractive and impressive.
 Simple and understandable
 Comparison becomes easy
 No need of mathematical knowledge
 Helpful in predictions.

Techniques of construction of Graph
We usually draw graph on a squared paper or a graph paper. To represent statistical data
graphically we take a point O generally at the left hand bottom of the graph paper. From O we
draw a horizontal line OX and a vertical line OY so that OY is perpendicular to OX. We call
OX the x-axis, OY the Y-axis and O is the origin. It is customary to take to take the independent
variable along the x-axis and the dependent variable along the Y-axis. A convenient scale is then
chosen for each of the axes. The scale for the two axes may be same or different according to
the nature of the data. The scale should be chosen in such a way that the entire data can be
neatly plotted on the graph paper. It is also important that the scale we chose make plotting easy
and reading easy. After choosing the scale, points corresponding to the different pairs of values
of X and Y are plotted on the graph paper . The points are then joined by straight line and the
graph is completed.

Graph of statistical data
Basically the graph of statistical data are broadly
classified into two types as:
1.Graph of time series or Historigram
2.Graph of frequency distribution
On the basis of our course of bbm we will be
discussing only about graph of frequency distribution.

Frequency Distribution graph
Frequency distribution graph can also be presented by the means
of graphs. Such graph facilitate comparative study of two or more
frequency as regards to their shape and pattern. A frequency
distribution graph can be presented in any of the following ways:
a) Histogram
b) Frequency Polygon
c) Frequency Curve or (Smoothed frequency curve)
d) Ogives or Cumulative frequency Curve.

a) Histogram
Histogram is most commonly used for graphical presentation of a
continuous frequency distribution. It consists of series of rectangle with no gap
between them. In histogram, we plot the class interval on the x-axis and their
respective frequencies on the Y-axis. Further we create a rectangle on each class
interval with its height proportional to the frequency density of the class. In
histogram we will consider two main cases in the construction of histogram i.e.
i. Histogram of equal class interval
ii. Histogram of unequal class interval.

i. Histogramof equal class interval
Histogram of equal class interval are those which are based on the
data with equal class interval. A series with equal class interval would
make a histogram including rectangles of equal width. Length of
rectangles would be different in proportion to the frequencies of the
class interval. It can be better understood with the help of following
example:

Example:
Weekly wages (Rs) No. of workers
0-20 4
20-40 8
40-60 12
60-80 6
80-100 3
100-120 1
Here, class interval of every class is equal.
Each class has the interval of 20 in their
class. So width of each class will be equal.
And its height will be proportional to the
corresponding frequencies. The histogram of
given table will be as:

0
2
4
6
8
10
12
14
0-20 20-40 40-60 60-80 80-100 100-120
No.
of
workers
weekly wages
weekly wages of the workers

ii. Histogram of unequal class interval
A histogram of unequal class interval is the one which is based on the data with unequal
class intervals. When the data of class interval are unequal; width of the rectangles would be
different. The width of the rectangles would increase or decrease depending upon the increase
or decrease in the size of the class interval. Before presenting the data in the form of graph,
frequencies of unequal class interval should be adjusted. First, we note a class of the smallest
interval. If the size of one class interval is twice the smallest size in the series, frequency of that
class is divided by ‘2’. Likewise, if the size of class interval is three times the size of the
smallest class interval in the series, frequency of that class is divided by ‘3’ and so on. For
adjusting unequal class interval this formula is used:
• Adjustment factor for any class=Class interval of the concerned class
Lowest class interval

Example:
Weekly wages(Rs) No. of workers
10-15 7
15-20 10
20-25 27
25-30 15
30-40 12
40-60 12
60-80 8
In this frequency distribution, minimum
class interval in 5. Other class interval is 10 and
20. Hence before drawing the graph, frequency
density should be calculated. It can be done by
dividing frequencies by an adjustment factors .
The above table is adjusted

Weekly Wages (Rs) Number of workers Adjustment factors Frequency Density
10-15 7 5/5=1 7/1=7
15-20 10 5/5=1 10/1=10
20-25 27 5/5=1 27/1=27
25-30 15 5/5=1 15/1=15
30-40 12 10/5=2 12/2=6
40-60 12 20/5=4 12/2=3
60-80 8 20/5=4 8/4=2
Adjustment of frequencies of unequal class intervals:
In the above table, the class interval for the first four class is 5. Fifth class however
has the interval of 10 (40-30=10), which is twice as much as the class interval of the first
four classes. So, frequency of the fifth class is divided by ‘2’. Further, class interval of the
sixth class is 20 (60-2=40) which is 4 times the minimum class interval of 5. So, frequency
of this class is divided by 4. Likewise, the frequencies of other classes have been adjusted.
On the basis of above adjusted table we will graph our histogram:

b) Frequency polygon
It is a graphical representation of data and its frequencies. This is similar to the
histogram in terms of displaying frequency distribution called class marks. It is formed by
joining mid-points of the tops of all rectangles in a histogram. However, a polygon can be
drawn even without constructing a histogram. For this, mid-values of the classes of a frequency
distribution are marked on X-axis of the graph; the corresponding frequencies are marked on
the Y-axis. Using a foot rule, all points indicating frequencies of the different classes are joined
to make a graph, called frequency polygon. Both the sides of the frequency polygon are
extended to meet the X-axis, at the mid-points of the immediately lower or higher imagined
class intervals of zero frequency. This is done to ensure that the area of a frequency polygon is
the same as that of the corresponding histogram.
Now we will see the example of the two way( i.e. frequency distribution with histogram
and frequency distribution without histogram) in which frequency polygon is drawn.

 Example of Frequency distributionwith histogram
Class frequency
15.5-20.5 2
20.5-25.5 7
25.5-30.5 14
30.5-35.5 5
35.5-40.5 3
Here, firstly we have to draw Histogram, then we have to plot
the middle point of each class interval in the top of each
rectangle. We find middle point by using the formula:
Midpoint/middle point=Lower class limit + Upper class limit
2
After that we have to join all the midpoint with the help of
scale and also the middle points of the class interval before and
after the first and last class in x-axis. Then the line which we
obtain is know as frequency Polygon with histogram. It can be
shown graphically as:

Frequency polygon with histogram

 Example of Frequency Polygon without histogram
Here, in frequency polygon without histogram we don’t
graph histogram. We find frequency polygon by
computing the mid points of each class interval. We
represent mid point along the x-axis and frequencies
along the y-axis. We plot all the mid –point of each
class interval corresponding to its frequency. The line
then we get after plotting the points of x and y-axis is
known as frequency polygon without histogram. It can
be shown graphically as:
Marks Obtained Number of Students
15.5-20.5 2
20.5-25.5 7
25.5-30.5 14
30.5-35.5 5
35.5-40.5 3

Frequency polygon without histogram

c) Frequency curve or SmoothedFrequency curve
It is just a variant of polygon. A Frequency Curve is a curve which is
plotted by joining the mid-points of all tops of a histogram by freehand
smoothed curves and not by straight lines. Area of a frequency curve is equal to
the area of a histogram or frequency polygon of a given data set. While drawing
a frequency curve, we should eliminate angularity of the polygon. Accordingly,
points of a frequency polygon are joined through a freehand smoothed curve
rather than straight lines.

Example:
Here, the given data set is first converted into a histogram.
Mid-point at the top of each rectangle is marked. Then
these points are joined through a freehand smoothed curve.
It is shown in the graph.
IQ Scores No .of students
10-20 20
20-30 25
30-40 30
40-50 7
50-60 3

Frequency curve or smoothed frequency curve

d) Cumulative frequency Curve (OGIVES)
Ogive or Cumulative Frequency Curve is the curve which is constructed
by plotting cumulative frequency data on the graph paper; in the form of a
smooth curve. A cumulative frequency curve or ogive may be constructed in two
ways:
I. Less than method
II. More than method

I. Less than method
In this method, beginning from upper limit of the 1st class interval we go
on adding the frequencies corresponding to every next upper limit of the series.
Thus in a series showing 0-5, 5-10 and 10-15 as different class intervals, we will
find the frequency for less than 5, for less than 10 and for less than 15. The
frequencies are added up to make Less than Ogive’.

II. More than method
In this method, we take cumulative total of the frequencies beginning with
lower limit of the 1st class interval. Thus, in a series showing 0-5, 5- 10 and 10-
15 as different class intervals, we find the frequency for more than 0, for more
than 5 and for more than 10. The frequencies thus presented make a 'More than
Ogive’.

Example:
Drawless thanand more than fromfollowing frequency distribution.
Marks No. of student
20-30 7
30-40 11
40-50 24
50-60 32
60-70 9
70-80 14
80-90 2
90-100 1
For less than Ogive:
Less than 30 7
Less than 40 7+11=18
Less than 50 18+24=42
Less than 60 42+32=74
Less than 80 83+14=97
Less than 100 99+1=100
For more than Ogive:
More than 20 93+7=100
More than 30 82+11=93
More than 40 58+24=82
More than 50 26+32=58
More than 60 17+9=26
More than 70 3+14=17
More than 80 2+1=3
More than 90 1

Graph of more than and less than ogive

Statistics

More Related Content

What's hot (20)

Similar to Statistics (20)

More from SophiyaPrabin (7)

Recently uploaded (20)

Statistics