## Saturday, October 27, 2018

Unit – 1: Introduction to Statistics
Q.1. Define the term Statistics. What are its Characteristics? Mention its Functions and Limitations.
Ans: Statistics: The word Statistics seems to have been derived from the Latin word “status” or the Italian word Statista. All word means a political state. In early year “statistics” equipped a collection of facts about the people in the state for administration or political purpose.
A comprehensive definition was given by Prof. Horace Secrist, which is a follows:  “By Statistics we mean aggregates of facts affected to a marked extent by multiplicity of causes, numerically expressed, enumerated or estimated according to a reasonable standards of accuracy, collected in a systematic manner for a predetermined purpose and placed in relation to each other.”
Characteristics of Statistics:
(i) Statistics are aggregates of facts. Single and isolated figures are not statistics because they cannot be compared.
(ii) Statistics must be numerically expressed. Statistical methods are applicable only to those data which can be numerically expressed.
(iii) Statistics should be capable of comparison and connected to each other.
(iv) Statistics should be collected in a systematic manner.
(v) Statistics should be collected for a definite purpose: The purpose should be specific and well defined.
Functions and Limitations of Statistics:
The functions of statistics are as follows:
(i) It presents fact in a definite form. Numerical expressions of data are convincing.
(ii) It simplifies mass of figures. The data presented in the form of table, graph or diagram, average or coefficients are simple to understand.
(iii) It facilitates comparison. Once the data are simplified they can be compared with other similar data.
(iv) It helps in prediction. Plans and policies of organisations are invariably formulated in advance at the time of their implementation.
(v) It helps in the formulation of suitable policies. Statistics provide the basic material for framing suitable policies.
Limitations of statistics are as follows:
(i) Statistics deals only with quantitative characteristics. Data Which cannot be expressed in numbers are incapable of statistical analysis. Qualitative characteristics like honesty, efficiency, intelligence etc. cannot be studied directly.
(ii) Statistics deals with aggregates not with individuals.
(iii) Statistical laws are not perfectly accurate.
(iv) Statistical results are only an average. Statistical results reveal only the average behavior.
(v) Statistics is only one of the methods of studying a problem. Statistical tools do not provide the best solution under all circumstances.
(vi) Statistics can be misused. The data placed to an inexperienced person may reveal wrong results. Only persons having fundamental knowledge of statistical methods can handle the data properly.
Q.2. What are various types of Statistical Data? Mention their merits and demerits.
Ans: Statistical data are of two types
(a) Primary data
(b) Secondary data.
Primary Data: Data which are collected for the first time for a specific purpose are known as Primary data. For example: Population census, National income collected by government, Textile Bulletin (Monthly), Reserve bank of India Bulletin (Monthly) etc.
Secondary Data: Data which are collected by someone else, used in investigation are knows as Secondary data. Data are primary to the collector, but secondary to the user. For example: Statistical abstract of the Indian Union, Monthly abstract of statistics, Monthly statistical digest, International Labour Bulletin (Monthly).
Merits and Demerits of Primary Data:
Merits:
(a) They are reliable and accurate.
(b) If during collection, the Data are wrong they can be checked again by cross examination.
(c)  It is more suitable if the field of enquiry is small.
Demerits:
(a) It the field of enquiry is too wide, it is not suitable.
(b) Collection of primary data is costly and time consuming.
(c) Personal Bias, prejudice and whims may affect the data.
Merits and Demerits of Secondary Data:
Merits:
(a) While using secondary data, time and labour are saved.
(b) It may also be collected from unpublished form.
(c) If secondary Data are available, they are much quicker to obtain than primary data.
Demerits:
(a) Degree of accuracy may not be acceptable.
(b) Secondary Data may or may not fit the need of the project.
(c) Data may be influenced by personal bias of investigator.
Q.3. Distinguish between Primary data and Secondary data.
Ans: Difference between Primary Data and Secondary Data:
(a) Primary data are those which are collected for the first time and thus original in character. While Secondary data are those which are already collected by someone else.
(b) Primary data are in the form of raw-material, whereas Secondary data are in the form of finished products.
(c) Primary data are collected directly from the people related to enquiry while Secondary data are collected from published materials.
(d) Data are primary in the hands of institutions collecting it while they are secondary for all others.
Q.4. What are various sources of Secondary Data? Mention the points which should be considered before using secondary data.
Ans: Sources of Secondary Data:
(a) Official publication by the central and state governments, district Boards. (b) Publication by research institutions, Universities etc. (c) Economic Journals. (d) Commercial Journals. (e) Reports of Committees, commissions. (f) Publications of trade associations, Chamber of Commerce etc.
Precautions in the use of Secondary Data:
The following aspects should be considered before use of secondary data:
(i) Suitability: The investigator must check before using secondary data that whether they are suitable for the present purpose or not.
(ii) Adequacy: The investigator has to determine whether they are adequate for the present purpose of investigators.
(iii) Dependability: Dependability of secondary data is determined by the following factors:-
(a) The authority which collected the data.
(b) Procedure of Sampling followed.
(c) Status of Investigator.
(iv) Units in which data are available.
Q.5. What are various essential qualities of Secondary data? Explain some effective methods of collecting primary data.
Ans: Qualities of Secondary Data:
(a) Data should be reliable
(b) Data should be suitable for the purpose of investigator.
(d) Data should be collected by trained investigator.
Methods of collecting primary Data
(a)  Direct Personal Observation: Under this method, the investigator collects the data personally from the persons concerned. The information obtained under this method is original in nature. This method is suitable when the field of enquiry is small.
(b) Indirect Oral Investigation: - Under this method, the investigator collects the data from third parties capable of supplying the necessary information. This method is suitable where the information to be obtained is of a complex nature and informants cannot be approached directly.
(c) Schedule and questionnaire: - A list of question regarding the enquiry is prepared and printed. Data are collected in any of the following ways:-
(i) By sending the questionnaire to the persons concerned with a request to answer the question and return the questionnaire.
(ii) By sending the questionnaire through enumerators for helping the informants.
(d) Local reports: - This method gives only approximate results at a low cost.
Q.6. What are various stages involved in statistical investigation? Explain them briefly.
Ans: Various stages in statistical investigation: There are five stages in a statistical investigation which are given below:
(i) Collection of Data: Utmost care must be exercised in collecting data as they are the foundation of statistical analysis. If the data are faulty, the conclusions drawn can never be reliable.
(ii) Organisation of Data: Data collected from published sources are generally in organised form but data collected from a survey frequently needs organisation. Organising of data involves three steps which are:
(a) Editing of data
(b) Classification of data according to some common characteristics
(c) Tabulation.
(iii) Presentation of Data: Organised data can be further presented in the form of Diagrams and Graphs.
(iv) Analysis: After collection, organisation and presentation, data are analysed by adopting various statistical methods such as measure of central tendency, measure of variation, correlation, regression etc. to dig out information useful for decision-making.
(v) Interpretation: The last stage is interpretation which is a difficult task and requires a high degree of skill, care and experience. If the data have been analysed and not properly interpreted, the whole object of investigation may be defeated and wrong conclusion be drawn.
Q.7. What is Questionnaire? What are its essential characteristics?
Ans: Questionnaire: A Questionnaire is simply a list of questions in a printed sheet relating to survey which the investigators asks to the informants and the answers of the informants are noted down against the respective questions on the sheet. Choice of questions is a very important parts of the enquiry whatever its nature.
Characteristics of an ideal Questionnaire:
(i) The Schedule of question must not be lengthy.
(ii) It should be clear and simple.
(iii) Questions should be arranged in a logical sequence.
(iv) Each question should be brief and must aim to some particular information necessary for the    investigation.
(v) Questions of personal matter like income of property should be avoided.
(vi) The Units of information should be Cleary shown in the sheet.
Q.8. What is Tabulation? Mention its Objectives. What are its importance and limitations?
Ans: Tabulation: Tabulation refers to the systematic arrangement of the information in rows and columns. Rows are the horizontal arrangement. In simple words, tabulation is a layout of figures in rectangular form with appropriate headings to explain different rows and columns. The main purpose of the table is to simplify the presentation and to facilitate comparisons.
According to Neiswanger, "A statistical table is a systematic organisation of data in columns and rows."
The principal objectives of tabulation are stated below:
(i) To make complex data simple: When data are arranged systematically in a table, such data become more meaningful and can be easily understood.
(ii) To facilitate comparison: When different data sets are presented in tables it becomes possible to compare them.
(iii) To economize space: A statistical table furnishes maximum information relating to the study in minimum space.
(iv) To make data fit for analysis and interpretation.
(v) To provide reference: A statistical table can be used as a source of reference for other studies of similar nature.
Importance of Tabulation:
a) Tabulation makes the data brief. Therefore, it can be easily presented in the form of graphs.
b) Tabulation presents the numerical figures in an attractive form.
c) Tabulation makes complex data simple and as a result of this, it becomes easy to understand the data.
d) This form of the presentation of data is helpful in finding mistakes.
e) Tabulation is useful in condensing the collected data.
f) Tabulation makes it easy to analyze the data from tables.
g) Tabulation is a very cheap mode to present the data. It saves time as well as space.
h) Tabulation is a device to summaries the large scattered data. So, the maximum information may be collected from these tables.
Limitations of Tabulation
a) Tables contain only numerical data. They do not contain details.
b) Qualitative expression is not possible through tables.
c) Tables can be used by experts only to draw conclusions. Common men do not understand them properly.
Q.9. What do you mean by Classification of Data? Mention its essential features.
Ans: Classification of Data: The process of arranging the data in groups or classes according to their common characteristics is technically known as classification. Classification is the grouping of related facts into classes. It is the first step in tabulation.
In the words of Secrist, "Classification is the process of arranging data into sequences and groups according to their common characteristics or separating them into different but related parts."
Essentials of classification
a) The classification must be exhaustive so that every unit of the distribution may find place in one group or another.
b) Classification must conform to the objects of investigation.
c) All the items constituting a group must be homogeneous.
d) Classification should be elastic so that new facts and figures may easily be adjusted.
e) Classification should be stable. If it is not so and is changed for every enquiry then the data would not fit for an enquiry.
f) The data must not overlap. Each item of the data must be found in one class.
Q.10. Define the term population and sample. What is sample and census survey? Distinguish between them.
Ans: Population and Sample
Population: Statistics is taken in relation to a large data. Single and unconnected data is not statistics. In the field of a statistical enquiry there may be persons, items or any other similar units. The aggregate of all such units under consideration is called “Universe or Population”.
Sample: If a part is selected out of the universe then the selected part or portion is known as sample. Sample is only a part of the universe.
Sample survey and Census Survey:
Sample survey: It is a survey under which only a part taken out of the universe is investigated. It is not essential to investigate every individual item of the Universe.
Census survey and complete enumeration: Under Census survey detail information regarding every individual person or item of a given universe is collected.
Difference between Census and Sample survey:
The following are the differences between Census and Sample method of investigation:
(a) Under Census method, each and every individual item is investigated whereas under sample survey only a part of universe is investigated.
(b) There is no chance of sampling error in census survey whereas sampling error cannot be avoided under sample survey.
(c) Large number of enumerators is required in census whereas less number of enumerators is required in sample survey.
(d) Census survey is more time consuming and costly as compared to sample survey.
(e) Census survey is an old method and it less systematic than the sample survey.
Q.11. Mention the Merits and Demerits of Census and Sample Survey.
Ans: Merits and Demerits of Census:
Merits:
(a) Since all the individuals of the universe are investigated, highest degree of accuracy is obtained.
(b) Since there is no possibility of personal bias affecting investigation, this method is free from sampling error.
(c)  It is more suitable if the field of enquiry is small.
(d) Since all the items of the universe are taken into consideration, all the characteristics of the universe
Demerits:
(a) It the field of enquiry is too wide, it is not suitable.
(b) Collection of primary data is costly and time consuming.
(c) Personal Bias, prejudice and whims may affect the data.
Merits and Demerits of sample survey:
Merits:
(a) While using secondary data, time and labour are saved.
(b) It may also be collected from unpublished form.
(c) If secondary Data are available, they are much quicker to obtain than primary data.
Demerits:
(a) Degree of accuracy may not be acceptable.
(b) Secondary Data may or may not fit the need of the project.
(c) Data may be influenced by personal bias of investigator.
Q.12. What are various types of diagrams and graphs? Distinguish between diagrams and graphs.
Ans: Types of diagrams and Graphs:
One of the most effective and interesting alternative way in which a statistical data may be presented is through diagrams and graphs. There are several ways in which statistical data may be displayed pictorially such as different types of graphs and diagrams. The commonly used diagrams and graphs to be discussed in subsequent paragraphs are given as under:
Types of Diagrams/Charts:
a)      Simple Bar Chart
b)      Multiple Bar Chart or Cluster Chart
c)       Staked Bar Chart or Sub-Divided Bar Chart or Component Bar Chart
d)      Simple Component Bar Chart
e)      Percentage Component Bar Chart
f)       Sub-Divided Rectangular Bar Chart
g)      Pie Chart
Types of Diagrams/Charts:
a)      Histogram
b)      Frequency Curve and Polygon
c)       Lorenz Curve
d)      Histogram
Difference Between Diagrams And Graphs
There is no clear-cut line of demarcation between a diagram and a graph yet:
a)      A graph needs a graph paper but a diagram can be drawn on a plain paper.
b)      As diagrams are attractive to look at, they are used for. Graphs on the other hand are more useful to statisticians and research workers for the purpose of further analysis.
c)       For representing frequency distribution, diagrams are rarely used when compared with graphs. For example, for the time series graphs are more appropriate than diagrams.
Q.13. Compare between Tabular and Diagrammatical Presentation of Data? Mention the uses and Limitations of diagrams and graphs.
Ans: Comparison Between Tabular And Diagrammatic Presentation
 Serial No: Diagrams and Graphs Tabulation 1. Diagrams and Graphs are meant for a lay man. Tables are meant for statisticians for the purpose of further analysis. 2. Diagrams give only an approximate idea. Tables contain precise figures. Exact values can be read from tables. 3. Diagrams can be more easily compared, and can be interpreted by a layman. Comparison and interpretations of tables can only be done by statisticians and it is a difficult task. 4. Diagrams and graphs cannot present much information. Tables can present more information. 5. Diagrams are more attractive and have a visual appeal. Tables are dry for a layman (may be attractive to a statistician.)
Uses of Diagrams and Graphs:
Diagrams and graphs are extremely useful due to the following reasons:
(i) Information presented though diagrams and graphs can be understood easily just in a bird’s eye view.
(ii) These are appealing and fascinating to the eyes; Scholars take greater interest in presenting data through these devices.
(iii) Diagrams and graphs produce a greater lasting impression on the mind of the readers than the figures presented in a table.
(iv) They facilitate ready comparison of data over time and space. Graphs study economic relationship between two variables.
However, graphic and diagrammatic presentations have some limitations:
(i)      Unlike a table a diagram or a graph does not show the exact value of a variable.
(ii)    Further, a limited set of facts can be presented through such devices like diagram and graph.
Q.14. What are the points that should be kept in mind while constructing a diagram or graph?
Ans: General Rules for Drawing Graphs and Diagrams
Following points must be kept in mind while constructing a diagram or graph. Every diagram or graph must have a serial number. It is necessary to distinguish one from the other.
1. Serial number: Every diagram or graph must have a serial number. It is necessary to distinguish one from the other.
2. Title: Title must be given to every diagram or graph. From the title one can know the idea contained in it. The title should be brief and self-explanatory. It is usually placed at the top.
3. Proper size and scale: A diagram or graph should be of normal size and drawn with proper scale. The scale in a graph specifies the size of the unit.
4. Cleanliness: Diagrams must be as simple as possible. Further they must be quite neat and clean. They should also be descent to look at.
5. Index: Every diagram or graph must be accompanied by an index. This illustrates different types of lines, shades or colors used in the diagram.
6. Footnote: Foot notes may be given at the bottom of a diagram if necessary. It clarifies certain points in the diagram.

Unit - 1: Measure of Central Tendency and Dispersion (Part – B)
Q.N.1.What is average? Why is it called a measure of central tendency?
Ans: - In the words of Croxton and Cowden, “An average value is a single value within the range of the data that is used to represent all the values in the series.”
The value of average lies between the maximum and minimum values of the series. That is why it is also called measure of central tendency.
Q.N.2.What is the desirable properties of a good average.
Ans: - The following are the important properties which a good average should satisfy
1.       It should be easy to understand.
2.       It should be simple to compute.
3.       It should be based on all the items.
4.       It should not be affected by extreme values.
5.       It should be rigidly defined.
6.       It should be capable of further algebraic treatment.
Q.N.3. Define Arithmetic Mean (A.M). What are its properties? Explain its merits and demerits.
Ans: - Arithmetic Mean: - It is a value obtained by adding together all the items and by dividing the total by the number of items. It is also called average. It is the most popular and widely used measure for representing the entire data by one value.
Arithmetic mean may be either:
(i)      Simple arithmetic mean, or
(ii)    Weighted arithmetic mean.
Properties of arithmetic mean:
1.       The sum of deviations of the items from the arithmetic mean is always zero i.e. ∑(X–X) =0.
2. The Sum of the squared deviations of the items from A.M. is minimum, which is less than the sum of the squared deviations of the items from any other values.
3. If each item in the series is replaced by the mean, then the sum of these substitutions will be equal to the sum of the individual items.
Merits of A.M.:-
(i)      It is simple to understand and easy to calculate.
(ii)    It is affected by the value of every item in the series.
(iii)   It is rigidly defined.
(iv)  It is capable of further algebraic treatment.
(v)    It is calculated value and not based on the position in the series.
Demerits of A.M.:-
(i)      It is affected by extreme items i.e., very small and very large items.
(ii)    It can hardly be located by inspection.
(iii)   In some cases A.M. does not represent the actual item. For example, average patients admitted in a hospital is 10.7 per day.
(iv)  A.M. is not suitable in extremely asymmetrical distributions.
Q.N.4. Define Geometric Mean (G.M). Mention its merits and demerits. What are its Uses?
Ans: - G.M:- It is defined as nth root of the product of n items or values. i.e., G.M. = n√ (x1. x2. x3 ……xn)
Merits of G.M.:-
(i)      It is not affected by the extreme items in the series.
(ii)    It is rigidly defined and its value is a precise figure.
(iii)   It is capable of further algebraic treatment.
(iv)  It is Useful in calculating index number.
Demerits of G.M.:-
(i)    It is difficult to understand and to compute.
(ii)   It cannot be computed when one of the values is 0 or negative.
Uses of G.M.:-
(i)      It is used to find average of the rates of changes.
(ii)    It is Useful in measuring growth of population.
(iii)   It is considered to be the best average for the construction of index numbers.
Q.N.5. Define Harmonic Mean (H.M). Mention its merits and demerits. What are its Uses?
Ans: - H.M.:- It is defined as the reciprocal of the arithmetic mean of the reciprocal of the individual observations.
 N H.M. = (1/x1 + 1/x2 + 1/x3 + ........ +1/xn)

Merits of H.M.:-
(i)      Like AM and GM, it is also based on all observations.
(ii)    It is most appropriate average under conditions of wide variations among the items of a series since it gives larger weight to smaller items.
(iii)  It is capable of further algebraic treatment.
(iv) It is extremely useful while averaging certain types of rates and ratios.
Demerits of H.M.:-
(i)      It is difficult to understand and to compute.
(ii)    It cannot be computed when one of the values is 0 or negative.
(iii)   It is necessary to know all the items of a series before it can be calculated.
(iv)  It is usually a value which may not be a member of the given set of numbers.
Uses of H.M.: If there are two measurements taken together to measure a variable, HM can be used. For example, tonne mileage, speed per hour. In the above example tonne mileage, tonne is one measurement and mileage is another measurement. HM is used to calculate average speed.
Q.N.6. Define Median. Mention its merits and demerits.
Ans: - Median : Median may be defined as the size (actual or estimated) to that item which falls in the middle of a series arranged either in the ascending order or the descending order of their magnitude. It lies in the centre of a series and divides the series into two equal parts. Median is also known as an average of position.
Merits of Median:-
(i)      It is simple to understand and easy to calculate, particularly is individual and discrete series.
(ii)    It is not affected by the extreme items in the series.
(iii)   It can be determined graphically.
(iv)  For open-ended classes, median can be calculated.
(v)    It can be located by inspection, after arranging the data in order of magnitude.
Demerits of Median:-
(i)      It does not consider all variables because it is a positional average.
(ii)    The value of median is affected more by sampling fluctuations
(iii)   It is not capable of further algebraic treatment. Like mean, combined median cannot be calculated.
(iv)  It cannot be computed precisely when it lies between two items.
Q.N.7. Define Mode. Mention its merits and demerits.
Ans: - Mode: Mode is that value a dataset, which is repeated most often in the database. In other words, mode is the value, which is predominant in the series or is at the position of greatest density. Mode may or may not exist in a series, or if it exists, it may not be unique, or its position may be somewhat uncertain.
Merits of Mode:-
(i)      Mode is the most representative value of distribution, it is useful to calculate model wage.
(ii)    It is not affected by the extreme items in the series.
(iii)   It can be determined graphically.
(iv)  For open-ended classes, Mode can be calculated.
(v)    It can be located by inspection.
Demerits of Mode:-
(i)      It is not based on all observations.
(ii)    Mode cannot be calculated when frequency distribution is ill-defined
(iii)   It is not capable of further algebraic treatment. Like mean, combined mode cannot be calculated.
(iv)  It is not rigidly defined measure because several formulae to calculate mode is used.
Q.N.8. What is the relationship between mean, median and mode? Give the formula.
Ans: - In a normal distribution Mean = Median = Mode. In an asymmetrical distribution median is always in the middle but mean and mode will interchange their positions or values. Mode = 3 Median - 2 Mean.
Q.N.9. What is the relation between arithmetic mean, geometric mean and harmonic mean?
Ans: - A.M. > G.M. > H.M.
GM = √ (ARITHMETIC MEAN * HARMONIC MEAN)
Q.N.10. What is Dispersion? What purpose does a measure of dispersion serve? What are its various types? Distinguish between absolute and relative measures of dispersion.
Ans: - Dispersion: Dispersion is the measure of variation of items. It measures the extent to which the items vary from central value. Dispersion is also known as average of the second order.  In the words of Brooks and Dick,” Dispersion is the degree of the scatter or variation of the variable about a central value.”
In the words Simpson and Kafka,” The measurement of the Scatterness of the mass of figures in a series about an average is called measure of variation or dispersion.”
Dispersion includes range, mean deviation, quartile deviation, and standard deviation. Mean, Median and Mode are the average of 1st order.
Purpose of Measure of Dispersion: Measures of Dispersion are needed for the following purposes:
(i)      To determine the reliability of an average.
(ii)    To serve as a basis for the control of the variability.
(iii)   To compare two or more series with regard to their variability.
(iv)  To facilitate the use of other statistical measures.
Measure of dispersion may be broadly classified into two types:-
a. Absolute measures of dispersion: It is classified into
i.      Range
ii.      Mean Deviation
iii.      Standard Deviation
iv.      Quartile Deviation
b.Relative measures of dispersion: It is classified into
i.      Coefficient of Range
ii.      Coefficient of Mean Deviation
iii.      Coefficient of Variation
iv.      Coefficient of Quartile Deviation.
Difference between absolute and relative measure of dispersion:
1.       Absolute measures are dependent on the unit of the variable under consideration whereas the relative measures of dispersion are unit free.
2.       For comparing two or more distributions, relative measures and not absolute measures of dispersion are considered.
3.       As compared to absolute measures of dispersion, relative measures of dispersion are difficult to compute and comprehend.
Q.N.11.What is the desirable properties of a good measure of dispersion (Variation).
Ans: - The following are the important properties which a good measure of dispersion should satisfy:
1.       It should be simple to understand and easy to compute.
2.       It should be simple to compute.
3.       It should be based on all the items.
4.       It should not be affected by extreme values.
5.       It should be rigidly defined.
6.       It should be capable of further algebraic treatment.
7.       It should have sampling stability.
Q.N.12. Define Range. Mention its merits and demerits.
Ans: - Range:  Range is defined as the difference between the value of the smallest item and the value of the largest item included in the distribution. It is the simplest method of measuring dispersion. Symbolically,
Range= Largest value (L) – Smallest Value (S)
The relative measure corresponding to range, called the coefficient of range, is obtained by applying the following formula: Coefficient of Range= (L- S)/ (L + S)
Merits of Range:-
(i)   It is simple to understand and easy to calculate.
(ii) It is less time consuming.
Demerits of Range:-
(i)      It is not based on each and every item of the distribution.
(ii)    It is very much affected by the extreme values.
(iii)   The value of Range is affected more by sampling fluctuations
(iv)  Range cannot be computed in case of open-end distribution.
Q.N.13. Define Quartile Deviation (Q.D). Mention its merits and demerits.
Ans: - Q.D: - The QD is half of the difference between the upper and lower quartiles. Symbolically, QD= ½ (Q3- Q1)
QD is an absolute measure of dispersion. The relative measure corresponding to QD, called the coefficient of QD, is obtained by applying the following formula: Coefficient of Range= (Q3 - Q1) / (Q3 + Q1)
Coefficient of QD can be used to compare the degree of variation in different distributions.
Merits of QD:-
(i)      It is based on 50% of the observations.
(ii)    It is not affected by the presence of extreme values.
(iii)   In case of open-end distribution, it can be computed.
Demerits of QD:-
(i)      It is not based on each and every item of the distribution.
(ii)    It is not capable of further algebraic treatments.
(iii)   The value of Range is affected more by sampling fluctuations
Q.N.14. Define Mean Deviation (M.D).Mention its merits and demerits.
Ans: - M.D: For a given set of observation, MD is defined as the arithmetic mean of the absolute deviation of the observations from an appropriate measure of central tendency. The formula for computing MD is
MD= ∑│D│/ N
MD is an absolute measure of dispersion. The relative measure corresponding to MD, called the coefficient of MD, is obtained by dividing mean deviation by the particular average used in computing mean deviation. Thus, if MD has been computed from median, the coefficient of mean deviation shall be obtained by dividing MD by median. Coefficient of MD = MD/ (Mean or Median)
Merits of MD:-
(i)      It is simple to understand and easy to compute.
(ii)    It is based on each and every item of the data.
(iii)   MD is less affected by the values of extreme items than the Standard deviation.
Demerits of MD:-
(i)      The greatest drawback of this method is that algebraic signs are ignored while taking the deviations of the items.
(ii)    It is not capable of further algebraic treatments.
(iii)   It is much less popular as compared to standard deviation.
Q.N.15. Define Standard Deviation (S.D). Mention its merits and demerits.
Ans: - S.D: The standard deviation commonly denoted by ‘σ’ (Sigma) is the most widely used measure of dispersion. It is the square root of the second moment of dispersion and is calculated from the arithmetic mean. In short, it may be defined as the root-mean-square deviation from the mean.
Merits of SD:-
(i)      It is based on each and every item of the data and it is rigidly defined.
(ii)    It is capable of further algebraic treatment. Combined SD of two or more groups can be calculated.
(iii)   It is less affected by fluctuations of sampling than most other measures of dispersion.
(iv)  For comparing variability of two or more series, co-efficient of variation is considered as most appropriate and this is based on SD and Mean.
(v)    SD is most prominently used in further statistical work.
Demerits of SD:-
(i)      It is not easy to calculate and to understand.
(ii)    It gives more weight to extreme items and less to those which are nearer to mean.
Q.N.16.Why Standard Deviation (S.D) is regarded as best measure of dispersion? Distinguish between SD and MD.
Ans: - SD is regarded as best measure of dispersion because: (Same as advantages of SD)
Distinguish between mean deviation and standard deviation
1. Algebraic signs are ignored while calculating mean deviations whereas standard deviation takes into account algebraic sign also.
2. Mean deviation can be computed either from median or mean whereas standard deviation is computed always from arithmetic mean.
Q.N.17.What is coefficient of variation? What purpose does it serve? Distinguish between variance and coefficient of variation.
Ans: - Coefficient of variation: It is the ratio of Standard deviation to the mean expressed as percentage.  Coefficient of variation can be defined as the coefficient of standard deviation with respect to mean which is generally expressed in terms of percentage. The coefficient of variation is also known as coefficient of variability. Symbolically, Coefficient of Variation (C.V.) = (S.D / Mean)*100.
Purpose of Coefficient of variation:
Coefficient of variation is used to compare the variability of two or more series. A series having greater coefficient of variation is said to have more variable, i.e., less uniform, less stable or less consistent. Again a series, having coefficient of variation lesser is said to be less variable, i.e., more uniform, more stable or more consistent.
Difference between variance and coefficient of variation: Coefficient of variation is the percentage variation in the mean while Variation is the total variation in the mean.