paper 3 business mathematics logical reasoning …

Vaibhav Jalan Classes

Where Knowledge values more than Marks……

PAPER 3 :

BUSINESS MATHEMATICS, LOGICAL

REASONING & STATISTICS

CHAPTER WISE MARKS WEIGHTAGE

Part C : Statistics (40 Marks)

S. No. Chapter Name Marks Weightage

1 Introduction to Statistics

Measure of Central Tendency and dispersion 18 - 20

2 Probability

Theoretical Distribution 10 - 12

3 Correlation and Regression 4 - 6

4 Index Numbers 4 - 6

Important :

Paper Pattern : All question are objective type of one mark each.

For every Correct answer 1 mark will be awarded and for every wrong 0.25 will be deducted.

All questions are compulsory. No Internal Choice.

A student is not required to pass in each Part but is required to pass in totality. i.e. he required

to secure 40 marks out of 100 irrespective of marks obtained in each Part.

Chapter 1Chapter 1Chapter 1Chapter 1 Page No. 2



STATISTICAL DESCRIPTION

OF DATA 1 Chapter

STATISTICS is derived from

Latin Word 'STATUS'

German Word ‘STATISTIK’

Italian Word ‘STATISTA'

French Word ‘STATISTIQUE'

DEFINITION of statistics

1. SINGULAR SENSE It is a scientific method of Collecting. Analyzing & Presenting data and

finally drawing Statistical Inferences. It is also known as “Science of Counting” or “Science of

Average”

2. PLURAL SENSE It is the collection of information, Qualitative as well as Quantitative, with a

view of having statistical analysis.

LIMITATION OF STATISTICS:

1. It deals with the aggregate.

2. It is concerned with quantitative data. (Qualitative information has to be converted into

quantitative data by providing an attribute.

3. Future projections can only be made under a specific set of conditions.

4. The theory of statistical inferences is built upon random sampling.

DATA: Quantitative Information of some particular characteristics which can be either Quantitative or

Qualitative

1. QUANTITATIVE CHARACTERISTICS:

Known as VARIABLE which can be DISCRETE or CONTINUOUS

Eg. of Discrete Variable: No. of Petals in a flower, No. of misprint in a book, No. of road

accident in a locality.

Eg. of Continuous Variable: Height, Weight, Sales, Profit




2. QUALITATIVE CHARACTERISTICS : Known as ATTRIBUTE

Eg.: Nationality of a person, Gender of a baby, Colour of a flower

SOURCE OF DATA

1. PRIMARY DATA: Data collected for the first time through Interview method,

Mailed Questionnaire and Observation method.

2. SECONDARY DATA:

Data being already collected used by different persons.

E.g. Data collected through International Agencies, National

Agencies, Private & Quasi Govt. Agencies and Unpublished

sources.

SCRUTINY OF DATA:

Checking the data for their accuracy and consistency with the help of related series.

CLASSIFICATION OF DATA:

Arranging the data on the basis of the characteristics under consideration into number of groups or

classes according to the similarities of the observation.

TYPES OF DATA SERIES

1. Chronological or Temporal or Time Series

2. Geographical or Spatial Series

3. Qualitative or Ordinal Data

4. Quantitative or Cardinal Data

PRESENTATION OF DATA

1. TEXTUAL

PRESENTATION :

Presenting the data in words through Paragraphs. Though it is simple to

understand but sometimes becomes dull and monotonous.

2. TABULAR

PRESENTATION :

It is considered as the Best Method of presentation as data is presented

in a neat, concise and condensed form eliminating unnecessary details. It

facilitates comparison between rows and columns. Complicated date can

also be represented using tabulation. It is must for diagrammatic

representation.




A table is divided into four parts:

Caption : It is the upper part of the table describing the columns and the

sub columns.

Box – Head : It is the entire upper part of the table describing the table,

units of measurement including Captions.

Stubs : It is the left part of the table providing the description of the rows.

Body : It is the main part of the table that contains the numerical figures.

3. DIAGRAMMATIC

PRESENTATION :

Presenting the data through graphs, diagrams, pictures and charts. It is

considered as a n attractive mode of Presentations and is generally

Useful to find hidden trends.

Line Diagram or Historiagram :

It is used to represent Time series Data.

Multiple Line diagram :

It is used to compare related time series data.

Logarithmic or Ratio Chart :

It is used to represent Time series data with wide fluctuations

Vertical Bar Diagram :

It is used to represent Time Series data as well as Quantitative data.

Horizontal Bar Diagram :

It is used to represent Geographical Data as well as Qualitative Data.

Multiple or Grouped Bar Diagram :

It is used to compare related series.

Component or Sub-Divided Bar Diagram or Divided Bar Diagram or

Percentage Bar Diagram :

It is used to represent data divided into various components and used to

compare various components with each other as well with the whole.

Pie chart or Pie diagram or Circle Diagram :

It is used to compare various components with each other as well with

the whole through a circle.

Line Diagram Bar Diagram Pie Chart




Classification of Diagrams on the basis of Dimensions :

1. One Dimensional Diagrams Line Diagram

Simple Bar Diagram

Sub-divided Bar Diagram

Multiple Bar Diagram

Percentage Bar Diagram

2. Two Dimensional Diagrams Rectangle Bar Diagram

Square diagram

Pie diagram

3. Three Dimensional Diagrams Cubic Diagram

FREQUENCY DISTRIBUTION Tabular representation of statistical data

The statistical data can be presented either through a Discrete Frequency distribution or a

Continuous Frequency Distribution. A continuous frequency distribution can be either Mutually

exclusive or Mutually Inclusive.

Mutually inclusive classification is generally meant for Discrete data whereas Mutually Exclusive

classification is generally meant for continuous data.

POINTS TO REMEMBER:

FREQUENCY (Tally Marks) :Number of times or how frequently a particular class occurs.

CLASS LIMITS : Minimum & Maximum value of the class interval.

CLASS BOUNDARY : Actual class limits of the class intervals

MID POINTS (CLASS MARK): 2

UCBLCBor

2

UCLLCL ++

CLASS WIDTH = UCB – LCB

FREQUENCY DENSITY : ClassWidth

Frequency Class

RELATIVE FREQUENCY : Frequency Total

Frequency Class

RANGE : Range ≅ Class Width ×Number of class Interval

Frequency density is used to prepare Histogram.

Cumulative Frequency is used to prepare Ogive.

Mid points are sed to prepare Frequency polygon.

Chapter Chapter Chapter Chapter 2222 Page No. 6



MEASURES OF CENTRAL

TENDENCY AND DISPERSION 2 Chapter

Measure of central tendency may be defined as the tendency of a given set of observation to cluster

around a single value.

Dispersion of a given set may be defined as the amount of deviation of the observation, usually, from

an appropriate measure of central tendency.

Absolute measure of dispersion are dependent on units of the variable under consideration whereas

the relative measures of dispersion are unit free. For comparing two or more distributions, we use

relative measure of dispersion.

PARTICULARS INDIVIDUAL DISCRETE CONTINUOUS

ARITHMETIC

MEAN n

xX

=

=f

fxX

ii

Replace ‘x’ with ‘m’

were ‘m’ is the mid point of

the respective class

interval.

GEOMETRIC

MEAN

G.M.=

n32.1.....xx.xx

n

G.M. =

( ) ( ) ( )n f

n

f

2

f

1n21 x........x.x

HARMONIC

MEAN H.M. =

( ) x

n

/1 H.M. =

( )

ii

i

/xf

f

MODE Highest frequency Highest frequency

Mo = Cff2f

ffL

110

101 ×

−−

−+

−

−

1L = Lower Class limit

0f = Class frequency of current class

1f = Class frequency of next class

1-f = Class frequency of previous class

C = Class width

1. COMBINED A.M. = 21

2211

nn

XnXn

+

+; 2. COMBINED H.M. =

2

2

1

1

21

H

n

H

n

nn

+

+

3. COMBINED G.M. = ( ) ( )( )21 21nn nnG.M.ofYG.M.ofX

+




POINTS TO REMEMBER:

• MEAN – MODE = 3 (MEAN – MEDIAN)

• 5052 PDQMe ===

• In case of distinct positive values, A.M. > G.M. > H.M.

• However if all values are same then A.M. = G.M. = H.M.

• In case of two variables 2)(GMHMAM =×

PARTICULARS INDIVIDUAL / DISCRETE CONTINUOUS

MEDIAN Me =

th

2

1n

+item

Me =

th

2

n

item

Me = CcN/2

L1 ×−

+f

f

QUARTILE ( ) th

34

1n3Q

+= item

th

34

3nQ

= item

Cc3N/4

LQ 13 ×−

+=f

f

DECILE ( ) th

710

1n7D

+= item

th

710

7nD

= item

Cc7N/10

LD 17 ×−

+=f

f

PERCENTILE ( ) th

63100

1n63P

+= item

th

63100

63nP

= item

Cc63N/100

LP 163 ×−

+=f

f

RANGE H – L H – L

COEFFICIENT OF

RANGE 100

LH

LH×

+

− 100

LH

LH×

+

−

INTER QUARTILE

RANGE 13 QQ −

13 QQ −

* 1L = Lower Class limit ; f = Frequency ; cf = Cumulative frequency upto previous interval ;

C = Class Width




PARTICULARS INDIVIDUAL / DISCRETE CONTINUOUS

QUARTILE DEVIATION

(SEMI INTER QUARTILE

RANGE) 2

13 QQ − Same

COEFFICIENT OF

QUARTILE DEVIATION 100

QQ

QQ

13

13 ×+

− Same

MEAN DEVIATION n

AX −

where A can be mean, median

or mode

Same

COEFFICIENT OF

MEAN DEVIATION 100

..×

A

DM Same

VARIANCE

( )n

XX2

− ;

( )

−

f

XXf2

Or

( )22

Xn

X−

; ( )2

2

Xf

fX−

Replace ‘x’ with ‘m’

STANDARD

DEVIATION

Variance Same

COEFFICIENT OF

VARIATION (CV) x100

X

S.D. Same

COMBINED S.D. = ( ) ( )

21

2

2

2

22

2

1

2

11

nn

dsndsn

+

+++

Where 2,111 XXd −= and

2,122

XXd −=




POINTS TO REMEMBER:

1. AM, Median and Mode are affected by shift in origin as well as by change in scale. However

Range, Standard Deviation, Mean Deviation and Quartile Deviation are not affected by shift

in origin but are only affected by change in scale.

If y = a + bx , then

1. xbay += 2. MeMe bxay +=

3. MoMo bxay += 4.

RangeRange xby =

5. .... DMDM xby = 6.

S..D.S..D. xby =

7. Q.D.Q.D. xby =

2. If all the values are same, then

→ The value of central tendencies (AM , GM , HM , Partition values) will be the same value

→ The absolute measure and relative measure of Dispersion (Range, MD,QD,SD and their

respective coefficients) will be ZERO.

→ There will be no Mode as there is no highest frequency.

3. ARITHMETIC MEAN:

• It is the best measure of central tendency as it rigidly defined, based on all observations,

easy to comprehend, simple to calculate and amenable to mathematical properties.

• The Algebric sum of deviation of a set of observations from their AM is zero.

• It is very much affected by sampling fluctuations.

• It can not be computed in case of distribution having open end classification.

4. GEOMETRIC MEAN:

• It is used in formulation of Index Numbers.

• It can not be used when the values are negative or zero.

• Log G.M. = nLogxLogxLogxLogx n /)..........( 321 +++

• If z = xy, then G.M. of z = (G.M. of x) × (G.M. of y)

• If z = x/y, then G.M. of z = (G.M. of x) ÷ (G.M. of y)

5. HARMONIC MEAN:

• It is the reciprocal of A.M. of the reciprocal of the numbers

• It can not be used if any observation is zero.

• It is useful to find averages if the observations are in ratios (E.g. Average of Different

speeds)




6. MODE

• It is the most frequent value of the given observations.

• It is not uniquely defined and can have multiple values.

• If all values are same, then there is no mode.

7. PARTITION VALUES:

• It is most appropriate for open end classification.

• It can be graphically computed through Ogive Curves.

• Quartiles divides the total observations into four equal parts. There are three quartiles.

First Quartile or Lower Quartile is denoted by Q1; Second Quartile or median is denoted

by Q2; and third quartile or upper quartile is denoted by Q3.

Similarly Deciles divides the total observations in ten equal parts. There are nine Deciles

represented as D1, D2, D3,…..D9.

Similarly Percentiles or centiles divides the total observations in hundred equal parts.

There are ninty nine Percentiles represented as P1, P2, P3,…..P99.

• Partition values are not affected by extreme values.

8. MEDIAN

• It is rigidly defined and easy to comprehend.

• It is most appropriate measure of central tendency in case of open end classification.

• It always lie between Mean and Mode

• For a moderately skewed data Mean – Mode = 3(Mean – Median)

• It is not based on all the observation and can not be used for further mathematical

treatments.

9. RANGE

• It is quickest to compute as it only based on extreme values.

• It is generally applied in quality control.

• It is highly affected by extreme values.

• For a grouped frequency distribution, range is defined as the difference between the two

extreme class boundaries

• Since it is based on only two observation, it is not regarded as an ideal measure of

dispersion

10. QUARTILE DEVIATION / SEMI INTER QUARTILE RANGE

• It is the best measure of dispersion for open end classification.

• It is based on the central fifty percent of the observations.




11. STANDARD DEVIATION

• It is defined as the root mean square deviation when the deviations are taken from the

AM of the observations.

• It is the Best measure of Dispersion.

• It is less affected by sampling fluctuations.

• It is used in various statistical analysis like computation of Correlation and regression

coefficients.

• The standard deviations of ‘n’ natural numbers is 12

1n 2 −

• For any two numbers, standard deviation is always half of the range.

• When all the values are same, standard deviation is Zero.

12. COEFFICIENT OF VARIATION (CV)

• It is useful in comparing the consistency in the observations

13. MEAN DEVIATION

• It is defined as the arithmetic mean of the absolute deviations of the observations from

an appropriate measure of central tendency.

• The value of Mean Deviation is minimum when calculated from Median.

• It is based on all observations.

14. CRITERIA FOR IDEAL MEASURE OF CENTRAL TENDENCY AND

MEASURE OF DISPERSION

• It should be properly and unambiguously defined.

• It should be easy to comprehend.

• It should be simple to compute.

• It should be based on all observations.




CORRELATION AND REGRESSION 3 Chapter

CORRELATION:

Establishing a relation between bi-variate data or multivariate data.

Denoted by ‘r’. It is a unit free measure.

POINTS TO REMEMBER:

• 11 ≤≤− r

• If r = – 1 , correlation between two variables is said to be perfectly negative

• If – 1 < r < 0 , correlation between two variables is said to be negative.

• If r = 0 , there is No Linear correlation between two variables

• If 0 < r < 1 , correlation between two variables is said to be positive.

• If r = 1 , correlation between two variables is said to be perfectly positive

If two variables move in the same direction i.e. an increase (or decrease) on the part of one variable

introduces an increase (or decrease) on the part of other variable, then the two variables are known to

be positively correlated.

However, If two variables move in the opposite direction i.e. an increase (or decrease) on the part of

one variable introduces a decrease (or an increase) on the part of other variable, then the two

variables are known to be negatively correlated.

The two variables are known to be uncorrelated if the movement on the part of one variable does

not produce any movement of the other variable in a particular direction.




METHODS OF COMPUTATION:

1. Scatter Diagram

Can be applied for any type of correlation : Linear as well as non – linear or curvilinear.

However this method fails to measure the extent of relationship between the variables.

r = – 1 – 1 < r < 0 r = + 1 0 < r < +1 r = 0

2. Karl Pearson’s correlation / Product Moment Correlation Coefficient.

Best measure of correlation.

Applicable only in case of a linear relationship between the two variables.

r = ( )

YX S.D.S.D.

yx,Cov

where ( ) =yx,Cov ( )( )yxn

xy−

or

n

)y-)(yx-(x

XS.D. = ( )22

xn

x−

or

n

)x-(x 2

YS.D. = ( )22

yn

y−

or

n

)y-(y 2

Alternatively it can be calculated as

( ) ( )

−−

2222ynxn

yx-xyn

yx

3. Spearman’s Rank Correlation Coefficient

Used to find correlation between two qualitative characteristics

r = nn

d61

3

2

−−

when values are different

r = nn

12

ttd6

13

32

−

−+

−

when values are in repitition




4. Coefficient of concurrent Deviations

A very simple and casual method of finding correlation when we are not serious about

the magnitude of the two variables.

r = ( )

m

m2c −±± ; where m = n – 1

NOTE:

1. If there is a linear relation between two variables, then correlation coefficient will either be

Perfectly positive or Perfectly Negative depending upon the coefficient of variables.

If y = a + bx or y = – a + bx , then the value of ‘r’ shall be +1

If y = a – bx or y = – a – bx , then the value of ‘r’ shall be –1

E.g. If y = 3 + 2x , in this case the correlation coefficient between x and y shall be +1

E.g. If y = – 3 + 2x , in this case the correlation coefficient between x and y shall be +1

E.g. If y = 3 – 2x , in this case the correlation coefficient between x and y shall be –1

E.g. If y = – 3 – 2x , in this case the correlation coefficient between x and y shall be –1

2. Correlation coefficient is neither affected by a shift in origin nor by change in scale except a sign.

E.g. If r (x,y) is the correlation between variables x and y, then

r(u,v) = r(x,y) if u = ± a + bx and v = ± c + dx or u = ± a – bx and v = ± c – dx

r(u,v) = – r(x,y) if u = ± a + bx and v = ± c – dx or u = ± a – bx and v = ± c + dx

Mathematically ),(),( yxrdb

bdvur =

E.g. Correlation coefficient between x and y is given to be 0.56

Case 1: u = 2 + 3x & v = 5 + 2y Correlation between u and v will be 0.56

Case 2: u = – 2 + 3x & v = 5 + 2y Correlation between u and v will be 0.56

Case 3: u = – 2 + 3x & v = – 5 + 2y Correlation between u and v will be 0.56

Case 4: u = 2 – 3x & v = 5 – 2y Correlation between u and v will be 0.56

Case 5: u = 2 – 3x & v = – 5 – 2y Correlation between u and v will be 0.56

Case 6: u = 2 + 3x & v = 5 – 2y Correlation between u and v will be – 0.56

Case 7: u = – 2 + 3x & v = 5 – 2y Correlation between u and v will be – 0.56

Case 8: u = – 2 – 3x & v = 5 + 2y Correlation between u and v will be – 0.56




REGRESSION:

Concerned with estimation of one variable based on the other variable.

EQUATION 1: Y = a + bX (Equation of Y on X)

Here Y is called as Dependent Variable or Explained Variable.

X is called Independent Variable or Predictor or Explanator.

‘a’ and ‘b’ are called the regression parameters

How to Solve: += XbnaY

+= 2XbXaXY

yxb = Regression Coefficient of y on x

yxb = ( )

( )2

XS.D.

yx,cov yxb =

X

Y

S.D.

S.D.r ×

EQUATION 2: X = a + bY (Equation of X on Y)

Here X is called as Dependent Variable or Explained Variable.

Y is called Independent Variable or Predictor or Explanator.

‘a’ and ‘b’ are called the regression parameters

How to Solve: += YbnaX

+= 2YbYaXY

xyb = Regression Coefficient of x on y

xyb =

( )( )2

yS.D.

yx,cov

xyb = S.D.y

S.D.r x×

Method of Least Square




POINTS TO REMEMBER:

• yxxy bb × = r

2

• If r > 0, then xyb &

yxb > 0

• If r < 0, then xyb & yxb < 0

• Coefficient of Determination (Percentage of variation accounted for) = r 2

• Coefficient of Non-Determination (Percentage of variation unaccounted for) = 1 – r2

• Coefficient of Alienation = 21 r−

• Standard Error (S.E.) = N

r1 2−

• Probable Error (P.E) = 0.6745 S.E.

• If r > 6 P.E., r is considered to be significant, otherwise r will be considered to be

insignifanct.

• Population Correlation Coefficient Range = r + P.E.

• If one of the regression coefficient is greater than 1, then other will be less than 1, such that

their multiplication can not exceed 1.

• Regression co-efficient is not affected by shift in origin but are affected by change in scale.

If u = a + bx and v = c + dy ,

then xyuv bd

bb = and yxvu b

b

db =

• Two regression equations are Non – Reversable.

However they becomes identical when r = + 1

They become perpendicular when r = 0

• If two variables x and y are independent or uncorrelated, then obviously the correlation

coefficient between x and y is zero. However, the converse may not be necessarily true. i.e. if

correlation coefficient between two variables comes out to be zero, we can only conclude that

there is no linear relation between two variables, however there may exist a non-linear relation

between them.




PROBABILITY AND EXPECTED

VALUE BY MATHEMATICAL

EXPECTATION 4 Chapter

POINTS TO CONSIDER:

■ P(A) = Events of No. Total

Events Favourable No.of ■ P( A′ ) =

Events of No. Total

Events ble UnfavouraNo.of

■ 0 ≤ P(A) ≤ 1 ■ P(A) + P( A′ ) =1

■ If P(A) = 0, Impossible Events ■ If P(A) = 1, Certain Events

■ Odds in favour of A = Events leUnfavourab

Events Favourable ■ Odds against A =

Events Favourable

Events leUnfavourab

■ P ( )BA ∪ [P(A or B)] = P(A) + P(B) – P ( )BA ∩

■ If the events are mutually Exclusive then P ( )BA ∩ = 0

■ If the events are Independent then P ( )BA ∩ = P(A) ×P(B)

■ P(A – B) = P(A) – P ( )BA ∩

■ Conditional Probability P ( )A

B = ( )

)(AP

BAP ∩

■ Expected Value ( )µ = 332211

pxpxpxxp ++= +…………………… nn px

■ Variance of Expectation ( ) ( )222

μpxσ −=

■ Properties of Expected values

1. Expectation of a constant ‘k’ is ‘k’ i.e. E(k) = k

2. Similarly E(x + y) = E(x) + E(y) ; E(kx) = kE(x) ; E(xy) = E(x) ×E(y)

If it is possible to express the probability (P) as a function of X, then the f(x) is called as Probability

mass function (Pmf) if X is a discrete variable and it is called as Probability density function (Pdf)

if X is a continuous variable. In both cases f(x) > 0




THEORETICAL DISTRIBUTIONS 5 Chapter

1. BINOMIAL DISTRIBUTION:

POINTS TO REMEMBER:

(a) Binomial Distribution is a Discrete Probability Distribution.

(b) Binomial Distribution is a Bi-Parametric Distribution having two parameters ‘n’ & ‘p’

(c) Binomial Distribution can be represented as X ~ B ( n , p )

where probability mass function of x is given by f(x) = P(X = x) = xnx

x

n qpC −

(d) p + q = 1

(e) Mean ( )µ = np

(f) Variance ( 2σ ) = npq

(g) np > npq

(h) Mode can be unimodal or bimodal

Unimodal when (n + l)p is a fraction : Value of Mode is highest integer in the fraction

Bi - Modal when (n + 1) p is an integer : Value of Mode is (n + l )p and (n + l )p – 1

(i) The value of variance is maximum at p = q = 0.5 and the maximum value is n/4

(j) Distribution is positively skewed when p < 0.5 or p < q

(k) Distribution is negatively skewed when p > 0.5 or p > q

(l) Distribution is symmetrical when p = q = 0.5

(m) P(x = 0) + P(x = 1) + P(x = 2) + ………….. P(x = n) = 1

(n) Method of Moments is used for fitting Binomial Distribution.

(o) Additive Property: If X and Y are two independent variables such that X ~ B ( 1n , p)

and Y ~ B ( 2n , p) , then X + Y ~ B ( 1n + 2n , p)




2. POISSON DISTRIBUTION:

POINTS TO REMEMBER:

(a) Poisson Distribution is a Discrete Probability Distribution.

(b) Poisson Distribution is used when trials are infinitely large & probability of success is

almost nil but the product of two is finite. (Poisson Distribution is also called as

distribution of ‘rare events’.

(c) Poisson Distribution is a Uniparametric Distribution having one parameter ‘m’

(d) Poisson Distribution can be represented as X ~ P(m)

where probability mass function of x is given by f(x) = P (X = x) =x!

me xm−

(e) The value of ‘e’ is 2.71828 approx.

(f) m = Mean = Variance = np

(g) Poisson Distribution is always Positively Skewed.

(h) Mode can be unimodal or bimodal

Unimodal when ‘m’ is a fraction : Value of Mode is highest integer in the fraction

Bi - Modal when ‘m’ is an integer : Value of Mode is m and (m – 1)

(i) Additive Property: If X and Y are two independent variables such that X ~ P ( 1m )

and Y ~ P ( 2m ) , then X + Y ~ P ( 1m + 2m )

3. NORMAL DISTRIBUTION:

POINTS TO REMEMBER:

( a ) Normal Distribution is a Continuous Probability Distribution.

( b ) Normal Distribution is a Bi-Parametric distribution having two parameters µ and 2σ

( c ) Normal Distribution can be represented as X ~ N ( µ , 2σ )

where probability mass function of x is given by f(x) =

( )2

2

2σ

μx

e2πσ

1−−

; ∞<<∞− x

Alternatively it can be express as f(x) = [ ]

2πσ

)2/1(Exp2

z−; ∞<<∞− x where Z =

σ

µ−x




( d ) Normal Curve is a Bell Shaped Curve.

( e ) Normal Curve is symmetrical about Mean

( f ) Mean = Median = Mode

( g ) The area under normal curve is 1.

( h ) Mean Divides the whole curve in two equal parts.

( i ) Area under the normal curve to the left of mean & to the right of mean is equal to 0.5

( j ) Normal Distribution is a uni Modal Distribution

( k ) Me = 2

QQ 31 +

( l ) Mean Deviation = π

2σ or 0.8σ

( m ) σµ 675.01 −=Q

( n ) σµ 675.03 +=Q

( o ) Q.D. = 0.675 σ or 2/3 σ

( p ) Additive Property: If X and Y are two independent variables such that

X ~ N (1µ ,

2

1σ ) and Y ~ N (2µ ,

2

2σ ) , then X + Y ~ N (1µ +

2µ , 2

2σ +2

1σ )

( q ) In case of Standard Normal variate Mean is equal to Zero and Standard Deviation is equal

to One.

( r ) Points of Inflexion are σµ + and σµ − . At these points, the normal curve changes its

curvature from concave to convex and from convex to concave.

( s ) Area under normal curves:

σµ ± = 0.6826 σµ 2± = 0.9545 σµ 3± = 0.9973




SKEWNESS :

When distribution is not symmetrical, it is said to be asymmetrical or skewed. Measures of skewness

helps us to know the direction of the variation.

If the distribution is positively skewed, the frequency of the distribution are spread over the greater

range of values on the right side.

If the distribution is Negatively skewed, the frequency of the distribution are spread over the greater

range of values on the Left side.

Measure of skewness :

Absolute measure of Skewness = Mean – Mode

Relative measure of Skewness

Karl Pearson's coefficient of Skewness = Deviation Standard

ModeMean −

MOMENTS :

The moments are used to describe the various characteristics of frequency distribution like central

tendency, variation, skewness and kurtosis.

The arithmetic mean of the various powers of these deviations in any distribution are called the

moments of the distribution

First Moment = ( )

N

XX −=

1µ Second Moment =

( )N

XX2

2

−=µ

Third Moment = ( )

N

XX −=

3

3µ Fourth Moment =

( )N

XX −=

4

4µ

Two important constants of a distribution are calculated from moments are Skewness and Kurtosis

represented by 1β and

2β .

1β (read as Beta one) = 3

2

2

3

µ

µ

2β (read as Beta two) = 2

2

4

µ

µ

CENTRAL MOMENTS ABOUT MEAN OF SOME DISTRIBUTION

Moments Binomial Distribution Poisson Distribution Normal Distribution

First Moment 0 0 0

Second Moment npq m σ2

Third Moment npq(q – p) m 0

Fourth Moment 3(npq)2 + npq(1–6pq) 3m2 + m 3σ4

KURTOSIS :

Kurtosis refers to degree of flatness or peakedness in the region about mode of frequency curve.

If the curve is more peaked than normal, it is called Leptokurtic. If the curve is normal, it is called

Mesokurtic. If the curve is more flat than normal, it is called Platokurtic.

The most important measure of kurtosis is the value of the coefficient of 2β .

The value of 2β for a normal curve is 3.

The greater the value of 2β , the more peaked is the distribution.




OTHER CONTINUOUS PROBABILITY DISTRIBUTION

1. CHI- SQUARE DISTRIBUTION: (2χ – distribution)

Chi – Square Distribution can be represented as X ~2

nχ

where probability density function of x is given by f(x) = ∞<<−−x0 ;xk.e

1 n/2x/2

POINTS TO REMEMBER:

(a) Mean = n

(b) Variance = 2n

(c) Chi – square distribution is always positively skewed

(d) Additive Property: If x and y are two independent chi – square distribution with m & n

degrees of freedom, such that x ~ 2

mχ and y ~ 2

nχ ,

then x + y ~ 2

nm+χ

2. T- DISTRIBUTION:

t – distribution can be represented as x ~nt , where probability density function of x is given by

f (t) =

( )/21n2

n

t1k

+−

+ ; ∞<<∞− t

POINTS TO REMEMBER:

(a) Mean = 0

(b) Variance = 2,2

>−

nn

n

(c) It is always symmetrical about t = 0

(d) If n > 30, t – distribution tends to a normal distribution

3. F- DISTRIBUTION:

F – distribution can be represented as x ~21 , nnF , where probability density function of x is given

by f (F) = ( ) ∞<<+ +−−

F0F/n)n(1k.F/2;nn

1

1 /2n 211

POINTS TO REMEMBER:

(a) Mean = 2n,2n

n2

2

2 >−

and Standard Deviation ( )

4n;4)(nn

2)n2(n

2n

n2

21

21

2

2 >−

−+

−

(b) F – distribution is always positively skewed.

(c) For large value of n1 and n2, F -distribution tends to Normal distribution with Mean = 0

and SD = 21

21

nn

)n2(n +




INDEX NUMBER 6 Chapter

• Relative Measure/Percentage Measure

• Used for Comparision of Variation arising out of difference due to time or due to place

POINTS TO REMEMBER:

01P is Index for time 1 on 0

10P is Index for time 0 on 1

METHODS OF COMPUTATION:

1. Simple Aggregate

2. Simple average of Relatives

3. Weighted Average

(a) Laspeyree (b) Paasche

(c) Marshal Edgeworth (d) Dorbish and Bowley

(e) Fisher

TEST OF ADEQUACY:

1. UNIT TEST: The Index Number should be independent of units.

Satisfied by all types of Index Numbers except Simple Aggregate.

2. TIME REVERSAL TEST: 1001 PP × = 1

Satisfied by Simple aggregate, Marshal and Fisher.

3. CIRCULAR TEST: Extension of Time Reversal Test. 1201201 =×× PPP

Satisfied only by Simple Aggregate

4. FACTOR REVERSAL TEST: 0101 QP × = V

01

Satisfied only by Fisher.




LIST OF FORMULAE:

S.NO. PARTICULARS PRICE QUANTITY

1 SIMPLE AGGREGATE

=0

1

01P

PP

=0

1

01Q

QQ

2 SIMPLE AVERAGE OF

RELATIVES n

P

P

P

=0

1

01

n

Q

Q

Q

=0

1

01

3 WEIGHTED AVERAGE

(A) LASPEYRE

=00

01

01QP

QPP

=00

01

01PQ

PQQ

(B) PASCHE

=10

11

01QP

QPP

=10

11

01PQ

PQQ

(C) MARSHAL

EDGEWORTH

( )( )

+

+=

100

101

01QQP

QQPP

( )( )

+

+=

100

101

01PPQ

PPQQ

(D) DORBISH & BOWLEY

2

QP

QP

QP

QP

P10

11

00

01

01

+

= 2

PQ

PQ

PQ

PQ

Q10

11

00

01

01

+

=

(E) FISHER (IDEAL)

×=10

11

00

01

01QP

QP

QP

QPP

×=10

11

00

01

01PQ

PQ

PQ

PQQ

Other Related Formulas :

Chain Index 100

year previous ofIndex Chain year Current of RelativeLink

×=

Deflated Value YearCurrent theofIndex Price

ValueCurrent =

Shifted Price Index 100shifted be tohasit on which year theofIndex Price

Index Price Original ×=

paper 3 business mathematics logical reasoning …

Documents