virtual university of pakistan lecture no. 3 statistics and probability by: miss saleha naghmi...
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Virtual University of Pakistan
Lecture No. 3 Statistics and Probability
By:Miss Saleha Naghmi Habibullah
IN THE LAST LECTURE, IN THE LAST LECTURE, YOU LEARNT:YOU LEARNT:
Concept of samplingConcept of sampling Random versus non-random Random versus non-random
samplingsampling Simple random samplingSimple random sampling A brief introduction to other A brief introduction to other
types of random samplingtypes of random sampling Methods of data collectionMethods of data collection
TOPICS FOR TODAYTOPICS FOR TODAY
Data RepresentationData Representation TabulationTabulation Simple bar chartSimple bar chart Component bar chartComponent bar chart Multiple bar chartMultiple bar chart Pie chartPie chart
The tree-diagram below presents an outline of the various techniques
TYPES OF DATA
QuantitativeQualitative
UnivariateFrequency
Table
Percentages
Pie Chart
Bar Chart
Bivariate Frequency
Table
MultipleBar
Chart
Discrete
Frequency Distribution
Line Chart
Continuous
Frequency Distribution
Histogram
Frequency Polygon
Frequency Curve
Component Bar Chart
In today’s lecture, we will be dealing with various techniques for summarizing and describing qualitative data.
Qualitative
UnivariateFrequency
Table
Percentages
Pie Chart
Bar Chart
Bivariate Frequency
Table
MultipleBar Chart
Component Bar Chart
We will begin with the univariate situation, and will proceed to the bivariate situation.
Suppose that we are carrying out a survey of the students of first year studying in a co-education. Suppose that in all there are 1200 students of first year in this large college. We wish to determine
What proportion of students have come from Urdu medium schools?
What proportion has come from English medium schools?
Example
Interview Results Interview Results We will have an array of observations as follows: U, U, E, U, E, E, E, U, ……
(U : URDU MEDIUM) (E : ENGLISH
MEDIUM)Question:
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 ofMedium ofInstitutionInstitution
No. of StudentsNo. of Students(f)(f)
UrduUrdu 719719
EnglishEnglish 481481
TotalTotal 12001200
Important: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.
Dividing the cell frequencies by the total frequency and multiplying by 100 we obtain the following:
Medium ofMedium ofInstitutionInstitution
ff %%
UrduUrdu 719719 59.9 = 60%59.9 = 60%
EnglishEnglish 481481 40.1 = 40%40.1 = 40%
12001200
100frequency
PercentageTotal No. of Students
Diagrammatical Representation of DataDiagrammatical Representation of Data
A pie chart consists of a circle which is divided into twoor more parts in accordance with the number of distinctcategories that we have in our data.
Medium Medium ofof
InstitutionInstitutionff AngleAngle
UrduUrdu 719719 215.7215.700
ENGLISHENGLISH 481481 144.3144.300
12001200
English40% Urdu
60%
Cell FrequencyDivision of Circle = 360
Total Frequency
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?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.
Diagrammatical Representation of DataDiagrammatical Representation of Data
SIMPLE BAR CHART
A simple bar chart consists of horizontal or vertical bars of equal width and lengths proportional to values they represent.
Example Example
Suppose we have available to us information regarding theturnover of a company for 5 years as given in the tablebelow:
YearsYears 19651965 19661966 19671967 19681968 19691969
TurnoverTurnover(Rupees)(Rupees)
35,00035,000 42,00042,000 43,50043,500 48,00048,000 48,50048,500
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.
0
10,000
20,000
30,000
40,000
50,000
1965 1966 1967 1968 1969
Next, against each year, we will draw vertical bars of equal width and different 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.
0
10,000
20,000
30,000
40,000
50,000
1965 1966 1967 1968 1969
When our values do not relate to time, they should be arranged in ascending or descending order before-charting.
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 alone variable of interest was ‘medium of schooling’. And in the second example, our one single variable of interest was turnover.
Example Example
Suppose that along with the enquiry Suppose that along with the enquiry
about the about the Medium of InstitutionMedium of Institution we are we are
also also recording the sexrecording the sex of the student. of the student.
Student No.Student No. MediumMedium GenderGender
11 UU FF
22 UU MM
33 EE MM
44 UU FF
55 EE MM
66 EE FF
77 UU MM
88 EE MM
:: :: ::
:: :: ::
Now this is a bivariate situation; we have two variables, medium of schooling and sex of the student.
Bivariate Frequency TableIn order to summarize the above information, we will construct a table called Bivariate Frequency Table, containing a boxhead and a stub as shown below:
SexSexMed.Med.
MaleMale FemaleFemale TotalTotal
UrduUrdu
EnglishEnglish
TotalTotal
Box Head
Stub
Next, we will count the number of students falling in each of the following four categories:
• Male student coming from an Urdu medium school.
• Female student coming from an Urdu medium school.
• Male student coming from an English medium school.
• Female student coming from an English medium school.
As a result, suppose we obtain the following figures:
SexSexMed.Med.
MaleMale FemaleFemale TotalTotal
UrduUrdu 202202 517517 719719
EnglishEnglish 350350 131131 481481
TotalTotal 552552 648648 12001200
Bivariate Frequency Table pertaining to two qualitative variables.
Let us now consider how we will depictthe above information diagrammatically
This can be accomplish by constructing the componentcomponent bar chart COMPONENT BAR CHARTcomponent bar chart is also known as the subdivided bar chart.
0
100
200
300
400
500
600
700
800
Male Female
Urdu
English
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.
The next diagram to be considered is the Multiple Bar Chart
MULTIPLE BAR CHARTUsed in a situation where we have two or more related sets of data.
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:
YearsYearsImportsImports
(Crores of Rs.)(Crores of Rs.)ExportsExports
(Crores of Rs.)(Crores of Rs.)
1970-711970-71 370370 200200
1971-721971-72 350350 337337
1972-731972-73 840840 855855
1973-741973-74 14381438 10161016
1974-751974-75 20922092 10291029
Source: State Bank of Pakistan
A A multiple multiple bar chart is a very useful andbar chart is a very useful andeffective way of presenting this kind ofeffective way of presenting this kind ofinformation.information.This kind of a chart consists of a set ofThis kind of a chart consists of a set ofgroupedgrouped bars, the lengths of which are bars, the lengths of which areproportionate to the values of ourproportionate to the values of ourvariables, and each of which is shaded orvariables, and each of which is shaded orcolored differently in order to aidcolored differently in order to aididentification.identification. With reference to the above example, weWith reference to the above example, weobtain the multiple bar chart shown ahead:obtain the multiple bar chart shown ahead:
0
500
1000
1500
2000
2500
1970-71 1971-72 1972-73 1973-74 1974-75
Imports
Exports
Multiple Bar Chart representing Imports & Exports of Pakistan ( 1970 - 71 to 1974 - 75)
Difference between Component Bar Chart Difference between Component Bar Chart
and Multiple Bar Chartand Multiple Bar Chart
Information available regarding Totals and their componentsFor Example:Total no. of male studentsi.e. English Medium and Urdu Medium
No Information regarding TotalsFor example:Imports and Exports do not addup to give you the totality ofsome one thing.
Component Bar Chart Multiple Bar Chart
Quantitative VariableQuantitative Variable
Quantitative Variable
Discrete Variable
Continuous Variable
• Frequency Distribution• Line Chart
• Frequency Distribution• Histogram• Frequency Polygon• Ogive
Example Example Suppose we walk in the nursery Suppose we walk in the nursery
class of a school and we count the no. class of a school and we count the no. of Books and copies that students have of Books and copies that students have in their bags. in their bags.
Suppose the no. of books and copies are Suppose the no. of books and copies are
3, 5, 7, 9 and so on. 3, 5, 7, 9 and so on.
Representation of Data in a Representation of Data in a Discrete Frequency DistributionDiscrete Frequency Distribution
XX TallyTally FrequencyFrequency
33 || 11
44 |||||| 33
55 |||| |||||||| |||| 99
66 |||| |||| ||||||| |||| ||| 1313
77 |||| |||||||| |||| 1010
88 |||||| 33
99 |||| ||||| | 66
TotalTotal 4545
Graphical Representation of Graphical Representation of Discrete DataDiscrete Data
8
10
12
2
4
6
03 4 5 6 7 8
X
14
9
No. of books and copies
No.
of
stu
den
ts
Relative Frequency DistributionRelative Frequency Distribution XX FrequencyFrequency Relative Relative
FrequencyFrequency
33 11 1/45 x 100 = 2.22%1/45 x 100 = 2.22%
44 33 3/45 x 100 = 6.67%3/45 x 100 = 6.67%
55 99 9/45 x 100 = 20%9/45 x 100 = 20%
66 1313 13/45 x 100 = 28.89%13/45 x 100 = 28.89%
77 1010 10/45 x 100 = 22.22%10/45 x 100 = 22.22%
88 33 3/45 x 100 = 6.67%3/45 x 100 = 6.67%
99 66 6/45 x 100 = 13.33%6/45 x 100 = 13.33%
TotalTotal 4545
Cumulative Frequency DistributionCumulative Frequency Distribution
XX FrequencyFrequency Cumulative Cumulative FrequencyFrequency
33 11 11
44 33 1+3 = 41+3 = 4
55 99 4+9 = 134+9 = 13
66 1313 13+13 = 2613+13 = 26
77 1010 26+10 = 3626+10 = 36
88 33 36+3 = 3936+3 = 39
99 66 39+6 = 4539+6 = 45
TotalTotal 4545
IN TODAY’S LECTURE, IN TODAY’S LECTURE, YOU LEARNTYOU LEARNT
Tabular and diagrammatic representation of Quantitative data
univariate Bivariate
Tabular and diagrammatic representation of Discrete Quantitative variable
IN THE NEXT TWO LECTURES, YOU WILL LEARN
Tabular and Diagrammatic representation of a Continuous Quantitative Variable.
Continuous Frequency Distribution Histogram Frequency polygon Frequency curve Cumulative frequency distribution (continuous) Cumulative frequency polygon (Ogive)
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