ECO - 07: ELEMENTS OF STATISTICS | IGNOU SOLVED ASSIGNMENT 2020 - 21 | B.COM | FREE SOLVED ASSIGNMENT

Q.1. Compare between mean, median and mode to point out their merits and limitations. (20) 

Ans: Central Tendency: One of the most important objectives of statistical analysis is to get one single value that describes the characteristic of the entire mass of unwieldy data. Such a value is called the central value or an ‘average’ or the expected value of the variable. The word average is very commonly used in day-to-day conversation. For example, we often talk of average boy in a class, average height or life of an Indian, average income, etc. When we say ‘he is an average student’ what it means is that he is neither vary good nor very bad, just a mediocre type of student. There are mainly three types of Central Tendency:

1. Mean or Arithmetic Mean

2. Median

3. Mode

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 Simple arithmetic mean, or Weighted arithmetic mean.

Merits of A.M.:

1. It is simple to understand and easy to calculate.

2. It is affected by the value of every item in the series.

3. It is rigidly defined.

4. It is capable of further algebraic treatment.

5. It is calculated value and not based on the position in the series.

Demerits of A.M.:

1. It is affected by extreme items i.e., very small and very large items.

2. It can hardly be located by inspection.

3. In some cases A.M. does not represent the actual item. For example, average patients admitted in a hospital is 10.7 per day.

4. A.M. is not suitable in extremely asymmetrical distributions.

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:

1. It is simple to understand and easy to calculate, particularly is individual and discrete series.

2. It is not affected by the extreme items in the series.

3. It can be determined graphically.

4. For open-ended classes, median can be calculated. 

5. It can be located by inspection, after arranging the data in order of magnitude.

Demerits of Median:

1. It does not consider all variables because it is a positional average.

2. The value of median is affected more by sampling fluctuations 

3. It is not capable of further algebraic treatment. Like mean, combined median cannot be calculated.

4. It cannot be computed precisely when it lies between two items.

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:

1. Mode is the most representative value of distribution, it is useful to calculate model wage.

2. It is not affected by the extreme items in the series.

3. It can be determined graphically.

4. For open-ended classes, Mode can be calculated. 

5. It can be located by inspection.

Demerits of Mode:

1. It is not based on all observations.

2. Mode cannot be calculated when frequency distribution is ill-defined 

3. It is not capable of further algebraic treatment. Like mean, combined mode cannot be calculated.

4. It is not rigidly defined measure because several formulae to calculate mode is used.

Q.2. Differentiate between dispersion and skewness. If Karl Persons’s coefficient of skewness (SKp) = 0.06; mean = 150; variance of distribution= 2500 Find: median; mode & coefficient of variation. (20) 

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Ans: Difference between Dispersion and Skewness

Measures of Dispersion: The various measures of central value discussed in the previous chapter give us one single figure that represents the entire data. But the averages alone cannot adequately describe a set of observations, unless all the observations are the same. It is necessary to describe the variability or dispersion of the observations. In two or more distributions the central value may be the same but still there can be wide disparities in formation of the distribution. Measures of dispersion help us in studying this important characteristic of a distribution.

Skewness: The term ‘SKEWNESS’ refers to lack of symmetry, i.e., when a distribution is not symmetrical (or is asymmetrical) it is called a skewed distribution. Any measure of skewness indicates the difference between the manner in which items are distributed in a particular distribution compared with a symmetrical (or normal) distribution. If, for example, skewness is positive, the frequencies in the distribution are spread out over a greater range of values on the high-value end of the curve (the right-hand side) than they are on the low value end. If the curve is normal spread will be the same on both sides of the centre point and the mean, median and mode will all have the same value. The concept of skewness gains importance from the fact that statistical theory is often based upon the assumption of the normal distribution. A measure of skewness is, therefore, necessary in order to guard against the consequences of this assumption. 

Solution of Practical Problems:

Q.3 a) Discuss the usefulness of statistics and explain the limitations of statistics. (10) 

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.”

Uses of Statistics:

(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.

b) What is tabulation? Differentiate between classification and tabulation? (10) 

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."

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."

Difference between Tabulation and Classification:

Q.4 a) Calculate the quartiles Q1 and Q3 for wages of the labours given below: (10)

Calculation of Q1and Q3

Solution: Calculation of Q1and Q3

b) The consumption of number of guava and orange on a particular week by a family are given below: (10)

Calculation of consistency in fruit eating. 

Since c.v. of consumption of Graves is less as compared to c.v. of consumption of Oranges . Therefore, we can say that family is more consistent in consuming graves. 

Q.5 Write short notes on the following: 
a) Law of statistical regularity 

Ans: Law of statistical regularity is derived from the concept of probability. According to this law, if a sample is collected at random from a population, it is probable that the sample posses the same properties which the population posses. Random selection means each and every item of the population has the equal chance of selection. A sample selected at a random from a large population will represent the whole universe. There is a very less chances of bias selection. This law is very important because quick conclusion can be drawn from a large universe. It reduces the necessary work before final conclusion is drawn.

b) Non sampling errors 

Ans: Non-sampling Errors: These non-sampling errors can occur in any survey, whether it is a complete, enumeration or sampling. Non-sampling errors include biases as well as mistakes. These are not chance errors. Most of the factors causing bias in complete enumeration are similar to the one described above under sampling errors. They also include careless definition of population, a vague conception regarding the information sought, inefficient method of interview and so on. Mistakes arise as a result of improper coding, computations and processing. More specifically, non-sampling errors may arise because of one or more of the following reasons:

i) Improper and ambiguous data specifications which are not consistent with the census or survey objectives.

ii) Inappropriate sampling methods, incomplete questionnaire and incorrect way of interviewing.

iii) Personal bias of the investigators or informants.

iv) Lack of trained and qualified investigators.

v) Errors in compilation and tabulation.

c) Factors Affecting Choice of Data 

Ans: Factors affecting choice of data (Primary or secondary data)

1. Nature, Scope and Object of inquiry: This constitutes the most important factor affecting the choice of a particular method.  The method selected should be such that it suits the type of inquiry that is to be conducted by the researcher.  This factor is also important in deciding whether the data already available are to be used or the data not yet available are to be collected.

2. Availability of Funds: It determines to a large extent the method to be used for the collection of data. When the funds at the disposal of researcher are very limited, he will have to select a comparatively cheaper method which may not be as efficient and effective as some other costly method.

3. Time Factor: Availability of time has also to be taken into account in deciding a particular method of data collection. Some methods take relatively more time, whereas with others the data can be collected in a comparatively shorter duration.

4. Precision required: Precision required is yet another important factor to be considered at the time of selecting method of collection of data.

d) Lorenz Curve

Ans: Lorenz curve is the form of a curve which is derived from the cumulative percentage of the given variables. This curve was given by Dr. Max O. Lorenz a popular Economic- Statistician. He studied distribution of Wealth and Income with its help. It is graphic method to study dispersion. It helps in studying the variability in different components of distribution especially economic. The base of Lorenz Curve is that we take cumulative percentages along X and Y axis. Joining these points we get the Lorenz Curve. Lorenz Curve is of much importance in the comparison of two series graphically. It gives us a clear cut visual view of the series to be compared.

Describe steps to plot 'Lorenz Curve'

1. Cumulate both values and their corresponding frequencies.

2. Find the percentage of each of the cumulated figures taking the grand total of each corresponding column as 100.

3. Represent the percentage of the cumulated frequencies on X axis and those of the values on the Y axis.

4. Draw a diagonal line designated as the line of equal distribution.

5. Plot the percentages of cumulated values against the percentages of the cumulated frequencies of a given distribution and join the points so plotted through a free hand curve.

Purpose of Lorenz curve: It is a cumulative distribution function especially referring to the income distribution of a population.  An often used coefficient of income inequality is the gini coefficient which is a measure to the deviation of the actual cumulative income distribution from what would be obtained if everyone had the same income.