quantitative method cp 102

8/18/2019 Quantitative Method Cp 102

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QUANTITATIVE METHOD-CP-102Q.1. What do you mean by measures of central tendency?

Measures of central tendency are very useful in Statistics. Their

importance is because of the following reasons:

(i) To find representative value:

Measures of central tendency or averages give us one value for the

distribution and this value represents the entire distribution. In this way

averages convert a group of figures into one value.

(ii) To condense data:

Collected and classified figures are vast. To condense these figures we

use average. Average converts the whole set of figures into just one

figure and thus helps in condensation.(iii) To make comparisons:

To make comparisons of two or more than two distributions, we have to

find the representative values of these distributions. These

representative values are found with the help of measures of the central

tendency.

(iv) Helpful in further statistical analysis:

Many techniques of statistical analysis like Measures of Dispersion,

Measures of Skewness, Measures of Correlation, and Index Numbers

are based on measures of central tendency. That is why; measures of

central tendency are also called as measures of the first order.

Seeing this importance of averages in statistics, Prof. Bowley said

"Statistics may rightly be called as science of averages."

Averages are very useful in Economics. It is because of the following

reasons:(i) Helpful in knowing the structure of any economy:

For studying the structure of any economy we use per capita income,

per capita consumption, per capita saving, per hectare production, per

worker production, etc. All these are averages.

(ii) Helpful in comparing different economies:

Suppose we are to compare the economies of Punjab, Haryana and

Himachal. For this purpose we shall use per capita income which is

nothing but an average.

(iii) Helpful in studying various economic problems:

These days the different economic problems are studied with the help

of Index numbers. For example problem of inflation is studied with the

help of price index number. Index numbers are nothing but special type

of averages.

(iv) Helpful in formulating and evaluating economic policy: Averages are used in formulating and evaluating economic policy. For

example, if we are to study the effect of economic planning on Indian

economy, we may use per capita income.

(v) Helpful in research:

Measures of central tendency are used in statistical analysis. Therefore,

these are used for research in Economics.

Limitations

In spite of this importance, measures of central tendency have many

limitations, which are as follows:

(i) It can be used properly only by skilled persons.

(ii) Sometimes, average is such value which is not in the distribution

hence is not true representative. For example mean of 100, 300, 100, 50

and 250 is 160 which are not in the distribution and hence not true

representative.(iii) Sometimes average gives absurd results. For example, we find

average number of members per family as 2.3.

(iv) Measures of central tendency do not describe the true structure of

the distribution. Two or more than two distributions may have same

mean but different structure.

Q.2.What is measure of dispersion state its range & standard deviation

?

Ans:- A measure of statistical dispersion is a nonnegative real

number that is zero if all the data are the same and increases as the

data become more diverse.

Most measures of dispersion have the same units as the quantity being

measured. In other words, if the measurements are in metres or

seconds, so is the measure of dispersion. Such measures of dispersio

include:

Sample standard deviation

Interquartile range (IQR)

Range

Mean absolute difference (also known as Gini mean absolute

difference)

Median absolute deviation (MAD)

Average absolute deviation (or simply called average deviation)

Distance standard deviation

These are frequently used (together with scale factors)

as estimators of scale parameters, in which capacity they arecalledestimates of scale. Robust measures of scale are those unaffec

by a small number of outliers, and include the IQR and MAD.

All the above measures of statistical dispersion have the useful prop

that they are location-invariant, as well as linear in scale.[clarification

needed ] So if a random variable X has a dispersion of S X then a linear

transformation Y = aX + b for real aand b should have

dispersion SY = |a|S X .

Other measures of dispersion are dimensionless. In other words, the

have no units even if the variable itself has units. These include:

Coefficient of variation

Quartile coefficient of dispersion

Relative mean difference, equal to twice the Gini coefficient

Entropy: While the entropy of a discrete variable is location-invariant and scale-independent, and therefore not a measure

dispersion in the above sense, the entropy of a continuous vari

is location invariant and additive in scale: If Hz is the entropy of

continuous variable z and y=ax+b, then Hy=Hx+log(a).

There are other measures of dispersion:

Variance (the square of the standard deviation) – location-

invariant but not linear in scale.

Variance-to-mean ratio – mostly used for count data when the

term coefficient of dispersion is used and when this ratio

is dimensionless, as count data are themselves dimensionless, n

otherwise.

Some measures of dispersion have specialized purposes, among them

the Allan variance and the Hadamard variance.

For categorical variables, it is less common to measure dispersion bysingle number; see qualitative variation. One measure that does so i

the discrete entropy.

6 properties of a good Measure of Dispersion

Since measures of dispersion are usually called as averages of

second order, they should possess all the qualities of a good aver

According to Yule and Kendall, they are as follows

1) It should be easy to calculate and simple to follow.

2) It should be rigidly defined: For the same data, all the methods sh

produce the same result.

3) It should be based on all the items so as to be more representative

4) It should be amenable to further algebraic treatment.

5) It should have sampling stability.

6) It should not be unduly affected by the extreme items.

Q.3. What is skewness state different co-efficient of variation .

Ans:- Comparison of skewness coefficient, coefficient of variat

and Gini coefficient as inequality measures within populations

Summary. The moment skewness coefficient, coefficient of varia

and Gini coefficient are contrasted as statistical measures of inequ

among members of plant populations. Constructed examples, real d

examples, and distributional considerations are used to illust

pertinent properties of these statistics to assess inequality. All th

statistics possess some undesirable properties but these properties

shown to be often unimportant with real data. If the underl

distribution of the variable follows the often assumed two-param

lognormal model, it is shown that all three statistics are likely to

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highly and positively correlated. In contrast, for distributions which are

not two-parameter lognormally distributed, and when the distribution is

not concentrated near zero, the coefficient of variation and Gini

coefficient, which are sensitive to small shifts in the mean, are often of

little practical use in ordering the equality of populations. The coefficent

of variation is more sensitive to individuals in the right-hand tail of a

distribution than is the Gini coefficient. Therefore, the coefficient of

variation may often be recommended over the Gini coefficient if a

measure of relative precision is selected to assess inequality. The

skewness coefficient is suggested when the distribution is either three-

parameter lognormally distributed (or close to such), or when a

measure of relative precision is not indicated.

Measures of Variance

Common Measures of Variance

Range

The range is the difference between the high and low values. Since it

uses only the extreme values, it is greatly affected by extreme values.

Procedure for finding

1. Take the largest value and subtract the smallest value

Formula

Variance

The variance is the average squared deviation from the mean. Itusefulness is limited because the units are squared and not the same as

the original data. The sample variance is denoted by s2, it is an unbiased

estimator of the population variance.


1. Find the mean of the data

2. Subtract the mean from each value to find the deviation from

the mean

3. Square the deviation from the mean

4. Total the squares of the deviation from the mean

5. Divide by the degrees of freedom (one less than the sample

size)

Formula

Standard Deviation

The standard deviation is the average deviation from the mean. It is

found by taking the square root of the variance and solves the problem

of not having the same units as the original data. The sample standard

deviation is denoted by s. It is not an unbiased estimator of the

population standard deviation.


1. Find the variance

2. Take the square root

Formula

Less Common Measures of Variance

Mean Absolute Deviation

The sum of the deviations from the mean will always be zero. We need

to make sure that none of the deviations are negative. We can do this

by squaring each deviation (as we do in the variance or standard

deviation) or by taking the absolute value (as we do in the mean

absolute deviation).



2.

Subtract the mean from each data value to get the deviation

from the mean

3.

Take the absolute value of each deviation from the mean

4. Total the absolute values of the deviations from the mean

5. Divide the total by the sample size.

Formula

Variation

The variation is the sum of the squares of the deviations from the m

It has units that are squared instead of the same as the original data

it does not take the sample size into account.Procedure for finding


2.

Subtract the mean from each value to find the deviation f

the mean

3. Square the deviation from the mean

4. Total the squares of the deviation from the mean

Formula

Range Rule of Thumb

The range rule of thumb says that the range is approximately four ti

the standard deviation. Alternatively, the standard deviation

approximately one-fourth the range. That means that most of the

lies within two standard deviations of the mean.


1. Find the range

2. Divide it by four

Formula

Pearson's Index of Skewness

Pearson's index of skewness can be used to determine whether the

is symmetric or skewed. If the index is between -1 and 1, then

distribution is symmetric. If the index is no more than -1 then

skewed to the left and if it is at least 1, then it is skewed to the right.Procedure for finding

1. Find the mean, median, and standard deviation of the data

2. Subtract the median from the mean.

3.

Multiply by 3

4. Divide by the standard deviation

Formula

Coefficient of Variation

The coefficient of variation is expressed as a percent and describes

standard deviation relative to the mean. It can be used to comp

variability when the units are different (the units will divide providing just a raw number).


1. Find the mean and standard deviation for the data

2.

Divide the standard deviation by the mean

3. Multiply by 100

Formula

Q.4. What is maximum & minimum & its application ?

Ans:-the Main Provisions

https://people.richland.edu/james/ictcm/2001/descriptive/helpcenter.htmlhttps://people.richland.edu/james/ictcm/2001/descriptive/helpcenter.htmlhttps://people.richland.edu/james/ictcm/2001/descriptive/helpcenter.htmlhttps://people.richland.edu/james/ictcm/2001/descriptive/helpcenter.htmlhttps://people.richland.edu/james/ictcm/2001/descriptive/helpcenter.htmlhttps://people.richland.edu/james/ictcm/2001/descriptive/helpcenter.htmlhttps://people.richland.edu/james/ictcm/2001/descriptive/helpcenter.htmlhttps://people.richland.edu/james/ictcm/2001/descriptive/helpcenter.htmlhttps://people.richland.edu/james/ictcm/2001/descriptive/helpcenter.htmlhttps://people.richland.edu/james/ictcm/2001/descriptive/helpcenter.htmlhttps://people.richland.edu/james/ictcm/2001/descriptive/helpcenter.htmlhttps://people.richland.edu/james/ictcm/2001/descriptive/helpcenter.htmlhttps://people.richland.edu/james/ictcm/2001/descriptive/helpcenter.html#medianhttps://people.richland.edu/james/ictcm/2001/descriptive/helpcenter.html#medianhttps://people.richland.edu/james/ictcm/2001/descriptive/helpcenter.html#medianhttps://people.richland.edu/james/ictcm/2001/descriptive/helpvariance.html#stdevhttps://people.richland.edu/james/ictcm/2001/descriptive/helpvariance.html#stdevhttps://people.richland.edu/james/ictcm/2001/descriptive/helpvariance.html#stdevhttps://people.richland.edu/james/ictcm/2001/descriptive/helpcenter.htmlhttps://people.richland.edu/james/ictcm/2001/descriptive/helpcenter.htmlhttps://people.richland.edu/james/ictcm/2001/descriptive/helpcenter.htmlhttps://people.richland.edu/james/ictcm/2001/descriptive/helpvariance.html#stdevhttps://people.richland.edu/james/ictcm/2001/descriptive/helpvariance.html#stdevhttps://people.richland.edu/james/ictcm/2001/descriptive/helpvariance.html#stdevhttps://people.richland.edu/james/ictcm/2001/descriptive/helpvariance.html#stdevhttps://people.richland.edu/james/ictcm/2001/descriptive/helpcenter.htmlhttps://people.richland.edu/james/ictcm/2001/descriptive/helpvariance.html#stdevhttps://people.richland.edu/james/ictcm/2001/descriptive/helpcenter.html#medianhttps://people.richland.edu/james/ictcm/2001/descriptive/helpcenter.htmlhttps://people.richland.edu/james/ictcm/2001/descriptive/helpcenter.htmlhttps://people.richland.edu/james/ictcm/2001/descriptive/helpcenter.htmlhttps://people.richland.edu/james/ictcm/2001/descriptive/helpcenter.html


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The following are the important provisions under Banking Regulation

Act, 1949 regarding control and regulation of Banking Sector in India.

The requirements regarding the minimum paid-up capital and reserves

for commence mint of banking business. Prohibition of charge on

unpaid capital.Payment of Dividends only after writing off all Capitalized

expenses.

Transfer to reserve fund out of Profits. (Minimum 20 per cent)

Maintenance of cash reserves by the non- scheduled banks. (Minimum 3

per cent) Restrictions on holding shares in other companies.

Restrictions on loans and advances to directors and others.Licensing of

banking companies.Licences for opening of new branches and transfer

of existing place of business. Maintenance of a percentage of liquid as

sets (SLR). (Minimum 25 per cent and maximum 40 per cent)

Maintenance of Assets in India By a banking company. (Minimum 75 per

cent of DTL) Submission of Return of unclaimed Deposits.

1. Power to call for and publish the information. Preparation of

Accounts and Balance Sheets.Audit of the Balance sheet and Profit &

Loss Account.Publication of Audited Accounts and Balance

Sheet.Inspection of books and accounts of banking companies by

RBI.Giving directions to banking companies.

2. Prior approval from RBI for appointment of managing directors.

3. Removal of managerial and any other persons from office.

4. Power of RBI to appoint additional directors

5. Moratorium under the orders of a High Court.

6. Winding up of banking companies.

7. Scheme of amalgamation to be sanctioned by the RBI.

8. Power of RBI to apply to the

9. Central Government for an order of mortal rim in respect of a banking

company and for a scheme of reconstruction or amalgamation.

10. Power of RBI to examine the record of proceedings and tender

advice in winding up proceedings.

11. Power of RBI to inspect and make its report to winding up.

12. Power of RBI to call for Returns and information from the Liquidator

of a Banking company.

13. Issue of No Objection Certificate for change of name.

14. Issue of No objection certificate for the Alteration of memorandum

of a banking company. Central Government to consult the RBI for

making rules regarding banking companies. Recommend to the Central

Government for exempting any bank from the provisions of the Banking

Regulation Act 1949.

Q.5. What do you mean by concept of probability & what is probability

distribution ?

Ans:-A probability is a number that reflects the chance or likelihood that a particular

event will occur. Probabilities can be expressed as proportions that range from 0

to 1, and they can also be expressed as percentages ranging from 0% to 100%. A

probability of 0 indicates that there is no chance that a particular event will

occur, whereas a probability of 1 indicates that an event is certain to occur. A

probability of 0.45 (45%) indicates that there are 45 chances out of 100 of the

event occurring.

The concept of probability can be illustrated in the context of a study of obesity

in children 5-10 years of age who are seeking medical care at a particular

pediatric practice. The population (sampling frame) includes all children who

were seen in the practice in the past 12 months and is summarized below.

Age (years)

5 6 7 8 9 10 Total

Boys 432 379 501 410 420 418 2,560

Girls 408 513 412 436 461 500 2,730

Totals 840 892 913 846 881 918 5,290

A probability function is a function which assigns probabilities to the values of a

random variable.

All the probabilities must be between 0 and 1 inclusive

The sum of the probabilities of the outcomes must be 1.

If these two conditions aren't met, then the function isn't a probability function.

There is no requirement that the values of the random variable only be between

0 and 1, only that the probabilities be between 0 and 1.

Probability Distributions

A listing of all the values the random variable can assume with t

corresponding probabilities make a probability distribution.

A note about random variables. A random variable does not mean that the va

can be anything (a random number). Random variables have a well defined s

outcomes and well defined probabilities for the occurrence of each outcome

random refers to the fact that the outcomes happen by chance -- that is,

don't know which outcome will occur next.

Q.6.What is sampling & what are its types & techniques

Ans:-sampling

A finite subset of the population, selected from it with the objectiv

investigating its properties is called a sample and the number of uni

the sample is known as the sample size.

Sampling is a tool which enables us to draw conclusions about

characteristics of the population after studying only those object

items that are included in the sample. The main objectives of

sampling theory are:

(i) To obtain the optimum results, i.e., the maximum information ab

the characteristics of the population with the available sources at

disposal in terms of time, money and manpower by studying the sam

values only.

(ii) To obtain the best possible estimates of the population paramete

The Essentials of good Sampling?

In order to reach at right conclusions, a sample must possess

following essential characteristics.

1. Representative:

The sample should truly represent the characteristics of the verse.

this investigator should be free from bias and the method of collec

should be appropriate.

2. Adequacy:

The size of the sample should be adequate i.e., neither too large

small but commensurate with the size of the population.

3. Homogeneity:

There should be homogeneity in the nature of all the units selected

the sample. If the units of the sample are of heterogeneous charact

will impossible to make a comparative study with them.

4. Independent ability:

The method of selection of the sample should be such that the item

the sample are selected in an independent manner. This means

lection of one item should not influence the selection of another ite

any manner d that each item should be selected on the basis of its merit.

Types of Purposive Sampling

Following are the main types of purposive sampling.

A. Quota Sampling:

It is a type of purposive sampling in which the whole universe is div

first into certain parts and the total sample is allocated among th

parts! Each part of' the population is assigned to an investigator

whom the quota of the units to be examined by him is fixed in adva

according to certain specified characteristics such as sex,

occupation, income group, political or religious affiliation.

The invigilator is asked to select the required number of units of

sample of his own accord and examine them to get the des

information as quickly as possible.

He is also authorized to substitute the new units in the quota if he fthat any unit of the sample so selected is not responding up to

mark. This method is very often used in opinion poll surveys, ma

surveys and political surveys.

B. Convenience Sampling

It is a type of purposive sampling in which the sample units are selec

purposively by the investigator to suit his convenience in the matte

location and contract with the units.

This method of selecting the sample is also called 'Chunk' since

samples under this method are selected neither on the basis of the r

of probability nor on the basis of the judgment of the investigator

on the basis of convenience on the part of the investigator.

For example, a sample obtained from a list of students in a college

enquiring into their educational problems, is a matter of convenie

sampling as in such a case it will be convenient for the investigator to


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Categories of Sampling

There are two major categories of sampling:

1.Random or Probability Sampling: Random sampling is also called

Probability Sampling because the laws of probability can be applied to it.

Note that the term 'random sample' is not used to describe data in a

sample; it is a process used to select the sample from a population.

Random Sampling does not depend upon the existence of detailed

information about the universe. It also provides such data as are

unbiased. Also, we can measure the relative efficiency of different

sample designs with random sampling methods.

Limitations of this type of sampling cannot be ignored either. It requires

high levels of skill. Also, it consumes a lot of time for planning the

process of actual sampling. The cost of execution of this sampling

method is very high.

2. Non-Random or Judgment Sampling: This is a process of sample

selection where we do not use random methods. A non-random sample

is selected on the basis of judgment or convenience. There is no

selection on the basis of probability considerations. The pattern of

sample variability in the process cannot be known.

Q.7.What is hypothesis testing? states its technique using chi-square

test?

Ans:-Hypothesis: Formulation, Types and Testing

In hypothesis testing, we must state the hypothesized value of the

population parameter before we begin sampling. The hypothesis we

wish to test is called Null Hypothesis and is denoted as H 0. Example: If

we want to test the hypothesis that the population mean is equal to

600, we can write it as follows: H0: p = 600 and read, "The null

hypothesis is that the population mean is equal to 600."

Hypothesis Testing Test of Means

1. By One-Tailed Test: Take an example of a drug, which is frequently

used by a hospital. The individual dose of this drug is 125 cc. There is no

harm when body takes excessive does of this drug. But on the other

hand, insufficient doses do not assist doctors in the necessary medical

treatment. The hospital has been purchasing the same drug from the

same manufacturer for many years and the population's standard

deviation is 4 cc. The hospital inspects 50 doses of this drug at random

from a very large consignment and calculates the mean of these doses

to be 99.5 cc. The data in this case are:

pH0 =125 (hypothesised value of the population mean) cr = 4 (population

standard deviation) n = 50 (sample size) x = 99.5 (sample mean)

The hospital sets a 0.10 significance level. We have to find out "whether

the dosages in this consignment are too small."

In order to find the answer, we can state the problem as follows: H 0: p =

125 (null hypothesis) H,:p


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The analysis consists of choosing and fitting an appropriate model, done

by the method of least squares, with a view to exploiting the

relationship between the variables to help estimate the expected

response for a given value of the independent variable. For example, if

we are interested in the effect of age on height, then by fitting a

regression line, we can predict the height for a given age..

1. Most of the variables show some kind of relationship. For instance,

there is relationship between price and supply, income and expenditure

etc. With the help of correlation analysis we can measure in one figure

the degree of relationship.

2. Once we know that two variables are closely related, we can estimate

the value of one variable given the value of another. This is known with

the help of regression.

3. Correlation analysis contributes to the understanding of economic

behavior, aids in locating the critically important variables on which

others depend.

4. Progressive development in the methods of science and philosophy

has been characterized by increase in the knowledge of relationship. In

nature also one finds multiplicity of interrelated forces.

5. The effect of correlation is to reduce the range of uncertainty. The

prediction based on correlation analysis is likely to be more variable and

near to reality.

Correlation

X : 5 10 15 20 25 30

Y : 10 13 18 17 21 29

Thus, from the above example it is clear that the ratio of change

between two variables is not same. Now, if we plot all these variables

on a graph, they would not fall on a straight line.

C. Number of Variables

According to the number of variables, correlation is said to be of the

following three types viz;

(i) Simple Correlation.

(ii) Partial Correlation.

(iii) Multiple Correlations.

(i) Simple Correlation:

In simple correlation, we study the relationship between two variables.

Of these two variables one is principal and the other is secondary? For

instance', income and expenditure, price_ and demand etc. Here

income and price are principal variables while expenditure and demand

are secondary variables.

(ii) Partial Correlation:

If in a given problem, more than two variables are involved and of these

variables we study the relationship between only two variables keeping

the other variables constant, correlation is said to be partial. It is so

because the effect of other variables is assumed" to be constant

(iii) Multiple Correlations:

Under multiple correlations, the relationship between two and more

variables is studied jointly. For instance, relationship between rainfall,

use of fertilizer, manure on per hectare productivity of maize crop.

quantitative method cp 102

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