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Elementary Mathematics

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MTH001 ­ Elementary Mathematics
Lecture # 23:
Tabulation
Simple bar chart
Component bar chart
Multiple bar chart
 Pie chart
As indicated in the last lecture, there are two broad categories of data ... qualitative data
and quantitative data. A variety of methods exist for summarizing and describing these two
types of data. The tree-diagram below presents an outline of the various techniques
TYPES OF DATA
Qualitative
Quantitative
Univariate
Bivariate Frequency
Discrete
Continuous
Frequency
Table
Table
Frequency
Frequency
Distribution
Distribution
Percentages
Component Multiple
Line Chart
Histogram
Pie Chart
Bar Chart
Frequency
Polygon
Bar Chart
Frequency Curve
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MTH001 ­ Elementary Mathematics
In today's lecture, we will be dealing with various techniques for summarizing and describing
qualitative data.
Qualitative
Univariate
Bivariate Frequency
Frequency
Table
Table
Percentages
Component
Multiple
Bar Chart
Bar Chart
Pie Chart
Bar Chart
We will begin with the univariate situation, and will proceed to the bivariate situation.
EXAMPLE:
Suppose that we are carrying out a survey of the students of first year studying in a
co-educational college of Lahore. Suppose that in all there are 1200 students of first year in
this large college.
We wish to determine what proportion of these students have come from Urdu medium
schools and what proportion has come from English medium schools.
So we will interview the students and we will inquire from each one of them about their
schooling.
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MTH001 ­ Elementary Mathematics
As a result, we will obtain a set of data as you can now see on the screen.
We will have an array of observations as follows:
U, U, E, U, E, E, E, U, ......
(U : URDU MEDIUM)
(E : ENGLISH MEDIUM)
Now, the question is what should we do with this data?
Obviously, the first thing that comes to mind is to count the number of students who said
"Urdu medium" as well as the number of students who said "English medium".
This will result in the following table:
Medium of
No. of Students
Institution
(f)
Urdu
719
English
481
1200
The technical term for the numbers given in the second column of this table is "frequency".
It means "how frequently something happens?"
Out of the 1200 students, 719 stated that they had come from Urdu medium schools.
So in this example, the frequency of the first category of responses is 719 whereas the
frequency of the second category of responses is 481.
It is evident that this information is not as useful as if we compute the proportion or
percentage of students falling in each category.
Dividing the cell frequencies by the total frequency and multiplying by 100 we obtain
the following:
Medium of
f
%
Institution
719
59.9 = 60%
Urdu
481
40.1 = 40%
English
1200
What we have just accomplished is an example of a univariate frequency table pertaining to
qualitative data.
Let us now see how we can represent this information in the form of a diagram.
One good way of representing the above information is in the form of a pie chart.
A pie chart consists of a circle which is divided into two or more parts in accordance with the
number of distinct categories that we have in our data.
For the example that we have just considered, the circle is divided into two sectors, the
larger sector pertaining to students coming from Urdu medium schools and the smaller
sector pertaining to students coming from English medium schools.
How do we decide where to cut the circle?
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MTH001 ­ Elementary Mathematics
The answer is very simple! All we have to do is to divide the cell frequency by the total
frequency and multiply by 360.
This process will give us the exact value of the angle at which we should cut the circle.
PIE CHART
Medium of
f
Angle
Institution
215.70
719
Urdu
144.30
481
ENGLISH
1200
Urdu
215.70
English
144.30
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MTH001 ­ Elementary Mathematics
SIMPLE BAR CHART:
The next diagram to be considered is the simple bar chart.
A simple bar chart consists of horizontal or vertical bars of equal width and lengths
proportional to values they represent.
As the basis of comparison is one-dimensional, the widths of these bars have no
mathematical significance but are taken in order to make the chart look attractive.
Let us consider an example.
Suppose we have available to us information regarding the turnover of a company
for 5 years as given in the table below:
Years
1965
1966
1967
1968
1969
Turnover
35,000
42,000
43,500
48,000
48,500
(Rupees)
In order to represent the above information in the form of a bar chart, all we have to do is to
take the year along the x-axis and construct a scale for turnover along the y-axis.
Next, against each year, we will draw vertical bars of equal width and different
50,000
40,000
30,000
20,000
10,000
0
1965
1966
1967
1968
1969
heights in accordance with the
turn-over figures that we have in our table.
As a result we obtain a simple and attractive diagram as shown below.
When our values do not relate to time, they should be arranged in ascending or descending
order before-charting.
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MTH001 ­ Elementary Mathematics
BIVARIATE FREQUENCY TABLE:
What we have just considered was the univariate situation.
In each of the two examples, we were dealing with one single variable.
In the example of the first year students of a college, our lone variable of interest was
`medium of schooling'.
50,000
40,000
30,000
20,000
10,000
0
1965
1966
1967
1968
1969
And in the second example, our one single variable of interest was turnover.
Now let us expand the discussion a little, and consider the bivariate situation.
Going back to the example of the first year students, suppose that alongwith the
enquiry about the Medium of Institution, you are also recording the sex of the
student.
Suppose that our survey results in the following information:
Student No.
Medium
Gender
1
U
F
2
U
M
3
E
M
4
U
F
5
E
M
6
E
F
7
U
M
8
E
M
:
:
:
:
:
:
Now this is a bivariate situation; we have two variables, medium of schooling and sex of the
student.
In order to summarize the above information, we will construct a table containing a box head
and a stub as shown below:
Sex
MALE
Female
Total
Med.
Urdu
English
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Total
The top row of this kind of a table is known as the boxhead and the first column of
the table is known as stub.
Next, we will count the number of students falling in each of the following four categories:
1.
Male student coming from an Urdu medium school.
2.
Female student coming from an Urdu medium school.
3.
Male student coming from an English medium school.
4.
Female student coming from an English medium school.
As a result, suppose we obtain the following figures:
Sex
MALE
Female
Total
Med.
202
517
719
Urdu
350
131
481
English
552
648
1200
Total
What we have just accomplished is an example of a bivariate frequency table pertaining to
two qualitative variables.
COMPONENT BAR CHAR:
Let us now consider how we will depict the above information diagrammatically.
This can be accomplished by constructing the component bar chart (also known as the
subdivided bar chart) as shown below:
Urdu
English
800
700
600
500
400
300
200
100
0
Male
Female
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MTH001 ­ Elementary Mathematics
In the above figure, each bar has been divided into two parts. The first bar represents the
total number of male students whereas the second bar represents the total number of
female students.
As far as the medium of schooling is concerned, the lower part of each bar
represents the students coming from English medium schools. Whereas the upper part of
each bar represents the students coming from the Urdu medium schools.The advantage of
this kind of a diagram is that we are able to ascertain the situation of both the variables at a
glance.
We can compare the number of male students in the college with the number of
female students, and at the same time we can compare the number of English medium
students among the males with the number of English medium students among the females.
MULTIPLE BAR CHART
The next diagram to be considered is the multiple bar chart.
Let us consider an example.
Suppose we have information regarding the imports and exports of Pakistan for the
years 1970-71 to 1974-75 as shown in the table below:
Imports
Exports
Years
(Crores of Rs.)
(Crores of Rs.)
1970-71
370
200
1971-72
350
337
1972-73
840
855
1973-74
1438
1016
1974-75
2092
1029
Source: State Bank of Pakistan
A multiple bar chart is a very useful and effective way of presenting this kind of information.
This kind of a chart consists of a set of grouped bars, the lengths of which are
proportionate to the values of our variables, and each of which is shaded or coloured
differently in order to aid identification.
With reference to the above example, we obtain the multiple bar chart shown below:
Multiple Bar Chart Showing
Imports & Exports
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MTH001 ­ Elementary Mathematics
of Pakistan 1970-71 to 1974-75
This is a very good device for the comparison of two different kinds of information.
2500
2000
1500
Imports
1000
Exports
500
0
If, in addition to information regarding imports and exports, we also had information
regarding production, we could
have compared them from year to year by grouping the three bars together.
The question is, what is the basic difference between a component bar chart and a multiple
bar chart?
The component bar chart should be used when we have available to us information
regarding totals and their components.
For example, the total number of male students out of which some are Urdu medium
and some are English medium. The number of Urdu medium male students and the number
of English medium male students add up to give us the total number of male students.
On the contrary, in the example of exports and imports, the imports and exports do not add
up to give us the totality of some one thing!
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Table of Contents:
  1. Recommended Books:Set of Integers, SYMBOLIC REPRESENTATION
  2. Truth Tables for:DE MORGANíS LAWS, TAUTOLOGY
  3. APPLYING LAWS OF LOGIC:TRANSLATING ENGLISH SENTENCES TO SYMBOLS
  4. BICONDITIONAL:LOGICAL EQUIVALENCE INVOLVING BICONDITIONAL
  5. BICONDITIONAL:ARGUMENT, VALID AND INVALID ARGUMENT
  6. BICONDITIONAL:TABULAR FORM, SUBSET, EQUAL SETS
  7. BICONDITIONAL:UNION, VENN DIAGRAM FOR UNION
  8. ORDERED PAIR:BINARY RELATION, BINARY RELATION
  9. REFLEXIVE RELATION:SYMMETRIC RELATION, TRANSITIVE RELATION
  10. REFLEXIVE RELATION:IRREFLEXIVE RELATION, ANTISYMMETRIC RELATION
  11. RELATIONS AND FUNCTIONS:FUNCTIONS AND NONFUNCTIONS
  12. INJECTIVE FUNCTION or ONE-TO-ONE FUNCTION:FUNCTION NOT ONTO
  13. SEQUENCE:ARITHMETIC SEQUENCE, GEOMETRIC SEQUENCE:
  14. SERIES:SUMMATION NOTATION, COMPUTING SUMMATIONS:
  15. Applications of Basic Mathematics Part 1:BASIC ARITHMETIC OPERATIONS
  16. Applications of Basic Mathematics Part 4:PERCENTAGE CHANGE
  17. Applications of Basic Mathematics Part 5:DECREASE IN RATE
  18. Applications of Basic Mathematics:NOTATIONS, ACCUMULATED VALUE
  19. Matrix and its dimension Types of matrix:TYPICAL APPLICATIONS
  20. MATRICES:Matrix Representation, ADDITION AND SUBTRACTION OF MATRICES
  21. RATIO AND PROPORTION MERCHANDISING:Punch recipe, PROPORTION
  22. WHAT IS STATISTICS?:CHARACTERISTICS OF THE SCIENCE OF STATISTICS
  23. WHAT IS STATISTICS?:COMPONENT BAR CHAR, MULTIPLE BAR CHART
  24. WHAT IS STATISTICS?:DESIRABLE PROPERTIES OF THE MODE, THE ARITHMETIC MEAN
  25. Median in Case of a Frequency Distribution of a Continuous Variable
  26. GEOMETRIC MEAN:HARMONIC MEAN, MID-QUARTILE RANGE
  27. GEOMETRIC MEAN:Number of Pupils, QUARTILE DEVIATION:
  28. GEOMETRIC MEAN:MEAN DEVIATION FOR GROUPED DATA
  29. COUNTING RULES:RULE OF PERMUTATION, RULE OF COMBINATION
  30. Definitions of Probability:MUTUALLY EXCLUSIVE EVENTS, Venn Diagram
  31. THE RELATIVE FREQUENCY DEFINITION OF PROBABILITY:ADDITION LAW
  32. THE RELATIVE FREQUENCY DEFINITION OF PROBABILITY:INDEPENDENT EVENTS