introduction of quantitave

Introduction to Quantitative Techniques, Statistics and

Presentation of Data.

COM-705 3(3-0) M.COM Quantitative Techniques in Business

Muhammad Salman Arshad(Visiting lecturer UAF)

Chapter No ONE: Introduction to QUANTITATIVE TECHNIQUESMeaning and Definition:

Quantitative techniques may be defined as “those techniques which provides the decision makes a systematic and powerful means of analysis, based on quantitative data”.

It is a scientific method employed for problem solving and

decision making by the management.With the help of quantitative techniques, the decision maker is

able to explore policies for attaining the predetermined objectives.

In short, quantitative techniques are inevitable in decision-making process.

Classification of Quantitative Techniques:There are different types of quantitative techniques. We can classify them into three categories. They are:

1. Mathematical Quantitative Techniques.2. Statistical Quantitative Techniques.3. Programming Quantitative Techniques.

Mathematical Quantitative Techniques:A technique in which quantitative data are used

along with the principles of mathematics is known as mathematical quantitative techniques. Mathematical quantitative techniques involve:

1) Permutations and CombinationsPermutation means arrangement of objects in a definite order. The number of arrangements depends upon the total number of objects and the number of objects taken at a time for arrangement.

Combination means selection or grouping objects without considering their order.

2: Set Theory:-Set theory is a modern mathematical device which solves various types of critical problems. Set theory is the branch of mathematical logic that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. 3: Matrix Algebra:Matrix is an orderly arrangement of certain given numbers or symbols in rows and columns. It is a mathematical device of finding out the results of different types of algebraic operations on the basis of the relevant matrices.

4: Determinants:It is a powerful device developed over the matrix algebra. This device is used for finding out values of different variables connected with a number of simultaneous equations.

5: Differentiation:It is a mathematical process of finding out changes in the dependent variable with reference to a small change in the independent variable. The derivative of a function of a real variable measures the sensitivity to change of a quantity (a function or dependent variable) which is determined by another quantity (the independent variable).

6. Integration:Integration is the reverse process of differentiation.7. Differential Equation:It is a mathematical equation which involves the differential coefficients of the dependent variables.In mathematics, the differential coefficient of a function f(x) is what is now called its derivative.

Statistical Quantitative Techniques:Statistical techniques are those techniques

which are used in conducting the statistical enquiry concerning to certain Phenomenon. They include all the statistical methods beginning from the collection of data till interpretation of those collected data.

Thus statistical techniques involves:

1. Collection of data:One of the important statistical methods is collection of data. There are different methods for collecting primary and secondary data.2. Measures of Central tendency,

dispersion, skewness and Kurtosis:Measures of Central tendency is a method used for finding the average of a series.While measures of dispersion is used for finding out the variability in a series.Measures of Skewness measures asymmetry of a distribution while measures of Kurtosis measures the flatness of peakedness in a distribution.

3. Correlation and Regression Analysis:Correlation is used to study the degree of relationship among two or more variables. Regression technique is used to estimate the value of one variable for a given value of another.4. Index Numbers:Index numbers measure the fluctuations in various Phenomena like price, production etc. over a period of time, They are described as economic barometers.5. Time series Analysis:Analysis of time series helps us to know the effect of factors which are responsible for changes.

6. Interpolation and Extrapolation:Interpolation is the statistical technique of estimating under certain assumptions, the missing figures which may fall within the range of given figures. Extrapolation provides estimated figures outside the range of given data.7. Statistical Quality ControlStatistical quality control is used for ensuring the quality of items manufactured. The variations in quality because of assignable causes and chance causes can be known with the help of this tool.

8. Ratio Analysis:Ratio analysis is used for analyzing financial statements of any business or industrial concerns which help to take appropriate decisions.9. Probability Theory:Theory of probability provides numerical values of the likely hood of the occurrence of events.10. Testing of Hypothesis:Testing of hypothesis is an important statistical tool to judge the reliability of inferences drawn on the basis of sample studies.

Programming Techniques:Programming techniques are also called operations

research techniques. Programming techniques are model building techniques used by decision makers in modern times.

It includes:Linear Programming:Linear programming technique is used in finding a solution for optimizing a given objective under certain constraints.Queuing Theory:Queuing theory deals with mathematical study of queues. It aims at minimizing cost of both servicing and waiting.

Game Theory:Game theory is used to determine the optimum strategy in a competitive situation. Decision Theory:This is concerned with making sound decisions under conditions of certainty, risk and uncertainty.Inventory Theory:Inventory theory helps for optimizing the inventory levels. It focuses on minimizing cost associated with holding of inventories.

Net work programming:It is a technique of planning, scheduling, controlling, monitoring and coordinating large and complex projects comprising of a number of activities and events. It serves as an instrument in resource allocation and adjustment of time and cost up to the optimum level. It includes CPM, PERT etc.Simulation:It is a technique of testing a model which resembles a real life situationsReplacement Theory:It is concerned with the problems of replacement of machines, etc. due to their deteriorating efficiency or breakdown. It helps to determine the most economic replacement policy.

Non Linear Programming:It is a programming technique which involves finding an optimum solution to a problem in which some or all variables are non-linear.Sequencing:Sequencing tool is used to determine a sequence in which given jobs should be performed by minimizing the total efforts.Quadratic Programming:Quadratic programming technique is designed to solve certain problems, the objective function of which takes the form of a quadratic equation.

Branch and Bound TechniqueIt is a recently developed technique. This is designed to solve the combinational problems of decision making where there are large number of feasible solutions. Problems of plant location, problems of determining minimum cost of production etc. are examples of combinational problems.

Functions of Quantitative Techniques:

The following are the important functions of quantitative techniques:

1. To facilitate the decision-making process2. To provide tools for scientific research3. To help in choosing an optimal strategy4. To enable in proper deployment of resources5. To help in minimizing costs6. To help in minimizing the total processing time required for performing a set of jobs

Use of Quantitative Techniques in BusinessQuantitative techniques render valuable

services in the field of business and industry.

Today, all decisions in business and industry are made with the help of quantitative techniques.Some important uses of quantitative techniques in the field of business and industry are given below:1. Quantitative techniques of linear

programming is used for optimal allocation of scarce resources in the problem of determining product mix.

2. Inventory control techniques are useful in dividing when and how much items are to be purchase so as to maintain a balance between the cost of holding and cost of ordering the inventory3. Quantitative techniques of CPM, and PERT(Project evaluation review technique) helps in determining the earliest and the latest times for the events and activities of a project. This helps the management in proper deployment of resources.4. Decision tree analysis and simulation technique help the management in taking the best possible course of action under the conditions of risks and uncertainty.

5. Queuing theory is used to minimize the cost of waiting and servicing of the customers in queues.6. Replacement theory helps the management in determining the most economic replacement policy regarding replacement of an equipment.Limitations of Quantitative Techniques:Even though the quantitative techniques are

inevitable in decision-making process, they are not free from short comings. The following are the important limitations of quantitative techniques:

1. Quantitative techniques involves mathematical models, equations and other mathematical expressions.

2. Quantitative techniques are based on number of assumptions. Therefore, due care must be ensured while using quantitative techniques, otherwise it will lead to wrong conclusions.

3. Quantitative techniques are very expensive.4. Quantitative techniques do not take into

consideration intangible facts like skill, attitude etc.

5. Quantitative techniques are only tools for analysis and decision-making. They are not decisions itself.

Chapter Two: INTRODUCTION TO STATISTICS

24

Statistics is a science, pure and applied, of creating, developing and applying techniques such that uncertainty of inductive inferences may be evaluated. OR

Statistics is a Mathematical Science of making decisions and drawing conclusions from data in situations of uncertainty.

25

STATICTICS DEALS WITH

• Collection and summarization of data.

• Designing experiments and sample survey.

• Measuring the magnitude of variation. • Estimating population parameters.

• Testing hypothesis about populations.

• Studying relationships among two or more variables.

• Forecasting.

26

Population:- The collection of all possible observations whether finite or infinite, relevant to some characteristic of interest is called a population. The number of observations in a finite population is called size of the population and is denoted by N.

INTRODUCTORY STATISTICS

Sample: A sample is a part of a population. Generally it consists of some of the observation. The number of observations in a sample is called size of the sample and is denoted by n.

27


Observation: The numerically recording of information is called observation/datum.

Data: The set of observations is called data.

Variable: A characteristics that varies from individual to individual is called a variable. For example age, plant height, weight, no of plants per plot etc are variables as they vary from individual to individual.

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Fixed or Mathematical Variable: A variable may be fixed or Mathematical when its value can be determined before hand. e.g. amount of fertilizer to be applied to a plot, amount of insecticide applied to control insect pests.

Random Variable: A variable may be random when its value cannot be exactly determined. e.g. yield from a plot Constant: Quantity which do not vary from individual toIndividual is called constant. e.g. e= 2.71828 , = 3.145

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Types of variables:

(1):- Quantitative variable (2):- Qualitative variable.

Quantitative variable:- A variable is called Quantitative variable when a characteristic can be expressed numerically such as weight, income, number of children.

Qualitative variable:- If a characteristic is non-numerical such as gender, colour education etc. the variable is called Qualitative variable.

30


Types of Quantitative variable

1:- Discrete variable 2:- Continuous variable

Discrete variable:- A variable which can assume some specific values within a given range is called a discontinuous or discrete variable. e.g. number of trees in a field, number of leaves in a tree. A discrete variable takes on values which are integers or whole numbers.Continuous variable:- A variable which can assume any value (fractional or integral) within a given range is called a continuous variable. For example Height of a plant, the temperature at a place.

32


Scales of Measurement

Measurement: Measurement refer to “Assigning of number to observations or objects.Scaling: Scaling is a process of measuring.

Four Scales of Measurements

1. Nominal Scale2. Ordinal Scale3. Interval Scale4. Ratio Scale

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Nominal Scale (Weakest form of measurement)

The classification or grouping of the observations into mutually exclusive qualitative categories or classes is said to constitute a nominal scale.

e.g. Sex , Race, Religion, Country

Rainfall may be classified as • Heavy• Moderate• Light

34


Ordinal Scale (When numbers are allocated in some order)

It includes the characteristics of nominal scale and in addition has a property of ordering or ranking of measurements.

• Attitude scale Strongly agree, agree, disagree • Social scale Upper, middle, lower

• Performance of players Excellent, good, fair, poor

35


Interval Scale

It has separate categories, like nominal scales and also hasordered categories like ordinal scales. But the interval measurements has no true zero point.• Temperature ( 80oF and 40oF is 26.7oC and4.4oC)

Ratio Scale

In this scale the intervals are consistent along the entire scale. The ratio measurements has true zero point. • Height of plant, weight of students, volume, length,

Chapter No Three: PRESENTATION OF DATA

36

When the suitable statistical data have been collected, the next step is the reduction or presentation of the data so that valid inferences can be drawn.

Methods for the presentation of data Frequency DistributionGraphical presentationStem and Leaf display

Tabular and Graphical Procedures

37

Data

Qualitative DataQualitative Data Quantitative Data

TabularMethods

TabularMethods

Graphical Methods

Graphical Methods

•Frequency Distribution•Rel. Freq. Dist.•% Freq. Dist.•Crosstabulation

•Bar Graph1. Simple2. Multiple3. Subdivided•Pie Chart

•Frequency Distribution•Rel. Freq. Dist.•Cum. Freq. Dist.•Cum. Rel. Freq. Distribution •Stem-and-Leaf Display•Cross Tabulation

•Dot Plot•Histogram•Ogive•Scatter Diagram

Quantitative Data

38

Quantitative Data

Discrete Data

• Frequency Distribution• Line Chart

Continuous Data

• Frequency Distribution• Histogram• Frequency Polygon• Ogive

Example Suppose we walk in the nursery

class of a school and we count the no. of Books and copies that students have in their bags.

Suppose the no. of books and copies are 3, 5, 7, 9 and so on.

39

Representation of Data in a Discrete Frequency DistributionX Tally Frequency3 | 14 ||| 35 |||| |||| 96 |||| |||| ||| 137 |||| |||| 108 ||| 39 |||| | 6

Total 4540

Graphical Representation of Discrete Data

41

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

Quantitative Data. Continuous DataFrequency Distribution

42

Definition

A frequency distribution is a tabular arrangement of the data, which shows the distribution of observations among different classes.

The number of observations falling in a particular class is referred to as class frequency or simply frequency and is denoted by "f".

In frequency distribution all the values falling in a class are assumed to be equal to the midpoint of that class.

Data presented in the form of a frequency distribution is also called grouped data. Data which have not been arranged in a systematic order are called raw data or ungrouped data.

Frequency Distribution

43

CLASS LIMITS: The class limits are defined as the number or the values of the variables which are used to separate two classes. The smaller number is called lower class limit and larger number is called upper class limit. For discrete variables, class boundaries are the same as the class limits. Sometimes classes are taken as 20--25, 25--30 etc In such a case, these class limits means " 20 but less than 25", "25 but less than 30" etc

CLASS MARKS OR MIDPOINTS: The class mark or the midpoint is that value which divides a class into two equal parts. It is obtained by dividing the sum of lower and upper class limits or class boundaries of a class by 2.

CLASS INTERVAL: Class interval is the length of a class. It is obtained by

(i)The difference between the upper class boundary and the lower class boundary.(Not the difference between class limits) OR

(ii)The difference between either two successive lower class limits or two successive upper class limits. OR

(iii) The difference between two successive midpoints.

A uniform class interval is usually denoted by "h".

Steps for Constructing a Frequency Distribution

44

Decide the number of classes: The number of classes is determine by the formula i.e. K=1+3.3 log(n)

Where K denotes the number of classes and n denotes the total number of observations.

Determine the range of variation of the data: The difference between the largest and smallest values in the data is called the range of the data. i.e. R = largest observation - smallest observation

Where R denote the range of the data. Determine the approximate size of class interval: The size of the

class interval is determine by dividing the range of the data by the number of classes i.e. h= R/K

Where h denotes the size of the class interval. In case of fractional results the next higher whole number is usually taken as the size of the class interval.

Steps for Constructing a Frequency Distribution

45

Decide where to locate the class limits: The lower class limit of the first class is started just below the smallest value in the data and then add class interval to get lower class limit of the next class, repeat this process until the lower class limit of the last class is achieved.

Distribute the data into appropriate classes: Take an observation and marked a vertical bar "I"(Tally) against the class it belongs.

Count the tally bars for each class and make a column for frequency

Case.1 Continuous Data.

46

Example.The following data is the Sale ($1000) of

thirty Stores of Faisalabad City. Construct a frequency distribution by considering all the steps involved.87 91 89 88 89 91 87 92 96

98 99 98 100 102 99 101 90 9895 97 96 100 101 105 103 107 105106 107 112

Solution

47

Step-1 : Number values in the data set=n

n = 30 Step-2 : Find maximum & minimum values

Max value = =112 Min value= = 87

Range: R = - = 112-87 =25 Step-3

Number of classes: K = 1+3.3 log(n)

= 1+3.3 log(30)=5.87 F 6 Step-4

Size of class interval: h = R/K

= 25/6=4.5 F 5

0Xm X0XXm

FREQUENCY DISTRIBUTION

48

Class limits Class

boundaries

Tally Frequency

F

Midpoin

t

X

c.f

F

r.f %

frequenc

y

Cumulative

% fre

86----90 85.5----90.5 IIII I 6 88 6 0.2000 20.00 20.00

91----95 90.5----95.5 IIII 4 93 10 0.1333 13.33 33.33

96----100 95.5----100.5 IIII IIII 10 98 20 0.3333 33.33 66.66

101----105 100.5----105.5 IIII I 6 103 26 0.2000 20.00 86.66

106----110 105.5----110.5 III 3 108 29 0.1000 10.00 96.66

111----115 110.5----115.5 I 1 113 30 0.0333 3.33 99.99

30 1.0000 100

BIVARIATE FREQUENCY DISTRIBUTION

49

A frequency distribution constructed by taking two variables at a time is called bivariate frequency distribution.

Example

50

Construct a bivariate frequency distribution by taking class interval of size 0.5 feet for heights and a class interval of size 5 kilograms for weights.

Height:

5.5 5.0 4.3 5.3 4.9 5.9 5.4 4.8

5.3 5.8 5.3 5.7 5.8 5.9 4.8 5.3

5.1 5.7 4.7 4.5 5.3 4.6 5.4 5.2

4.7

Weight:

60 55 46 67 48 69 67 55

57 67 57 65 63 65 49 55

60 65 50 50 60 53 62 59

55

BIVARIATE FREQUENCY DISTRIBUTION

51

Weights(kilograms) 4.0—4.4 4.5—4.9 5.0—5.4 5.5—5.9

45—49 1 2 - -

50---54 - 3 - -

55---59 - 2 5 -

60---64 - - 3 2

65---69 - - 2 5

Types of Graphs for Quantitative Variable

52

Graph of time series or Historigram

Histogram

Frequency polygon & Frequency curve

Cumulative Frequency polygon or Ogive

Percentage cumulative frequency polygon

Historigram

53

Historigram is constructed by taking Time along X-axis and Value of the variable along Y-axis Plot the points Connect the points by straight line segments to get the

Historigram.

Example: The data represent the records of a company’s savings

over the years. Construct a time series plot to represent it.

Year 1950 1951 1952 1953 1954 1955 1956 1957

Saving(Rs)

1000 2000 3000 1900 2300 1200 1500 1100

Historigram

54

1 2 3 4 5 6 7 8

1000

1500

2000

2500

3000

3500

Index

Sa

vin

g(R

s)

Histogram

55

A Histogram consists of a set of adjacent rectangles whose bases are marked off by

Class boundaries along the X-axis

Frequency along Y-axis

Draw rectangles whose height are proportional to the frequencies with respective classes

HISTOGRAMA histogram consists of a set of adjacent

rectangles whose bases are marked off by class boundaries along the X-axis, and whose heights are proportional to the frequencies associated with the respective classes.

Class Limit

Class Boundaries

Frequency

30.0 – 32.9 29.95 – 32.95 2 33.0 – 35.9 32.95 – 35.95 4 36.0 – 38.9 35.95 – 38.95 14 39.0 – 41.9 38.95 – 41.95 8 42.0 – 44.9 41.95 – 44.95 2

Total 30

56

0

2

4

6

8

10

12

14

29.95 32.95 35.95 38.95 41.95 44.95

Miles per gallon

Nu

mb

er

of

Car

s

X

Y

57

0

2

4

6

8

10

12

14

Miles per gallon

Nu

mb

er o

f C

ars

X

Y The frequency of the second

class is 4. Hence we draw a rectangle of

height equal to 4 units against the

second class, and thus obtain the

following situation:

58

0

2

4

6

8

10

12

14

Miles per gallon

Nu

mb

er

of

Car

s

X

YThe frequency of the third class is 14. Hence

we draw a rectangle of height equal to 14

units against the third class, and thus obtain

the following picture:

59

0

2

4

6

8

10

12

14

Miles per gallon

Nu

mb

er

of

Car

s

X

Y

60

0

2

4

68

10

12

14

16

Miles per gallon

Nu

mb

er

of

Ca

rs

X

Y

61

Frequency Distribution

62

For Ford Auto Repair, if we choose six classes:

49.5-59.5 59.5-69.5 69.5-79.5 79.5-89.5 89.5-99.5 99.5-109.5

2 13 16 7 7 5Total 50

C. Boundaries Frequency

Approximate Class Width = (109 - 52)/6 = 9.5 10

50-59 60-69 70-79 80-89 90-99 100-109

Class Interval

Histogram

63

22

44

66

88

1010

1212

1414

1616

1818Fre

qu

en

cy

Fre

qu

en

cy

49.5-59.5 -69 .-79 .5 -89.5 -99 .5 -109.5 49.5-59.5 -69 .-79 .5 -89.5 -99 .5 -109.5

Tune-up Parts Cost

PartsCost ($) PartsCost ($)

Histogram (Common categories)

64

SymmetricLeft tail is the mirror image of the right tailExamples: heights and weights of people

Rela

tive F

req

uen

cyR

ela

tive F

req

uen

cy

.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

00

Histogram

65

Moderately Skewed LeftA longer tail to the leftExample: exam scores

Rela

tive F

req

uen

cyR

ela

tive F

req

uen

cy

.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

00

Histogram

66

Moderately Right SkewedA Longer tail to the rightExample: housing values

Rela

tive F

req

uen

cyR

ela

tive F

req

uen

cy

.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

00

Histogram

67

Highly Skewed RightA very long tail to the rightExample: executive salaries

Rela

tive F

req

uen

cyR

ela

tive F

req

uen

cy

.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

00

This diagram is known as the histogram,

and it gives an indication of the overall pattern of

our frequency distribution.

Next, we consider another graph which is

called frequency polygon.

68

FREQUENCY POLYGON

A frequency polygon is obtained by plotting the class frequencies against the mid-points of the classes, and connecting the points so obtained by straight line segments.

In our example of the EPA mileage ratings, the classes were:

ClassBoundaries

Mid-Point(X)

Frequency(f)

26.95 – 29.95 28.4529.95 – 32.95 31.45 232.95 – 35.95 34.45 435.95 – 38.95 37.45 1438.95 – 41.95 40.45 841.95 – 44.95 43.45 244.95 – 47.95 46.45

69

02468

10121416

28.4

531

.45

34.4

537

.45

40.4

543

.45

46.4

5

Miles per gallon

Nu

mb

er

of

Car

s

X

Y

70

TYPES OF FREQUENCY Distribution

71

Symmetrical distributionA frequency distribution or curve is symmetrical if values

equidistant from a central maximum have the same frequencies.

Skewed distributionA frequency distribution or curve is skewed when it

departs from symmetry.

Cumulative Distributions

72

Ford Auto Repair

< 49.5 < 59.5 < 69 .5 < 79.5 < 89.5 < 99.5< 109.5

Cost ($) CumulativeFrequency

CumulativeRelativeFrequency

CumulativePercent Frequency

0 2 15 31 38 45 50

0 .04 .30 .62 .76 .90 1.00

0 4 30 62 76 90 100

2 +13

15/50 .30(100)

Ogive

An ogive is a graph of a cumulative distribution. The data values are shown on the horizontal axis. Shown on the vertical axis are the:• cumulative frequencies, or• cumulative relative frequencies, or• cumulative percent frequencies

The frequency (one of the above) of each class is plotted as a point.

The plotted points are connected by straight lines.

73

• Because the class limits for the parts-cost data are 50-59, 60-69, and so on, there appear to be one-unit gaps from 59 to 60, 69 to 70, and so on.

Ogive

• These gaps are eliminated by plotting points halfway between the class limits.

• Thus, 59.5 is used for the 50-59 class, 69.5 is used for the 60-69 class, and so on.

Ford Auto Repair

74


1010

2020

3030

4040

50 50

Cu

mu

lati

ve P

erc

en

t Fr

eq

uen

cyC

um

ula

tive P

erc

en

t Fr

eq

uen

cy

49.5 <59.5 <69.5 <79.5 <89.5 <99.5 <109.549.5 <59.5 <69.5 <79.5 <89.5 <99.5 <109.5

(89.5, 38)

Ogive with Cumulative Frequencies

Tune-up Parts CostTune-up Parts Cost

75


2020

4040

6060

8080

100100

Cu

mu

lati

ve P

erc

en

t Fr

eq

uen

cyC

um

ula

tive P

erc

en

t Fr

eq

uen

cy

50 <59.5 <69.5 <79.5 <89.5 <99.5 <109.550 <59.5 <69.5 <79.5 <89.5 <99.5 <109.5

(89.5, 76)

Ogive with Cumulative Percent Frequencies

Tune-up Parts CostTune-up Parts Cost

76

Scatter Diagram

77

Scatter diagram is used to see the relationship between variables.

Example:

X Y

8 18

10 17

11 23

13 27

15 30

20 35

201510

35

25

15

X

Y