Revision Of All Topics1.Coefficient of . Range . 2.Coefficient of . QD . 1.Coefficient of MD...

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Revision Of All Topics

Quantitative Aptitude & Business Statistics

Quantitative Aptituide & Business Statistics: Revision Of All Toics

Def:Measures of Central Tendency

• A single expression representing the whole group,is selected which may convey a fairly adequate idea about the whole group.

• This single expression is

known as average.

Averages are central part of distribution and, therefore ,they are also called measures of central tendency.

Types of Measures central tendency:

There are five types ,namely 1.Arithmetic Mean (A.M) 2.Median 3.Mode 4.Geometric Mean (G.M) 5.Harmonic Mean (H.M)

Arithmetic Mean (A.M)

The most commonly used measure of central tendency. When people ask about the “average" of a group of scores, they usually are referring to the mean.

• The arithmetic mean is simply dividing the sum of variables by the total number of observations.

Arithmetic Mean for Ungrouped data is given

xxxx n

∑=++++ == 1......321

Arithmetic Mean for Discrete Series

=++++ =++++

xfxfxfxf

ffffX nn

......

....332211

Arithmetic Mean for Continuous Series

fdAX ×+= ∑

Weighted Arithmetic Mean

• The term ‘ weight’ stands for the relative importance of the different items of the series. Weighted Arithmetic Mean refers to the Arithmetic Mean calculated after assigning weights to different values of variable. It is suitable where the relative importance of different items of variable is not same

• Weighted Arithmetic Mean is specially useful in problems relating to

• 1)Construction of Index numbers. • 2)Standardised birth and death rates

• Weighted Arithmetic Mean is given by

∑∑=

Mathematical Properties of Arithmetic Mean

• 1.The Sum of the deviations of the items from arithmetic mean is always Zero. i.e.

• 2.The sum of squared deviations of the items from arithmetic mean is minimum or the least

( ) 0=−∑ XX

( ) 02

≤−∑ XX

• 3.The formula of Arithmetic

mean can be extended to

compute the combined average

of two or more related series

• 4.If each of the values of a

variable ‘X’ is increased or decreased by some constant C, the arithmetic mean also increased or decreased by C .

• Similarly When the value of the variable ‘X’ are multiplied by constant say k,arithmetic mean also multiplied the same quantity k .

• When the values of variable are divided by a constant say ‘d’ ,the arithmetic mean also divided by same quantity

Median

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Median = 5 Median = 5

Median for raw data

• When given observation are even • First arrange the items in ascending

order then

• Median (M)=Average of Item

Median for raw data

• When given observation are odd • First arrange the items in ascending

order then

• Median (M)=Size of Item

Median for continuous series

−+= 2

Where M= Median; L=Lower limit of the Median Class,m=Cumulative frequency above median class f=Frequency of the median class N=Sum of frequencies

Merits of Median

• 1.Median is not affected by extreme values .

• 2.It is more suitable average for dealing with qualitative data ie.where ranks are given.

• 3.It can be determined by graphically.

Limitations of Median

1.It is not based all the items of the series .

2.It is not capable of algebraic treatment .Its formula can not be extended to calculate combined median of two or more related groups.

• A measure of central tendency • Value that occurs most often • Not affected by extreme values • Used for either numerical or

categorical data • There may be no mode or

several modes 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Mode = 9

0 1 2 3 4 5 6

No Mode

• A distribution that consists of only one of each score has n modes.

• When there are ties for the most frequent score, the distribution is bimodal if two scores tie or multimodal if more than two scores tie.

Mode for Continuous Series

ffLZ ×

−−

2Where Z= Mode ;L=Lower limit of the Mode Class f0 =frequency of the pre modal class f1=frequency of the modal class f2=frequency of the post modal class C=Class interval of Modal Class

Relationship between Mean, Median and Mode

• The distance between Mean and Median is about one third of distance between the mean and the mode.

Karl Pearson has expressed the relationship as follows.

Mean –Mode=(Mean-Median)/3 Mean-Median=3(Mean-Mode)

Mode =3Median-2Mean Mean=(3Median-Mode)/2

Limitations of Mode

• 1.In case of bimodal /multi modal series ,mode cannot be determined.

• 2.It is not capable for further algebraic treatment, combined mode of two or more series cannot be determined.

Geometric mean

xxxxx/1

Computation of G.M -Discrete Series

• Take the logarithms of each item of variable and multiply with the respective frequencies obtain their total

i.e ∑ f .log X • Calculate G M as follows

AntiMGlog.

Weighted Geometric Mean

∑∑

AntiMGlog.

• It is useful for averaging ratios and percentages rates are increase or decrease

Harmonic Mean (H.M)

• Harmonic Mean of various items of a series is the reciprocal of the arithmetic mean of their reciprocal .Symbolically,

NMH1.......111

• 4.It is useful for averaging measuring the time ,Speed etc

Quartiles

1NSizeQ1+

= Item

thNSizeQ4

+= Item

Octiles

thNjSizeOj 8)1( +

= Item

thNSizeO8

+= Item

Deciles

thNjSizeDj 10)1( +

= Item

thNSizeD10

+= Item

Relation Ship Between Partition Values

1.Q1=O2=P25 value of variate which exactly 25% of the total number of observations

2.Q2=D5=P50,value of variate which exactly 50% of the total number of observations.

3. Q3=O6=P75,value of variate which exactly 75% of the total number of observations

Measures of Dispersion

Why Study Dispersion?

• An average, such as the mean or the median only locates the centre of the data.

• An average does not tell us anything about the spread of the data.

Properties of Good Measure of Dispersion

• Simple to understand and easy to calculate

• Rigidly defined • Based on all items • A meanable to algebraic

treatment • Sampling stability • Not unduly affected by Extreme

items.

Absolute Measure of Dispersion

Based on selected

items Based on all items

1.Range 2.Inter Quartile Range

1.Mean Deviation 2.Standard Deviation

Relative measures of Dispersion

Based on Selected items

Based on all items

1.Coefficient of Range 2.Coefficient of QD

1.Coefficient of MD 2.Coefficient of SD & Coefficient of Variation

The Range

• The simplest measure of dispersion is the range.

• For ungrouped data, the range is the difference between the highest and lowest values in a set of data.

• RANGE = Highest Value - Lowest Value

Coefficient of Range

• Coefficient of Range =

Interquartile Range

• The interquartile range is used to overcome the problem of outlying observations.

• The interquartile range measures the range of the middle (50%) values only

• Inter quartile range = Q3 – Q1 • It is sometimes referred to as

the quartile deviation or the semi-inter quartile range.

Lower Quartile Deviation

LQ1 ×

Upper Quartile Deviation

• Inter Quartile Range=Q3-Q1

• Coefficient of Quartile

Deviation

Mean Deviation

• The mean deviation takes into consideration all of the values

• Mean Deviation: The arithmetic mean of the absolute values of the deviations from the arithmetic mean

∑ −=

Where: X = the value of each observation X = the arithmetic mean of the values

n = the number of observations

|| = the absolute value (the signs of the deviations are disregarded)

• If the data are in the form of a frequency distribution, the mean deviation can be calculated using the following formula:

Where f = the frequency of an observation x

n = Σf = the sum of the frequencies

Frequency Distribution Mean Deviation

∑∑ −

Standard Deviation

• Standard deviation is the most commonly used measure of dispersion

• Similar to the mean deviation, the standard deviation takes into account the value of every observation

−=σ

∑∑

Thus SD is never less than QD and MD

In a normal distribution there is fixed relationship

Properties of Standard Deviation

• Independent of change of origin • Not independent of change of

Scale. • Fixed Relationship among

measures of Dispersion.

µ−3σ µ−2σ µ−1σ µ µ+1σ µ+2σ µ+ 3σ

. and between iprelationsh the showing Curve Shaped-Bell µσ

68.27%

95.45%

99.73%

• Minimum sum of Squares; The Sum of Squares of Deviations of items in the series from their arithmetic mean is minimum.

• Standard Deviation of n natural numbers

121N 2 −

• Combined standard deviation

12 NNdNdNNN

+++σ+σ

• Where =Combined standard Deviation of two groups

• =Standard Deviation of first group

• N1=No. of items of First group • N2=No. of items of Second group • = Standard deviation of Second

1211 XXd −=

1222 XXd −=Where is the combined mean of two groups

Limitations of Standard Deviation

• It can’t be used for comparing the variability of two or more series of observations given in different units. A coefficient of Standard deviation is to be calculated for this purpose.

• It is difficult to compute and compared

Variance • Variance is the arithmetic mean

of the squares of deviations of all the items of the distributions from arithmetic mean .In other words, variance is the square of the Standard deviation=

• Variance=

iancevar=σ

Interpretation of Variance

• Smaller the variance ,greater the uniformity in population.

• Larger the variance ,greater the variability

The Coefficient of Variation

• The coefficient of variation is a measure of relative variability It is used to measure the changes that have taken place in a population over time

• Formula: Where: X = mean = standard deviation

CV ×σ

Correlation

• Correlation is the relationship that exists between two or more variables.

• If two variables are related to each other in such a way that change increases a corresponding change in other, then variables are said to be correlated.

Methods of studying correlation

Method of studying

Correlation

Graphic Algebraic

1.Karl Pearson 2.Rank method

3.Concurrent Deviation

Scatter Diagram Method

• Scatter diagrams are used to demonstrate correlation between two quantitative variables.

Scatter Plots of Data with Various Correlation Coefficients

r = -1 r = -Ve r = 0

r = +Ve r = 1

The value of r lies between -1 and +1

• If r=0 There exists no relationship between the variables

• If +0.75 ≤r ≤ +1 There exists high positive relationship between the variables .

• If -0.75 ≥ r ≥ -1 There exists high negative relationship between the variables

• If +0.5 ≤r ≤ 0.75 There exists Moderate positive relationship between the variables .

• If -0.50 ≥ r >-0.75 There exists moderate negative relationship between the variables.

• If r > -0.50 There exists low negative relationship between the variables

• If r <0.5 There exists low positive relationship between the variables .

• If +0.5 ≤r ≤ 0.75 There exists Moderate positive relationship between the variables .

• If -0.50 ≥ r >-0.75 There exists moderate negative relationship between the variables.

• If r > -0.50 There exists low negative relationship between the variables

• If r <0.5 There exists low positive relationship between the variables .

Covariance • Definition : Given a n pairs of

observations (X1,Y1),(X2,Y2) .,,,,,, (Xn,Yn) relating to two variables X and Y ,the Covariance of X and Y is usually represented by Cov(X,Y)

( )( )

−−=

Properties of Co-Variance

• Independent of Choice of origin • not Independent of Choice of

Scale. • Co-variance lies between negative

infinity to positive infinity. • In other words co-variance may

be positive or negative or Zero.

Coefficient of Correlation • Measures the strength of the

linear relationship between two quantitative variables

( )( )

( ) ( )1

i ii i

X X Y Yr

X X Y Y

− −=

− −

∑ ∑

Properties of KralPear son’s Coefficient of Correlation

• Independent of choice of origin • Independent of Choice Scale • Independent of units of

Measurement

Assumptions of Karl Pearson’s Coefficient of Correlation

• Linear relationship between variables.

• Cause and effect relationship. • Normality.

• The correlation coefficient lies between -1 and +1

• The coefficient of correlation is the geometric mean of two regression coefficients.

• The correlation coefficient lies between -1 and +1

• The coefficient of correlation is the geometric mean of two regression coefficients.

Correlation for Bivariate analysis

( )( )

( ) ( )∑ ∑∑ ∑

∑ ∑∑

−−

Standard error

• Standard error of co efficient of correlation is used foe ascertaining the probable error of coefficient of correlation

• Where r=Coefficient of correlation • N= No. of Pairs of observations

21−=

Probable Error

• The Probable error of coefficient of correlation is an amount which if added to and subtracted from value of r gives the upper and lower limits with in which coefficients of correlation in the population can be expected to lie. It is 0.6745 times of standard error.

Uses of Probable Error

• PE is used to for determining reliability of the value of r in so far as it depends on the condition of random sampling.

Case Interpretation

1.If |r |< 6 PE

2. 1.If |r | >6 PE

The value of r is not at all significant. There is no evidence of correlation. The value of r is significant. There is evidence of correlation

Spearman’s Rank Correlation

Spearman’s Rank Correlation uses ranks than actual observations and make no assumptions about the population from which actual observations are drawn.

−−= ∑

Spearman’s Rank Correlation for repeated ranks

• Where m=the no of times ranks are repeated

• n=No of observations • r= Correlation Coefficient

.....12

−=∑

Features of Spearman’s Rank Correlation

• Spearman’s Correlation coefficient is based on ranks rather than actual observations .

• Spearman’s Correlation coefficient is distribution –free and non-parametric because no strict assumptions are made about the form of population from which sample observation are drawn.

Merits of Spearman’s Rank Correlation

• Simple to understand and easy to apply

• Suitable for Qualitative Data • Suitable for abnormal data. • Only method for ranks • Appliacble even for actual

Limitations of Spearman’s Rank Correlation

• Unsuitable data • Tedious calculations • Approximation

When is used Spearman’s Rank Correlation method

• The distribution is not normal • The behavior of distribution is

not known • only qualitative data are given

Meaning of Concurrent Deviation Method

• Concurrent Deviation Method is based on the direction of change in the two paired variables .The coefficient of Concurrent Deviation between two series of direction of change is called coefficient of Concurrent Deviation .

• rc=Coefficient of Concurrent deviation • C= no of positive signs after multiplying

the change direction of change of X- series and Y-Series

• n=no. of pairs of observations computed

−±±=

Limitations of Concurrent Deviation Method

• This method does not differentiate between small and big changes .

• Approximation

Merits of Concurrent Deviation

• Simple to understand and easy to calculate.

• Suitable for large N

Regression • Regression is the measure of

average relationship between two or more variables in terms of original units of the data.

Regression lines

• Regression line X on Y

• Where X= Dependent Variable Y =Independent variable a=intercept and b= slope

bYaX +=

( )YYbXX xy −=−• Another way of regression line X

( )YYrXXy

x −=−σσ

Calculate bxy

( )∑ ∑

∑ ∑ ∑

YbXa −=

Regression coefficients

• There are two regression coefficients byx and bxy

• The regression coefficient Y on X is

yyx .rb

The regression coefficient X on Y is

xxy .rb

Regression coefficients

The regression coefficient X on Y is

xxy .rb

• Regression line Y on X

• Where Y= Dependent Variable • X =Independent variable • a=intercept and b= slope

bXaY +=

• Another way of regression line Y on X

( )XXrYYx

y −=−σσ

( )XXbyxYY −=−

Calculate byx

( )∑ ∑

∑ ∑ ∑

XbYa −=

Properties of Linear Regression

• Two Regression Equations. • Product of regression

coefficient. • Signs of Regression Coefficient

and correlation coefficient. • Intersection of means. • Slopes .

• Angle between Regression lines

Value of r Angle between Regression Lines

a) If r=0

b) If r=+1 or -1

Regression lines are perpendicular to each other. Regression lines are coincide to become identical .

Properties of regression coefficients

1.Same Sign. 2.Both cannot greater than one . 3.Independent of origin but not of

scale . 4.Arithmetic mean of regression

coefficients are greater than Correlation coefficient.

5.r,bxy and byx have same sign. 6 .Correlation coefficient is the

Independent of origin but not of scale.

• This property states that if the original pairs of variables is (x,y) and if they are changed to the pair (u,v), where x=a + p u and y=c +q v

andbpq

Measure of Variation: The Sum of Squares

SST = SSR + SSE

Total Sample

Variability

= Explained Variability

+ Unexplained Variability

Measure of Variation: The Sum of Squares

• SST = Total Sum of Squares – Measures the variation of the Yi

values around their mean Y • SSR = Regression Sum of Squares

– Explained variation attributable to the relationship between X and Y

• SSE = Error Sum of Squares – Variation attributable to factors

other than the relationship between X and Y

Coefficient of determination(r2)

• The coefficient of determination is the square of the coefficient of correlation. It is equal to r2.

• The maximum value of r2 is unity and in the case of all the variation in Y is explained by the variation in X ,it is defined as

• Coefficient of determination( r2 )

nceTotalVariainacevarExplained

Coefficient of non-determination(k2)

• Coefficient of non-determination(k2)=1-r2

nceTotalVaria

inacevarlainedexpUn=

Population or Universe refers to the aggregate of statistical information on a particular character of all the members covered by an investigation/enquiry. For example, constitute population.

SAMPLE

• Sample refers to the part of aggregate statistical information (i.e. Population) which is actually selected in the course of an investigation/enquiry to ascertain the characteristics of the population.

SAMPLE SIZE

• Sample size refers to the number of members of the population included in the sample.

• Usually, the sample size is denoted by 'n'

METHODS OF SAMPLING

1.Deliberate, Purposive or Judgment Sampling. 2. Block or Cluster Sampling 3. Area Sampling 4. Quota Sampling 5. Random (or Probability) Sampling 6. Systematic Sampling

METHODS OF SAMPLING

• 7. Stratified Sampling • 8. Multi Stage Sampling

Standard Error Meaning • Standard Error of a given statistic is the

standard deviation of sampling distribution of that statistic. In other words, standard error of a given 'statistic is the standard deviation of all possible values of that statistic in repeated sample of a fixed size from given populatiot.It is a measurer of the divergence between‘ the‘ statistic and parameter values. This divergence varies with the sample size (n).

HOW TO COMPUTE STANDARD ERROR OF THE

MEAN Statistic Standard Error

sample mean SRSWR

HOW TO COMPUTE STANDARD ERROR OF THE MEAN

Statistic Standard Error

sample mean SRSWOR X

n −−σ

PROPORTION

Observed sample Proportion ‘P’ SRS WR

PROPORTION

Observed sample Proportion ‘P’ SRS WOR n

PQ1NnN.

−−

1.Problem of Estimation

• This problem arises when no information is available about the parameters of the population from which the sample is drawn. Statistics obtained from the sample are used to estimate the unknown parameter of the population from which the sample is drawn.

MEANING OF ESTIMATION

• In the context of statistics, an estimation is a statistical technique of estimating unknown population parameters from the corresponding sample statistic. Two ways - A population parameter ,can be estimated in two ways:

• 1. Point Estimation, and • 2. Interval Estimation

POINT ESTIMATION

• It provides a single value of a statistic that is used to estimate an unknown population parameter.

• Estimator -The statistic which is used to obtain a point estimate is called estimator.

• Criteria for a Good Estimator - According to R A: Fisher, the criteria for good estimator are:

• (a) Unbiased ness, (b) Consistency, (c) Efficiency and (d) Sufficiency

INTERVAL ESTIMATION

• Provides an interval of finite width centered ,if the point estimate of the parameter, within which unknown parameter is expected to lie with a specified probability. Such an interval is called a confidence interval for population parameter.

• Confidence Limits - The lower and upper limits of the confidence interval are called confidence limits.

• Confidence Coefficient – The probability with which the confidence interval will include the true value of the parameter.

Calculation of Confidence 'Limits

• The calculation of confidence limits is based on the appropriate statistic. If the population is normal or the sample size (n) is large (say more than 30), percentage of area Under the standard normal curve maybe used to find confidence limits) corresponding to any specified percentage of confidence.

Why hypothesis is used?

• Hypothesis Testing is a process of making a decision on whether to accept or reject an assumption about the population parameter on the basis of sample information at a given level of significance.

NULL HYPOTHESIS (Ho)

• Null hypothesis is the assumption which we wish to test and whose validity is tested for possible rejection on the basis of sample information.

• It asserts that there is no significant difference between the sample statistic

NULL HYPOTHESIS (Ho)

• Acceptance - The acceptance of null hypothesis implies that we have no evidence to believe otherwise and indicates that the difference is not significant and is due to sampling fluctuations.

• Rejection - The rejection of null hypothesis implies that it is false and indicates that the difference is significant.

ALTERNATIVE HYPOTHESIS(H1)

• Alternative hypothesis is the hypothesis which differs from the null hypothesis. It is not tested

• Rejection of one implies the acceptance of the other.

• Symbol - It is denoted by H1 • Acceptance - Its depends on the rejection of

the null hypothesis. Rejection - Its rejection depends on the acceptance of the null hypothesis.

Example for ALTERNATIVE HYPOTHESIS(H1)

• If population mean is 50, an alternative hypothesis may be anyone of the following three:

H1 : µ ≠50,H1 : µ > 50,H1: µ < 50 • Ho and HI are mutually exclusive

statements in the sense that both cannot hold good simultaneously.

• Rejection of one implies the acceptance of the other.

LEVEL OF SIGNIFICANCE(α )

• Level of significance is the maximum probability of rejecting the null hypothesis when it is true.

• Symbol - It is usually expressed as % and is denoted by symbol α (called 'Alpha ')

• Usefulness - It is used as a guide in decision-making. It is used to indicate the upper limit of the size of the critical rejection.

Example of LEVEL OF SIGNIFICANCE(α )

• At 5% level of significance implies that there are about 5 chances in 100 of rejecting the Ho when it is true or in other words, we are about 95% confident that we will make a correct decision.

Usefulness

• It is used as a guide in decision making regarding acceptance or rejection of Ho' If the value of the test statistic falls in the critical region, the null hypothesis is rejected. If the value of the test statistic does not fall in the critical region, the null hypothesis is accepted.

TEST STATISTIC

Test-Statistic Used for i) Z-test

ii) t-test

For test of Hypothesis involving large sample i.e. n> 30 Hypothesis involving small sample i.e. n≤ 30 and if SD is unknown.

TEST STATISTIC

Test-Statistic Used for iii) Chi-square test iv) F-test

Test For testing the discrepancy between

observed frequencies and expected frequencies

without any reference to population parameter

Testing the sample variances.

Two-Tailed Tests level of significance(α )

Critical Region

Lower Critical Value

Upper Critical Value

Critical Region

Two –tailed tests Level of significance α

Acceptance Region (1-α)

• by one tail (or side) under the curve when the null hypothesis is tested against 'one sided alternative' right or left. The tests of hypothesis which are based on the critical region represented by one tail (on right side or on left side)

One-Tail Test (left- tail)

Two-Tail Test

One-Tail Test (right- tail)

Acceptance Region

Critical Values of test-statistic

Types of tests

Level of Significance 1% 5%

1.Two Tailed 2.One tailed

a. Right tailed b. Left tailed

± 2.58

+ 2.33 - 2.33

± 1.96

+ 1.645 - 1.645

Type I and Type II Errors

• The decision to accept or reject null hypothesis Ho is made on the basis of the information supplied by the sample data. There is always a chance of committing an error. There are two possible types of error in the test of a hypothesis.

Type I and Type II Errors • Type I Error - This is the error committed by

the test in rejecting a true null hypothesis. The probability of committing Type I Error is denoted by α , the level of significance. .

• Type II Error - This is the error committed by the test in accepting a false null hypothesis. The probability of committing Type II Error is denoted by β.

POWER OF THE TEST

• Power of the Test is the probability of rejecting a false null hypothesis. It can be calculated as follows:

• Power of the Test = I-Probability of Type II Error.=1-β

Decision Statement Reject H0 AcceptH0

H0 is True Wrong (Type I Error)

Correct

H0 is False Correct

Wrong (Type II Error)

Confidence Limits • The confidence limits of

population parameter are calculated

• 95% Confidence Limits 99% Confidence Limits are

• Sample Statistic± 1.96 S. E. Sample Statistic ± 2.58 S. E.

TEST OF SIGNIFICANCE OF LARGE SAMPLES

• A sample is regarded as large only if its size exceeds 30.

• The following assumptions are made while dealing with problems relating to large samples:

• The random sampling distribution of a statistic is approximately normal and Values given by the samples are approximately close to the population value.

Testing the significance of Mean of Random Sample

If Population SD is known

XX.E.S

σµ−

µ−=

SampleSizenSDPopulationMeanPopulation

SampleMeanX

==σ=µ=

Test for difference between the Standard deviations of two

samples

andSSUseareunknown

andIf σσ

Test for number of Success

npqPpZ −

Test for Proportion of Success

PpZ −=

Test for difference between the two Proportions

PPWhere

nnpnpnP̂

THE END

Revision Of All Topics

Revision Of All Topics1.Coefficient of . Range . 2.Coefficient of . QD . 1.Coefficient of MD...

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