This document provides an overview of key concepts in statistics including measures of central tendency, measures of dispersion, probability, correlation, time series analysis, and network planning methods like CPM and PERT. It defines terms like mean, median, mode, standard deviation, variance, probability, random experiment, correlation, time series, network, critical path, slack time, and differences between CPM and PERT. It also lists advantages and limitations of using PERT/CPM for project management.

1. Module -1
Introduction to Statistics 9 hours
Introduction to Statistics: Meaning and Definition, functions, scope and limitations, Collection
and presentation of data, frequency distribution, measures of central tendency - Mean, Median,
Mode, Geometric mean, Harmonic mean. Measures of dispersion: Range – Quartile Deviation –
Mean Deviation -Standard Deviation – Variance-Coefficient of Variance - Comparison of
various measures of Dispersion.
WHAT IS STATISTICS? ENUMERATE THE IMPORTANCE AND LIMITATION OF IT.
“Statistics may be defined as the collection, presentation, analysis and interpretation of
numerical data”.
WHAT IS THE IMPORTANCE OF STATISTICS?
1. To collect and present facts in a systematic manner.
2. Helps in formulation and testing of hypothesis.
3. Helps in facilitating the comparison of data.
4. Helps in predicting future trends.
5. Helps to find the relationship between variable.
6. Simplifies the mass of complex data.
7. Help to formulate polices.
8. Helps to take decisions.
DEFINE CENTRAL TENDENCY
A measure of central tendency is a single value that attempts to describe a set of data
by identifying the central position within that set of data.
WHAT ARE REQUISITES (ESSENTIALS) OF A GOOD MEASURE OF CENTRAL
TENDENCY?
1. It should be rigidly defined.
2. It should be simple to understand & easy to calculate.
3.It should be based upon all values of given data.
4.It should be capable of further mathematical treatment.
5.It should have sampling stability.
6.It should be not be unduly affected by extreme values.

2. EXPLAIN VARIOUS METHODS (measures) OF CENTRAL TENDENCY
Methods of central tendency.
1. Mean
2. Mode
3. Median
4. Geometric Mean
Harmonic Mean
DEFINE MEAN? WHAT ARE THE MERITS & DEMERITS OF MEAN.
Arithmetic Mean: It is the most common type of measures of central tendency.
It is obtained by dividing the sum of all observation in a series by the total
number of observation. The mean is the arithmetic average of all the observations in
the data.
Merits of Arithmetic Mean:
1. Easy to calculate
2. Based on all observations
3. Capable of further mathematical calculations.
4. It is rigidly defined.
5. It is easy to understand & easy to calculate.
6. It is based upon all values of the given data.
7. It is capable of further mathematical treatment.
8. It is not much affected by sampling fluctuations.
Demerits:
1. Affected by extreme values.
2. Cannot be calculated in open-end series.
3. Cannot be graphically determined.
4. Sometimes misleading or absurd result
5. It cannot be calculated if any observations are missing.
6. It cannot be calculated for the data with open end classes.
7. It may be number which is not present in the data.
8. It can be calculated for the data representing qualitative characteristic.
DEFINE MEDIAN:

3. The point or the value which divides the data into two equal parts., or when
the data is arranged in numerical order. The median is the middle value of an ordered
set of data.
Merits of Median
1. It is rigidly defined.
2. It is easy to understand & easy to calculate.
3. It is not affected by extreme values.
4. Even if extreme values are not known median can be calculated.
5. It can be located just by inspection in many cases.
6. It can be located graphically.
7. It is not much affected by sampling fluctuations.
8. It can be calculated for data based on ordinal scale
Demerits:
1. Not based on all observations.
2. It requires arrangement of data.
3. Not capable o further algebraic treatment.
DEFINE MODE
The Mode is simply the most frequently occurring observation ( score )in a distribution.
The Mode is the most frequently occurring value in a set of values.
DEFINE QUARTILE:
The values of the variate which divide the total frequency into four equal parts are called
quartiles.
DEFINE DECIBLES
The values of the variate which divide the total frequency into ten equal parts are called
deciles.
DEFINE PERCENTILES
The values of the variate which divide the total frequency into hundred equal
parts, arte called percentiles

4. Define measure of dispersion
Measures of Dispersion
Dispersion in statistics is a way of describing how spread out a set of data is. When a data set
has a large value, the values in the set are widely scattered; when it is small the items in the set
are tightly clustered. Very basically, this set of data has a small value.The measure of
dispersion shows the deviation/scatterings of the data. It tells the variation of the data
from one another and gives a clear idea about the distribution of the data. The measure
of dispersion shows the homogeneity or the heterogeneity of the distribution of the
observations.
Methods of dispersion
1. Range and Mean Deviation
2. Quartiles, Quartile Deviation and Coefficient of Quartile Deviation
3. Standard deviation and Coefficient of Variation
List the Characteristics of Measures of Dispersion
1. A measure of dispersion should be rigidly defined
2. It must be easy to calculate and understand
3. Not affected much by the fluctuations of observations
4. Based on all observations
DEFINE S.D
Standard Deviation: The square root of the variance is known as the standard
deviation
Standard deviation is a measure of the dispersion of a set of data from its mean..
Standard deviation (or S.D.) is the positive square root of the arithmetic
mean of the square deviations of various values from their arithmetic mean M.
DEFINE VARIANCE
Variance is the expectation of the squared deviation of a random variable from its mean.

5. Define probability
It refers to “the chances of occurrence of an event among a large number of
possibilities”
Define Random Experiment:
If an experiment or trial is repeated under the same conditions for any number of times
and it is possible to know the number of outcomes is called as “Random Experiment”.
Define Sample Space:
The set of all possible outcomes of a random experiment is known as “Sample Space”
and denoted by set S.
Define Event:
An ‘event’ is an outcome of a trial meeting a specified set of conditions
Define Exhaustive Events:
The total number of all possible elementary outcomes in a random experiment is known
as‘exhaustive events’.
Define Mutually Exclusive Events:
Events are said to be ‘mutually exclusive’ if the occurrence of an event totally prevents
occurrence of all other events in a trial.

6. Define Equally likely or Equi-probable Events:
Outcomes are said to be ‘equally likely’ if there is no reason to expect one outcome to
occur in preference to another. i.e., among all exhaustive outcomes, each of them has
equal chance of occurrence.
Define Independent Events:
Two or more events are said to be ‘independent’, in a series of a trials if the outcome of
one event is does not affect the outcome of the other event or vise versa.
Example: When a coin is tossed twice, the result of the second toss will in no way
be affected by the result of the first toss.
Explain Bays theorm.
In other words, it is used to calculate the probability of an event based on its association
with another event. Bayes’ Theorem is a way of finding a probability, when we know
certain other probabilities.
For example, If we know that it’s cloudy, than we can easily judge the possibilities of
happening of rain.
For example, if the probability that someone has cancer is related to their age, using
Bayes’ theorem the age can be used to more accurately judge the probability of cancer
than can be done without knowledge of the age.

7. Definition: coorelation
Correlation is the degree of inter-relatedness(relationship) among the two or more
variables. Correlation analysis is a process to find out the degree of relationship
between two or more variables by applying various statistical tools and techniques
It defines how two variables are closely related with each other …
In a distribution if the change in one variable effects a change in the other variable, the
variable are said to be correlated(or there is a correlation between the variables)
Explain Types of Correlation:
The important ways of classifying the correlation are:
1. Positive and Negative
2. Simple , Partial and Multiple
3. Linear and non-Linear.
WRITE A NOTE THE VARIOUS TYPES OF CORRELATION.
Positive correlation: If two related variables are such that when one increases (decreases), the
other also increases (decreases).
If one variable is increasing and with its impact on average other variable is also increasing
that will be positive correlation.
Negative correlation: If two variables are such that when one increases (decreases), the
other decreases (increases) . if one variable is increasing and with its impact on average

8. other variable is also decreasing.
Positive correlation: If two related variables are such that when one increases
(decreases), the other also increases (decreases).
if one variable is increasing and with its impact on average other variable is also
increasing that will be positive correlation.
Negative correlatioIf two variables are such that when one increases (decreases),
the other decreases (increases) . if one variable is increasing and with its impact on
average other variable is also decreasing
Simple correlation
Correlation is said to be simple when only two variables are analyzed.
For example :
Correlation is said to be simple when it is done between demand and supply or we can
say income and expenditure etc
Partial correlation :
When three or more variables are considered for analysis but only two influencing
variables are studied and rest influencing variables are kept constant.
For example :
Correlation analysis is done with demand, supply and income. Where income is kept
constant.
Multiple correlation :
In case of multiple correlation three or more variables are studied simultaneously.
For example :
Rainfall, production of rice and price of rice are studied simultaneously will be known
are multiple correlation
Linear correlation :
If the change in amount of one variable tends to make changes in amount of other
variable bearing constant changing ratio it is said to be linear correlation
Non linear correlation

9. If the change in amount of one variable tends to make changes in amount of other
variable but not bearing constant changing ratio it is said to be non - linear correlation.
Define Time series analysis.
“In other words, the arrangement of data in accordance with their time of occurrence is
a time series. It is the chronological arrangement of data. Here, time is just a way in
which one can relate the entire phenomenon to suitable reference points. Time can be
hours, days, months or years.

10. Ex: Values taken by a variable over time (such as daily sales revenue, weekly orders,
monthly overheads, yearly income) and tabulated or plotted as chronologically ordered
numbers or data points.
List out uses Of Studying Time Series Analysis
1. I t helps us to predict the future behaviour of the variable based on past
experience
2. It is helpful for business planning as it helps in comparing the actual current
performance with the expected one
3. study the past performance and behaviour of the phenomenon or the
variable under consideration.
4. We can compare the changes in the values of different variables at different
times or places, etc.
EXPLAIN THE VARIOUS COMPONENTS FOR TIME SERIES ANALYSIS
The four categories of the components of time series are
 Trend
 Seasonal Variations
 Cyclic Variations
 Random or Irregular movements
Trend
The trend shows the general tendency of the data to increase or decrease during a long
period of time. A trend is a smooth, general, long-term, average tendency. It is not
always necessary that the increase or decrease is in the same direction throughout the
given period of time.It is observable that the tendencies may increase, decrease or are
stable in different sections of time. But the overall trend must be upward, downward or
stable. The population, agricultural production, items manufactured, number of births
and deaths, number of industry or any factory, number of schools or colleges are some
of its example showing some kind of tendencies of movement.

11. Seasonal Variations: The variations in a time series data which operate themselves over
less than a span of one year are the Seasonal Variations. These are the rhythmic forces
which operate in a regular and periodic manner over a span of less than a year. They
have the same or almost the same pattern during a period of 12 months. This variation
will be present in a time series if the data are recorded hourly, daily, weekly, quarterly, or
monthly.
For example, it is commonly observed that the consumption of ice-cream during
summer is generally high and hence an ice-cream dealer’s sales would be higher in
some months of the year while relatively lower during winter months. Employment,
output, exports, etc., are subject to change due to variations in weather. Similarly, the
sale of garments, umbrellas, greeting cards and fire-works are subject to large variations
during festivals like Valentine’s Day, Eid, Christmas, New Year’s, etc. These types of
variations in a time series are isolated only when the series is provided biannually,
quarterly or monthly. These variations come into play either because of the natural
forces or man-made conventions. The various seasons or climatic conditions play an
important role in seasonal variations. Such as production of crops depends on seasons,
the sale of umbrella and raincoats in the rainy season, and the sale of electric fans and
A.C. shoots up in summer seasons.
Cyclic Variations: The variations in a time series which operate themselves over a span
of more than one year are the cyclic variations. This oscillatory movement has a period
of oscillation of more than a year. One complete period is a cycle. This cyclic movement
is sometimes called the ‘Business Cycle’.
It is a four-phase cycle comprising of the phases of prosperity, recession, depression,
and recovery. The cyclic variation may be regular are not periodic. The upswings and the
downswings in business depend upon the joint nature of the economic forces and the
interaction between them.
Random or Irregular Movements: There is another factor which causes the variation in
the variable under study. They are not regular variations and are purely random or
irregular. These fluctuations are unforeseen, uncontrollable, unpredictable, and are
erratic. These forces are earthquakes, wars, flood, famines, and any other disasters.

12. CPM & PERT(THEORY)
DEFINE NETWORK
“A network is, then, a graphical representation of a project plan, showing the inter-
relationship of the various activities.
DEFINE PROJECT
A project is an interrelated set of activities that has a definite starting and ending point
and that result in a unique product or service.
DEFINE PROJECT MANAGEMENT
Project management is a scientific way of planning, implementing, monitoring &
controlling the various aspects of a project such as time, money, materials, manpower &
other resources.
LIST OUT METHODS USED FOR NETWORK PLANNING ARE:
 CPM
 PERT
DEFINE DUMMY ACTIVITY
–An activity which does not consume any kind of resource or time but merely shows
the technological dependence is called a dummy activity.
DEFINE MERGE EVENT
–When more than one activity comes and joins an event such an event is known as
merge event
DEFINE BURST EVENT
–When more than one activity leaves an event such an event is known as burst
event.
DEFINE SLACK TIME:
Slack time for an activity is the difference between its earliest(Ei)and latest start time(Li)
or between the earliest and latest finish time.
DEFINE CRITICAL PATH

13. Critical path is the sequence of activities between a projects’ that takes the longest time
to complete. Critical Path is “A path in a project network is called critical if it is the
longest path. The activities lying on the critical path are called the critical activities.”
Define
1. Optimistic time (to) – It is the shortest time in which the activity can be
completed.
2. Most likely time (tm) – It is the probable time required to perform the activity.
3. Pessimistic time (tp) – It is the longest estimated time required to perform an
activity.
DIFFERENCE B/W
CPM PERT
THERE IS ONL ONE TIME ESTIMATE 3 TIME ESTIMATES
DETERMINISTIC MODEL Probabilistic Model
PREDICTABLE ACTIVITIES UNPREDICTABLE ACTIVITIES
Non-research projects like civil
construction, ship building etc
RESEARCH PROJECTS
EVENTS ACTIVITES
Crashing concept applicable Crashing concept not applicable
Related with activities of certain time Related with activities of uncertain time
List out advantages and limitations of PERT/CPM
 Advantages:
Simple to understand and use
 It provides a graphical display of project activities that helps the users understand
the relationships among the activities.
 Show whether the project is on schedule; or behind/ ahead of the schedule.
 Identify the activities that need closer attention (critical).
 Determine the flexibility available with activities
 Show potential risk with activities (PERT)
 Provide good documentation of the project activities

14.  Help to set priorities among activities and resource allocation as per priority
Limitations:
 Uncertainly about the estimate of time and resources.
 It is not suitable for relatively simple and repetitive processes such as assembly line
work which are fixed-sequence jobs
 It is also difficult to estimate the activity completion time in a multidimensional
project.
 Overemphasis on Critical path
 Activity time estimates are subjective
 The allocation of resources cannot be properly monitored.
 The project managers have to spend a lot of time to calculate it carefully.
 Cost of crashing an activity may not be linear
 It requires a lot of information as input to generate an effective plan. This may prove
too expensive.
WHAT ARE THE COMMON ERRORS
 Looping
 Dangling
 Redundancy
Looping: Looping error is also called as cycling error in a network diagram. Making an
endless loop in a network is called as error of looping.
Dangling: Whenever an activity is disconnected from the network it is called dangling
error. To disconnect an activity before the completion of all activities in a network
diagram is known as dangling
Redundancy: When the dummy activity is introduced and it is not required, it is called
redundancy errors. Unnecessarily inserting the dummy activity in network logic is known
as the error of redundancy
DEFINE ACTIVITY “An activity is any portion of a project which consumes time or
resources and has a definable beginning and ending
EXPLAIN THE BASIC STEPS IN PERT / CPM

15. Project scheduling by PERT / CPM consists of four main steps
1.PLANNING
2.SCHEDULING
3.ALLOCATION OF RESOURCES
4.CONTROLLING
1.PLANNING
The planning phase is started by splitting the total project in to small projects. These
smaller projects in turn are divided into activities and are analyzed by the department or
section.
The relationship of each activity with respect to other activities are defined and
established and the corresponding responsibilities and the authority are also stated.
Thus the possibility of overlooking any task necessary for the completion of the project
is
reduced substantially.
2.SCHEDULING
The ultimate objective of the scheduling phase is to prepare a time chart showing the
start
and finish times for each activity as well as its relationship to other activities of the
project.
Moreover the schedule must pinpoint the critical path activities which require special
attention if the project is to be completed in time.
For non-critical activities, the schedule must show the amount of slack or float times
which can be used advantageously when such activities are delayed or when limited
resources are to be utilized effectively.
3.ALLOCATION OF RESOURCES
Allocation of resources is performed to achieve the desired objective. A resource is a
physical variable such as labour, finance, equipment and space which will impose a
limitation on time for the project.
 When resources are limited and conflicting, demands are made for the same type of
resources a systematic method for allocation of resources become essential.
 Resource allocation usually incurs a compromise and the choice of this compromise

16. depends on the judgment of managers.
4.CONTROLLING
The final phase in project management is controlling. Critical path methods facilitate
the
application of the principle of management by expectation to identify areas that are
critical to the completion of the proj ect.
 By having progress reports from time to time and updating the network continuously,
a
better financial as well as technical control over the project is exercised.
 Arrow diagrams and time charts are used for making periodic progress reports. If
require d, a new course of action is determined for the remaining portion of the
DEFINE SLACK TIME FOR AN EVENT :
“The slack time or slack of an event in a network is the difference between the
latest event time and the earliest event time.
DEFINE TOTAL FLOATS:“The total activity float is equal to the difference between the
earliest and latest allowable start or finish times for the activity in question. Thus, for an
activity (i-j), the total float is given by
DEFINE INDEPENDENT FLOAT: It is computed by subtracting the tail event slack from
the free float of an activity.
Transportation
Define Basic feasible solution:
A feasible solution to a transportation problem is said to be a basic feasible solution if it
contains no more than m + n – 1 non – negative allocations, where m is the number of
rows and n is the number of columns of the transportation problem.
Define Optimal solution:
A feasible solution (not necessarily basic) that minimizes (maximizes) the transportation
cost (profit) is called an optimal solution.
Define Non -degenerate basic feasible solution:

17. A basic feasible solution to a (m x n) transportation problem is said to be non –
degenerate if, the total number of non-negative allocations is exactly m + n – 1 (i.e.,
number of independent constraint equations), and these m + n – 1 allocations are in
independent positions.
Define Degenerate basic feasible solution:
A basic feasible solution in which the total number of non-negative allocations is less
than m + n – 1 is called degenerate basic feasible solution. In a transportation problem
with m origins and n destinations if a basic feasible solution has less than m + n – 1
allocations (occupied cells), the problem is said to be a degenerate transportation
problem.
Define Linear Programming
Mathematical programming or modeling technique which is used to find the best or
optimal solution to a problem that requires a decision or set of decisions about how
best to use a set of limited resources to achieve a state goal of objectives
It is a mathematical modeling technique used to determine a level of operational activity
in order to achieve an objective.
What are ESSENTIALS OF LINEAR PROGRAMMING MODEL( read more)
1. limited resources
2. objective
3. linearity
4. homogeneity
5. divisibility
1. Limited resources : limited number of labour, material equipment and finance
2. Objective : refers to the aim to optimize (maximize the profits or minimize the costs).
3. Linearity : increase in labour input will have a proportionate increase in output.
4. Homogeneity : the products, workers' efficiency, and machines are assumed to be
identical.
5. Divisibility :it is assumed that resources and products can be divided into fractions. (in
case the fractions are not possible, like production of one-third of a computer, a
modification of linear programming called integer programming can be used).

18. Explain the steps in formulation of LPP
Steps in Formulation of LP
• Identify the decision variables;
• Formulate the objective function; and
• Identify and formulate the constraints.
Objective function:
The objective of the problem is identified and converted into a suitable objective
function. The objective function represents the aim or goal of the system (i.e., decision
variables) which has to be determined from the problem. Generally, the objective in
most cases will be either to maximize resources or profits or, to minimize the cost or
time.
Constraints:
When the availability of resources are in surplus, there will be no problem in making
decisions. But in real life, organizations normally have scarce resources within which the
job has to be performed in the most effective way. Therefore, problem situations are
within confined limits in which the optimal solution to the problem must be found.
Non-negativity constraint
Negative values of physical quantities are impossible, like producing negative number of
chairs, tables, etc., so it is necessary to include the element of non-negativity as a
constraint.
What are the Advantages & disadvantages LP
 It helps decision - makers to use their productive resource effectively.
 The decision-making approach of the user becomes more objective and less
subjective.
 In a production process, bottle necks may occur.
Disadvantages of LP
1. Linear programming deals with only single objective, whereas in real life
situations may have multiple and conflicting objectives

19. 2. Not used for more decision variables or factors
3. LP is used only when constraints and objective function are linear i.e., where they
can be expressed as equations which represent straight lines.
4. Constraints or objective functions are not linear, this technique cannot be used.
5. Factors such as uncertainty and time are not taken into consideration.
6. Parameters in the model are assumed to be constant but in real life situations
they are not constants.