sbaoc statistical methods & their applications i unit : i -...

SBAOC

Statistical Methods &

their Applications I

Unit : I - V

SBAOC – Statistical Methods & their Applications -I

Unit I - Syllabus

Statistics definition

Characteristics and functions of Statistics

Classification & tabulation

Diagrams & graphs

Lorenz Curve

2

Statistics Definition

“By statistics we mean aggregates of facts affected to a marked

extent by multiplicity of causes numerically expressed,

enumerated or estimated according to reasonable standards of

accuracy, collected in a systematic manner for a pre-

determined purpose and placed in relation to each other.”

- PROF.HORACE SECRIST

SBAOC – Statistical Methods & their Applications -I 3

CHARACTERISTICS/FEATURES

Statistics are aggregate of facts

Statistics are affected to a marked extent of multiplicity of causes.

Statistics must be Numerically Expressed

Statistics must be enumerated or estimated according to

reasonable standard of accuracy.

Statistics should be collected in a systematic manner.


FUNCTIONS OF

STATISTICS1. Statistics presents the fact in definite form

2. It simplifies mass of data.

3. It facilitates for comparison

4. It enlarges individual experience

5. It facilitates in formulating policies

Statistics laws are true on average. Statistics are aggregates

of facts. So single observation is not a statistics, it deals with

groups and aggregates only.

Statistical methods are best applicable on quantitative data.

Limitations of Statistics


A Taxonomy of Statistics


Classification

The process of grouping a large number of individual facts or

observations on the basis of similarity among the items, is called

classification.

OBJECTIVES OF CLASSIFICATION

To condense the mass of data, and to present the facts in a

simple form.

To facilitate comparison between data.

To eliminate unnecessary details so that main features of the

collected data are easily understandable.

To facilitate further statistical treatment of the data


TYPES OF CLASSIFICATION

CLASSIFICATION TYPES

GEOGRAPHICAL

CHRONOLOGICAL

QUANTITATIVE

QUALITATIVE


TABULATION:

The mass of data collected has to be arranged in some kind of concise

and logical order.

Tabulation summarizes the raw data and displays data in form of some

statistical tables.

Tabulation is an orderly arrangement of data in rows and columns.

OBJECTIVE OF TABULATION:

1. Conserves space & minimizes explanation and descriptive statements.

2. Facilitates process of comparison and summarization.

3. Facilitates detection of errors and omissions.

4. Establish the basis of various statistical computations.


TYPES OF TABLE

ON THE BASIS OF

COVERAGE

ON THE BASIS OF

OBJECTIVE

ON THE BASIS OF

ORGINALITY

SIMPLE AND COMPLEX TWO – WAY TABLE

MANIFOLD OR

HIGHER ORDER TABLETHREE WAY TABLE

GENERAL PURPOSE SPECIAL PURPOSE


A diagram is a visual form for presentation of statistical data,

highlighting their basic facts and relationship. If we draw diagrams

on the basis of the data collected they will easily be understood and

appreciated by all. It is readily intelligible and save a

considerable amount of time and energy.

A graph is a visual form of presentation of statistical data. A

graph is more attractive than a table of figure. Even a common

man can understand the message of data from the graph.

Comparisons can be made between two or more phenomena very

easily with the help of a graph.

Diagrams

Graphs


TYPES OF DIAGRAMS

ONE DIMENSIONAL OR

BAR DIAGRAMS

TWO DIMENSIONAL OR

AREA DIAGRAMS

THREE DIMENSIONAL OR

VOLUME DIAGRAMS

PICTOGRAMS AND

CARTOGRAMS


DEVIATION

BAR

PERCENTAGE

BAR

SUBDIVIDED

BAR

MULTIPLE BAR

SIMPLE BAR

TYPES OF

ONE

DIMENSIONAL


TYPES OF GRAPHS

GRAPHS FOR FREQUENCY DISTRIBUTION ARE AS FOLLOWS:

1. Histogram

2. Frequency polygon

3. Frequency curve

4. Ogives or cumulative frequency curve


LORENZ CURVE

LORENZ CURVE is a graphical method of studying dispersion.

It is used to study the variability in the distribution of profits, wages,

revenue etc.,

To study the degree of inequality in the distribution of income and

wealth between countries or different periods.



Unit II - Syllabus

Sampling Definition

Simple random sampling

Systematic random sampling

Stratified random sampling

Sampling & non-sampling errors

Methods of reducing sampling errors

16

Definition of Sampling

Sampling is the process of selecting a small number of

elements from a larger defined target group of elements such

that the information gathered from the small group will allow

judgments to be made about the larger groups.


• Sampling have various benefits to us. Some of the

advantages are listed below:

• Sampling saves time to a great extent by reducing the volume

of data. You do not go through each of the individual items.

• Sampling Avoids monotony in works. You do not have to

repeat the query again and again to all the individual data.

• When you have limited time, survey without using

sampling becomes impossible. It allows us to get near-

accurate results in much lesser time


Advantages of Sampling

18

• Every coin has two sides. Sampling also have some demerits.

Some of the disadvantages are:

• Since choice of sampling method is a judgmental task, there

exist chances of biasness as per the mindset of the person

who chooses it.

• Improper selection of sampling techniques may cause the

whole process to defunct.

• Selection of proper size of samples is a difficult job.


Disadvantages of Sampling

19

Sample size

• The number of cases used for the research analysis

• Sample size may be determined by using:

– Subjective methods (less sophisticated methods)

• The rule of thumb approach: eg. 5% of population

• Conventional approach: eg. Average of sample sizes of

similar other studies;

• Cost basis approach: The number that can be studied with

the available funds;

– Statistical formulae (more sophisticated methods)

• Confidence interval approach.


Simple random sampling

A sample of size n draw from a finite population of size N is said

to be a random sample if it is chosen in such a way that each of the

possible sample has the same probability of being selected.


Systematic random sampling

Systematic random sampling is a method of probability sampling in

which the defined target population is ordered and the sample is

selected according to position using a skip interval

K = N

n

K = Sampling Interval

N = Universe Size

n= Sample Size


Stratified random sampling

• This method is used when the population distribution of items is

skewed.

• It allows us to draw a more representative sample. Hence if there

are more of certain type of item in the population the sample has

more of this type and if there are fewer of another type, there are

fewer in the sample.


Sampling & Non-Sampling Errors

• In statistics, the word „Error‟ is used to denote the difference

between the true value and the estimated value.

• The term error should be distinguished from mistake or inaccuracies

which may be committed in the course of making observations,

counting calculations etc.,

• The error, mainly arise at the stage of ascertainment and

processing of data in complete enumeration and sample

surveys.

• Errors are classified into sampling errors and non-sampling

errors.

Slide number / Total slidesSBAOC – Statistical Methods & their Applications -I 28

Methods of Reducing Sampling Errors

Once the absence of bias has been ensured, attention should be

given to the random sampling errors. Such errors may be reduced to

the minimum to attain the desired accuracy.

Apart from reducing errors of bias, the simplest way of increasing

the accuracy of a sample is to increase its size.

Slide number / Total slidesSBAOC – Statistical Methods & their Applications -I 29


Unit III - Syllabus

Central tendency

Dispersion

Coefficient of variation

Skewness

Kurtosis

30

Measures of Central Tendency

The value or the figure which represents the whole series is neither the

lowest value in the series nor the highest it lies somewhere between

these two extremes.

1. The average represents all the measurements made on a group,

and gives a concise description of the group as a whole.

2. When two are more groups are measured, the central tendency

provides the basis of comparison between them.


KINDS OF AVERAGES

CENTRAL

TENDENCY

(AVERAGES)

AVERAGES OF

LOCATION

(POSITION)

COMMERCIAL

AVERAGE

(BUSINESS)

MATHEMATICAL

AVERAGES

3. COMPOSITE

AVG

2. PROGRESSIVE

AVG

1.MOVING AVG

2. MODE

1. MEDIAN

4. QUADRATIC

MEAN

3. HARMONIC

MEAN

2. GEOMETRIC

MEAN

1.ARITHMETIC

MEAN

2. WEIGHTED

1. SIMPLE


Mean

The mean (arithmetic mean or average) of a set of data is found

by adding up all the items and then dividing by the sum of the

number of items.

x

The mean of a sample is denoted by (read “x bar”).

The mean of a complete population is denoted by (the lower

case Greek letter mu).

The mean of n data items x1, x2,…, xn, is given by the formula

or


Example:

Ten students were polled as to the number of siblings in their

individual families.

The raw data is the following set: {3, 2, 2, 1, 3, 6, 3, 3, 4, 2}.

Find the mean number of siblings for the ten students.

siblings


Median

Another measure of central tendency, is the median.

The median is not as sensitive to extreme values as the mean.

This measure divides a group of numbers into two parts, with half

the numbers below the median and half above it.

To find the median of a group of items:

Rank the items.

If the number of items is odd, the median is the middle item

in the list.

If the number of items is even, the median is the mean of the

two middle numbers.


Example:

Ten students in a math class were polled as to the number of

siblings in their individual families and the results were:

3, 2, 2, 1, 1, 6, 3, 3, 4, 2.

Find the median number of siblings for the ten students.

Data in order: 1, 1, 2, 2, 2, 3, 3, 3, 4, 6

Median

Median

= (2+3)/2 = 2.5

Position of the median: 10/2 = 5

Between the 5th and 6th values

siblings


The mode of a data set is the value that occurs the most often.

If a distribution has two modes, then it is called bimodal.

Mode

In a large distribution, this term is commonly applied even

when the two modes do not have exactly the same frequency

Ten students in a math class were polled as to the number of siblings

in their individual families and the results were: 3, 2, 2, 1, 3, 6, 3, 3, 4,

2. Find the mode for the number of siblings.

Example:

3, 2, 2, 1, 3, 6, 3, 3, 4, 2

The mode for the number of siblings is 3.


Geometric mean:

Geometric mean is the nth root when you multiply n numbers

together.

Harmonic mean:

• Harmonic mean as the reciprocal of the arithmetic mean of

the reciprocals.

• So far we have looked at ways of summarising data by

showing some sort of average (central tendency).

• But it is often useful to show how much these figures differ

from the average. This measure is called dispersion.


METHODS OF MEASURING DISPERSION

• Range

• Quartile Deviation

• Mean Deviation

• Standard Deviation


Range:

The range is the difference between the maximum and minimum

values.

Quartile deviation (or) inter-quartile range :

The inter-quartile range is the range of the middle half of the values.

Mean deviation:

The mean of the absolute values of the numerical differences between

the numbers of a set (such as statistical data) and their mean or

median.

Standard deviation:

A statistic used as a measure of the dispersion or variation in a

distribution, equal to the square root of the arithmetic mean of the

squares of the deviations from the arithmetic mean.


Video Link: Quartile Deviation

https://www.youtube.com/watch?v=N_YnzL8mEqw

Video Link: Quartile Deviation

https://www.youtube.com/watch?v=K3wsOqIqA6k

Video Link: Mean Deviation

https://www.youtube.com/watch?v=z9AJk7TvdpQ


https://www.youtube.com/watch?v=N_YnzL8mEqw











Coefficient of Variation (C.V.):

The variance and the standard deviation are useful as

measures of variation of the values of a single variable for a

single population (or sample).

If we want to compare the variation of two variables we

cannot use the variance or the standard deviation because:

1. The variables might have different units.

2. The variables might have different means.


We need a measure of the relative variation that will not

depend on either the units or on how large the values are. This

measure is the coefficient of variation (C.V.) which is defined by:

%100*.x

VC

(free of unit or unit less)

Mean S.D C.V.

%100.1

11

xVC

%100.2

22

xVC

11x

2x 2

1st data

set

2nd data

set

The relative variability in the 1st data set is larger than the relative

variability in the 2nd data set if C.V1> C.V2 (and vice versa).


Skewness

• Skewness is the measure of the shape of a nonsymmetrical

distribution

• Two sets of data can have the same mean & SD but different

skewness

Two types of skewness:

– Positive skewness

– Negative skewness


Kurtosis

A measure of whether the curve of a distribution is:

Bell-shaped -- Mesokurtic

Peaked -- Leptokurtic

Flat -- Platykurtic



Unit IV - Syllabus

Probability

Additive Rule

Independent events

Conditional probability

Multiplicative Rule for Intersections

Bayes' Rule

47

TM

Slide number / Total slides

Probability:

Definition: The numerical chance that a specific outcome will

occur (i.e.)., it is a mathematical measure of “Measuring the

certainty or uncertainty of an event”.

Sometimes you can measure a probability with a number: "10%

chance of rain", or you can use words such as impossible,

unlikely, possible, even chance, likely and certain.

Example: "It is unlikely to rain tomorrow".


TM

Approaches to Probability

1. Relative frequency event probability = x/n, where x= no.of

occurrences of event of interest, n = total no. of observations

2. Subjective probability

Individual assigns prob. based on personal experience, anecdotal

evidence, etc

3.Classical approach: Every possible outcome has equal probability

(more later)

The probability for the occurrence of an event A is defined as the

radio between the number of favourable outcomes for the occurrence

of the event and the total number of possible outcomes,

Probability of an event =

Number of favourable outcomes / Total number of outcomes

P(E) = m/n


TM

Basic Definitions

Experiment: act or process that leads to a single outcome that cannot

be predicted with certainty

Sample space: all possible outcomes of an experiment. Denoted by S

Event: any subset of the sample space S;typically denoted A, B, C, etc.

Simple event: event with only 1 outcome

Certain event: S

Laws of Probability

1. 0 P(A) 1 for any event A

2. P() = 0, P(S) = 1 and P(A‟) = 1 – P(A)

4. If A and B are disjoint events, then P(A B) = P(A) + P(B)

5. If A and B are independent events, then P(A B) = P(A) × P(B)

6. For any two events A and B, P(A B) = P(A) + P(B) – P(A B)


TM

Examples

1. Toss a fair coin. P(Head) = ½

2. Toss a fair coin twice. What is the probability of observing at least

one head

P(at least 1 head) = P(E1) + P(E2) + P(E3) = 1/4 + 1/4 + 1/4 = ¾

3. Select a student from the classroom and record his/her hair color

and gender.

A: student has brown hair

B: student is female

C: student is male Mutually exclusive; B = CC

What is the relationship between events B and C?

•AC: Student does not have brown hair

•BC: Student is both male and female =

•BC:Student is either male and female = all students = S


TM

The Additive Rule for Unions:

For any two events, A and B, the probability of their union, P(A B), is

When two events A and B are mutually exclusive, P(AB) = 0

and P(AB) = P(A) + P(B).

)()()()( BAPBPAPBAP AUB BA


TM

Example: Additive Rule

Example: Suppose that there were 120 students in the classroom,

and that they could be classified as follows:

A: Statistics Student

P(A) = 50/120

B: Female

P(B) = 60/120

Allied Statistics Non

Statistics

Male 20 40

Female 30 30

P(selecting a statistics female student) = P(AB) = P(A) + P(B) – P(AB)= 50/120 + 60/120 - 30/120 = 80/120 = 2/3


TM

Independent Event & Conditional Probabilities

1.Two events, A and B, are said to be independent if the occurrence or

nonoccurrence of one of the events does not change the probability of the

occurrence of the other event. In terms of conditional, Two events A and B

are independent if and only if P(A|B) = P(A) or P(B|A) = P(B)

Otherwise, they are dependent.

2. The probability that A occurs, given that event B has occurred is called

the conditional probability of A given B and is defined as

0)( if )(

)()|(

BP

BP

BAPBAP

“given”


TM

Example 1

1. Toss a fair coin twice. Define

A: head on second toss

B: head on first toss

P(A|B) = ½

P(A|not B) = ½

P(A) does not change, whether B

happens or not…

A and B are

independent!

2.Toss a pair of fair dice. Define

A: red die show 1

B: green die show 1

P(A|B) = P(A and B)/P(B)

=1/36 / 1/6=1/6=P(A)


TM

The Multiplicative Rule for

IntersectionsFor any two events, A and B, the probability that both A and B occur is

= P(A) P(B given that A occurred)

If the events A and B are independent, then the probability that both A and

B occur P(A B)=P(A)P(B/A) if A & B are independent,

Example:

In a certain population, 10% of the people can be classified as being high

risk for a heart attack. Three people are randomly selected from this

population. What is the probability that exactly one of the three are high

risk? Define H: high risk N: not high risk

P(exactly one high risk) = P(HNN) + P(NHN) + P(NNH)

= P(H)P(N)P(N) + P(N)P(H)P(N) + P(N)P(N)P(H)

= (.1)(.9)(.9) + (.9)(.1)(.9) + (.9)(.9)(.1)= 3(.1)(.9)2 = .243

( ) ( ). ( )P A B P A P B


TM

The Law of Total Probability

Let S1 , S2 , S3 ,..., Sk be mutually exclusive and exhaustive events

(that is, one and only one must happen). Then the probability of

any event A can be written as

P(A) = P(A S1) + P(A S2) + … + P(A Sk)

= P(S1)P(A|S1) + P(S2)P(A|S2) + … + P(Sk)P(A|Sk)


TM

Bayes’ Rule

,...k, i SAPSP

SAPSPASP

ii

iii 21for

)|()(

)|()()|(

)|()(

)|()(

)(

)()|(

)|()()()(

)()|(

Proof

ii

iiii

iii

i

ii

SAPSP

SAPSP

AP

ASPASP

SAPSPASPSP

ASPSAP


TM

ExampleSuppose a rare disease infects one out of every 1000 people in a

population. And suppose that there is a good, but not perfect, test for this

disease: if a person has the disease, the test comes back positive 99% of

the time. On the other hand, the test also produces some false positives:

2% of uninfected people are also test positive. And someone just tested

positive. What are his chances of having this disease?

Solution:

Define A: has the disease B: test positive

We know:

P(A) = .001 P(Ac) =.999

P(B|A) = .99 P(B|Ac) =.02


TM

Example:

( ) ( | )( | )

( ) ( | ) ( ) ( | )

.001 .99.0472

.001 .99 .999 .02

P A P B AP A B

c cP A P B A P A P B A



Unit V - Syllabus

Correlation

Methods of studying Correlation

Scatter diagram

Regression

Least squares method

Partial Correlation & Multiple Correlation

61

TM

Correlation coefficient is a statistical measure of the degree to

which changes to the value of one variable predict change to

the value of another.

When the fluctuation of one variable reliably predicts a similar

fluctuation in another variable, there‟s often a tendency to think

that means that the change in one causes the change in the

other.

CORRELATION


TM

Coefficient of Correlation lies between -1 and +1.

Coefficients of Correlation are independent of Change of

Origin.

Coefficients of Correlation possess the property of symmetry.

Coefficient of Correlation is independent of Change of Scale.

Co-efficient of correlation measures only linear correlation

between X and Y.

Properties of Correlation Coefficient


TM

Types of correlation

Correlation are of three types:

Positive Correlation

Negative Correlation

No correlation

In correlation, when values of one variable increase with the increase

in another variable, it is supposed to be a positive correlation.

On the other hand, if the values of one variable decrease with the

decrease in another variable, then it would be a negative correlation.

There might be the case when there is no change in a variable with

any change in another variable. In this case, it is defined as no

correlation between the two.


TM

Scatter diagram.

Karl pearson's coefficient of correlation.

Spearman's Rank correlation coefficient.

Methods of studying Correlation


TM

The scatter diagram graphs pairs of numerical data, with one

variable on each axis, to look for a relationship between them. If the

variables are correlated, the points will fall along a line or curve. The

better the correlation, the tighter the points will hug the line.

The following video explains the scatter diagram with an example

https://www.youtube.com/watch?v=NcgRa0uotXs

Scatter Diagram





TM

Karl pearson's coefficient of correlation

The formula to calculate Karl pearson‟s coefficient correlation

is as follows:

The following video explains the correlation problem.

2222 )()()()(

))((

yynxxn

yxxynr

https://www.youtube.com/watch?v=pOKWxNTterg


https://www.youtube.com/watch?v=pOKWxNTterg

TM

A rank correlation coefficient measures the degree of similarity

between two rankings, and can be used to assess the significance of

the relation between them.

The following video explains the rank correlation problem

Spearman's Rank correlation coefficient.

https://www.youtube.com/watch?v=KK-wuL7WN20





TM

In statistical modeling, regression analysis is a set of statistical

processes for estimating the relationships among variables.

It includes many techniques for modeling and analysing several

variables, when the focus is on the relationship between a

dependent variable and one or more independent variables (or

'predictors').

Regression


TM

Anyone who wants to make predictions or inferences based on a

particular data set. Companies,

Example, Companies, use regression to determine their profit,

revenue, and cost functions and scientists use regression to

analyze and predict all sorts of time series phenomenon including

the population of certain bacteria, crop data, planetary orbits, and

projectile tracking. These days, regression is the most used method

for data analysis.

Uses of Regression


TM

If Y depends on X then the regression line is Y on X. Y is

dependent variable and X is independent variable.

The regression equation Y on X is Y = a + bx, is used to estimate

value of Y when X is known.

If X depends on Y, then regression line is X on Y and X is

dependent variable and Y is independent variable.

The regression equation X on Y is X = c + dy is used to estimate

value of X when Y is given

Here a, b, c and d are constant.

Regression Lines


TM

Formula to calculate Regression lines


TM

Least-squares Method

If a straight line is fitted to the data it will serve as a satisfactory

trend, perhaps the most accurate method of fitting is that of least

squares.

The formula for a straight-line trend can most simply be expressed

as

Yc = a + bX

where X represents time variable, Yc is the dependent variable for

which trend values are to be calculated and a and b are the

constants of the straight tine to be found by the method of least

squares.


TM

https://www.youtube.com/watch?v=T7LauV6AnRg

Regression

The following link explains the regression problem in detail .


https://www.youtube.com/watch?v=T7LauV6AnRg

Correlation between two statistical variables under the condition

that all other relevant variables are fixed.

Partial Correlation:

Correlation involving two or more independent mathematical

variables.

Multiple Correlation:

SBAOC – Statistical Methods & their Applications -ISBAOC – Statistical Methods & their Applications -I

Partial correlation & Multiple Correlation

75

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