Quantitative techniques in Economic: Basic statistics.
6.1 Review of Basic statistic:
definition, Scope, Importance and limitation
6.1.1 Meaning of Statistics
The word ‘Statistics’ has developed from the Latin word ‘status’ and Italian word ‘Statista’. Even though it is liked with different languages, the scholars have tried to explain this in their own ways. Now, its scope and use have become wider than before. Different writers and scholars have expressed their ideas about statistics as follows.
A.L. Bowley has defined statistics as “Statistic is the numerical statements of facts in any department of inquiry in relation to each other”.
Boddington has defined it as- “Statistic is the science of estimates and probabilities”.
Levin has defined is as –“Statistics is the science which deals with classification and tabulation of numerical facts as the basic for explanation, description, and comparison of phenomena.
Yule and Kendall have defined statistic as –“By statistics we mean quantitative data affected to a marked extent by multiplicity of causes”.
Crotona and Cowden have defined it as “statistics may be defined as the science of collection, presentation, analysis and interpretation of numerical data.
The credit for defining statistic in clear and wider sense goes to Horace Secrets. He has defined statistics as- “Statistics may be defined as the aggregates of facts, affected to marked extent by a multiplicity of causes, numerically expressed, enumerated according to reasonable standards of accuracy, collected in a systematic manner of a predetermined purpose and placed in relation to each other.”
The following conclusions/facts can be found out from the analysis of the Horace Secrets definition.
1. Statistics are aggregate of facts.
2. Statistics are affected to a marked extent by a multiplicity of causes,
3. Statistics are numerically expressed,
4. Statistic are collected in a systematic manner,
5. Statistics are enumerated according to reasonable standard of accuracy,
6. Statistic are allocated for a pre determined purpose,
7. Statistics should be placed in relation to each other.
Avoiding the weakness in the definitions of different writers and scholars M.R Spiegel has defined statistic as “ statistics as –“ Statistic is concerned with scientific method for collecting, organizing, summarizing presenting and analyzing data as well as drawing valid conclusion and making reasonable decision on the basic of such analysis.
According to this definition, five stages should be passed through from collection to presentation of data. They are as:
1. Collection of data
2. Organization of data
3. Summarization of data
4. Analysis of data and
5. Presentation
6.1.2 Characteristics of statistics:
Different scholars have explained statistic. It is a kind of science. Most of its facts remain in mathematical system. It helps in quantitative explanation and analysis of different economic and social activities. It has become wider because of its expanded nature is definitions have become wider. Because of its expanded nature its definitions have become incomplete. However, its overall characteristics can be discussed as follows-
1. Statistic is aggregate of facts: Statistic is aggregate of facts: Statistics are aggregate presentation. It does not include a single item or unit. For example if a person x earns Rs. 500/-, this can not constitute statistics. But the subject matter of statistics. But the incomes of the person x, y, and z are rs. 500, 600 and 700 respectively, constitute statistic.
2. Statistics are affected to a marked extent by a multiplicity of cause: Statistic is affected to a marked extent by a multiplicity of causes:
Statistics become helpful in presentation of social science; many facts of social science are affected by different causes; so the subject matters of statistic cannot remain unaffected. Quantitative analysis of such effects becomes the subject of statistic. For example production of paddy is paddy is affected by rain, weather, seeds, size of land etc.
3. Statistics are numerically expressed: If different activities and processes are expressed or presented in figure, it is statistic but, if it is expressed in words or language, it cannot be statistics. For example- the statement ‘paddy did not grow due to draught’ is not statistic. But the paddy grew 25% less than that of year due to draught is statistic.
4. Statistics are enumerated or estimated according to reasonable standard of accuracy:
The data about any event or phenomenon can be obtained either by enumeration or by estimation. Where or when numbers are very large, enumeration is not possible and in such cases figures can only be stained. For example, the number of students in a class can be obtained by enumeration and this figure will be exact, it may be exact, it may be a few hundreds or thousands less, in many statistical works 100% accurate. But the number of people in the mass demonstration can be estimated which may not be exact, it may be a few hundred or thousands more or less, in many statically works 100% accurate. But the number of people in the mass demonstration can be obtained by enumeration and this figure will be extract. It may be a few hundreds or thousands more or less, in many statically works 100% accuracy is rare. The degree of accuracy expected depends open the nature and object of the enquiry. For example in measuring the heights of individuals, even the fraction of a centimeter is material whereas in measuring the distance between two cities biratnagar and Kathmandu the fraction of a kilometer can be neglected However, it is important that reasonable standards of accuracy should be attained for drawing valid conclusion.
5. Statistics are collected in a systematic manner: Certain process is applied in collecting statistics. For this, method, procedure and purpose are predetermined. Haphazardly collected statistics cannot give true information.
6. Statistic is collected for a predetermined purpose: Collection and presentation of statistics is purposeful or predetermined purposeless statistics is useless. How much area to be covered, which method to be applied, statistics is useless. How much area to be covered which method to be applied, statistics of which period should be collected etc. are predetermined.
7. Statistics are related to each other: Statistics are related to each other and from one period to the other. They can be compared on the basic of period and place.
8. Statistic has no independent existence: Statistics are related with economic, social activities for scientific researches or analysis in one or the other way; they work as the basis of analysis being related with other subjects.
6.1.3 Scope and Important of Statistic: The scope of statistic is related with its use and practice. Its scope is expanding together with its use and practice in different areas. So, its importance is increasing ever more. Statistic has been playing an important role every sector of society in the use of scientific analysis of economic activities, in presentation of events etc. Now, it has taken an important place in presentation of events etc. Now, it has taken a campuses and universities, so the importance of statistics in different sectors of human life can be presented as follows:
1. State and statistics: Statistic has remained as the backbone of state system since the beginning. Functions of states have widened much more than before, for example, at first a state collects statistics in order to get clear information about public health and social security, economic transactions, transport and communication land and agriculture related matters, and other social welfare works and then only adopts policy and formulates programs on the basis of the study of the statistics, so statistic can be called as the eye of a state.
2. Planning and statistic: A work done without good planning cannot be successful. In the present days every country or organization lunches any program only after formulating proper/good plan, but what were done to make good plans in the past and what result could be found at percent from them? What is good to do at present? What is good to do at present to get satisfactory result in future? It becomes possible only from the formal use of statistics. Formulating programs or plans without accurate information and knowledge is like walking kin darkness. So in the lack of statistics, it is impossible to prepare any suitable plan.
3. Business and statistic: Statistics is very important in industry and business sector. A skilled producer determines the size of production of any goods on the basis of the past production, demand and estimation of the possibility of future demand. Besides demand, it is also necessary to pay proper attention to price, market situation and other factors, businessmen or industrialists become successful only when their estimation is real or nearest to the reality. For this, Use of statistics is unavoidably necessary.
4. Economics and statistic: Statistic is important in economics the importance of statistics in economic can be discussed as follow:
a. Consumption: Different kinds of people live in society. They consume various goods and services according to their life standard in comparison to the consumption? How is their life standard in comparison to the consumption? How wide is inequality among them? The answers to such questions can be found only with the help of statistic.
b. Production: The answers to different questions such as what is the production how much quantity of necessary means of production, how much in what manner, and up to what time, can necessary goods for services be produced etc. can be found only with the help of statistic.
c. Exchange: statistics is very important in the exchange sector market demand, supply, price, price situation, import and export situation can be clearly found out with the help statistic.
d.Distribution: Problems relating to national income and distribution of payment for the means of production etc. can be solved with the help of statistics, statistic are necessary determine to quality of allocation to find out and determine the equality or inequality in the distribution of income and also to find out real situation of saving.
e. Public finance: With the help of statics methods revenue source, distribute of public expenditure, debt quantity, its pace present economic condition the future situation etc, can be understood. Governments need to take help of statistic in preparing budget and imposing taxes.
Statistic is very important for the analysis of various subjects including economics. Most of the concepts analysis and policies based on the statistics become appropriate. Evaluation of economic and social activities and their results and analysis are also made with the help of statistics. Even then, it is not applicable to make decisions. So there are also limitations of problems in the use of statistic as follows:
1. Statistic does not deal with individuals: Statistic helps to explain, analyze and present in group. Study of individual unit is beyond its scope. For example it does not study height or obtained marks of any student in a class. But it analyzes or examples average height or average obtained marks.
2. Statically laws are true only in average: Statistics cannot explain anything completely or cent percent true. It can explain only on average. It does not help to make totally true laws, for explain the probability of getting a head in a single toss of a coin is ½. But this does not imply that a 4 heads will be obtained if a coin tossed 8 times. The outcomes may be only one head or no head. So, it may not be equally useful for all and every subject.
3. It is not able to explain all subjects: It cannot explain all events and subjects that exist in the society. it is scarcely used in explaining religious, culture and other social activities that are impossible to be represented in quantities from similarly, it cannot be used to explain nor analyze the abstract things.
4. Problems in data collection: Analysis and conclusion made out with the help of statistic becomes difficult to collects data relating to behavior, personal habit, nature etc. of a man due to which statistics cannot be used in analyzing such things.
5. Statistics can be misused: Only the knowledgeable and skilled persons should are statistics, but this may not be always so, all cannot collect sufficient data. The conclusion drawn out from insufficient and incomplete data cannot be true and sometimes they may be misused.
6.1.5 Function of Statistics: Statistics works are the backbone in the analysis of various subjects. Some of its important functions can be discussed as follow:
1. Statistics presents the facts in define form:
The facts presented understandable for all rather than those presented in descriptive form. Statistic makes any fact short and certain giving them in quantities or numerical form and making easy to understand. For example- only the statement ‘what production increased by 200 tons this year gives real picture.
2. Statistic simple’s complexities: The collected data are in disordered form. They are the raw materials and heap of complexities. The statistical methods such as classification, tabulated, figure, graphs, average etc. Present the hug mass collected data in short, simple and ordered form in such a way that even a simple person can understand the information. For example the record of individual incomes of entire population of our county becomes complex task. But, everyone only easily understands per capita income through comparative study.
3. Statistics facilities comparison: Without comparatively studying two or more observation obtained from two or more places at different times, no conclusion can be drawn, for example, if the average age of a country is 45 years, whether this is high, low, or general, nothing can be said without comparing it with the average age of other countries statistic is very important to study such facts and find out conclusion through comparative study.
4. Statistics helps in formulating and testing hypothesis: Statistical methods are very useful in making hypothesis, testing them and developing new principles. For example, statistical methods are very usefully in testing the issues like whether Urea Fertilizer is beneficial in paddy production or not, whether budget deflects may be minimized through control over administrative expenses or not, whether students may get more facility by operating extra bus services or not, etc. Similarly it is equality useful in testing truth of some sciencetific facts.
5. Statistics helps in formulation of policies: it is important and necessary to analyze available data and information in order to formulate necessary social, economic and political policies of a country. Which goods need to be imported and which to be exported, or restricted etc. also can be decided with the help of statistic. Beside this food policy pricing policy, monetary policy, banking policy and others can be decided with the help of statistic.
6. Statistics helps to eliminate present and forecast future situation: Future trend and situation can be predicted on the basic of the analysis of present situation with the help of statically methods. Nowadays, it becomes very difficult to estimate future trend of planning, trade and commerce, and other sectors, but it becomes easy to forecast future trend and situation with the help of statistic.
6.2 Data Collection:
6.2.1 Source of Data:
Necessary data for the analysis of economic, social, scientific sectors and many others can be collected from different sources. Such sources can be classified in to two parts, in other word; there are mainly two sources of data.
1. Primary data: The data collected by a writer or a researcher for his own purpose are called primary data. Such data are totally new until they are published; only the collectors know about such data. Many researchers collect data from such sources, generally field level studies are concerned with primary data.
2. Secondary data: The data which are collected and published by others and are used by a researcher are called secondary data. For collection of such data, researcher does not need to travel to any places. Sources of secondary data are also divided in two types-
1. Government publications
2. Reports published by government of government formed commissions,
3. Data given in different materials published by international organization,
4. Data/Statistic published by Nepal Rastra Bank, NIDC, RNAC, telecommunication, etc.
5. Different publications containing data published by different writers,
6. Trade and business published publications,
7. Different financial, economic and demographic publications,
8. Magazines and newspapers published by different agencies,
9. University, campuses and school publications,
10. Other periodicals.
B. Unpublished Source: Unpublished source include data collected by persons, organizations, offices etc. but not published yet. To obtain such data, direct contact should be made with offices, persons, or organizations.
6.2.2 Methods of Primary Data collection
A researcher can collect data from primary or secondary sources. For collecting data/information from secondary sources the researcher needs to go to libraries, related persons, organizations and offices. In the way to go to libraries, related persons, organization and office they as follow:
1. Direct personal interviews: In this method the researcher goes to the concerned persons taking questionnaire. The interview and interviewee (known as informants) stand face to face,, in the in this methods questions are put or asked by researcher and the interviewee gives answers. Merits of this method are as follows-
a. As the researcher himself goes to take interview, the information will be true or near to reality.
b. As the researcher himself involves in taking information additional information also can be obtained.
c. Answers to any sensitive question can be obtained from this method.
d. Collected data become original and correct,
e. the method becomes effective in limited area,
f. Uniformity can be maintained between asking question and given answers.
g. Since researcher himself goes to ask questions to the concerned person and questions are set by the researcher promptly asked interview. They are as follows:
a. Since direct contact is kept will the respondents and researcher, this method cannot be appropriate for wider
B The informant my conceal several facts,
C it takes more time and fund.;
d. If the study area is extensive, answers cannot be obtained by asking all questions.
e. There may remain possibility of discrimination by researcher.
a. It takes less than fund.
b. Information to nearest t the truth since more than single person are asked question.
c. It becomes easy to study an area widely.
d. Suggestions and advice of export/specialists can be collected.
Beside the above merits, this method has demerits too. They are-
a.The informants hesitate to give true information about other.
b. The information obtained from information may not be totally true; they may give wrong information if they have prejudice against the concerned person.
c. The information collection collectors may twist the information.
3. Information from correspondents: In this method the researchers may appoint local correspondents or assistant is sent to the researcher. The researcher tabulates the information and presents it in orderly manner. The merits of these methods are as follows:
a. Study and collection of data become possible from larger areas.
b. Study becomes possible with less expense and less time.
c. This method is appropriate and useful for regular data collection,
d. As the corresponding or agents live in the same place it becomes easy to collect real data.
4. Mailed questionnaire: In this method the researcher prepares questionnaire and send to the concerned persons through post. The concerned person writes answers to the question and sends back to the researcher.
a. It helps to study covering wide areas.
b. Labor, fund and time are saved in this method.
c. In this method the informant is not given pressure, so they can give information freely.
d. Uneducated or illiterate informant cannot fill answer in the questionnaire,
e. Additional information cannot be obtained since the researcher does not have direct contract with the respondents.
5. Questionnaire sent through enumerators: According to this method the researcher sends questions to the enumerator enumerators and informants have face to face talks. The enumerators ask questions and write the answers given by the informants. Merits of the method are as follows.
a. Informed can be obtained from even illiterate or uneducated persons.
b. Enumerators and informant have direct contact/face to face talks answers to many questions can be obtained.
c. Enumerators can ask several additional and cross questions to respondents by which more reliable information can be obtained.
Beside the above merits, this method has demerits also. They are as follows-
a. as enumerators themselves have to go to informants it takes time labor and money,
b. Enumerators and training, so it is costly,
c. Enumerators may collect data with discrimination due to which the information may be wrong.
d. It becomes difficult to cover wider geographical area.
Different methods as mentioned above can be applied for primary data collection. All of them have both merits and demerits. But to same extent demerits can be controlled. At the above methods can be used in collecting information. But the researcher should decide appropriate method. The selection of the method depends on the nature, purpose and areas of the research or study, available means budget time expected accuracy etc.
6.2.3 Census and Sampling methods
A. Census Method:
In this method, all information of population can be personally collected.All the area covered by population is studied. Information is collected covering all respondents in the census method. As all population is consulted, the detail information can be collected so that analysis would be appropriate. This method has different merits and demerits. They are mentioned as follows-
Merits
1. All the necessary descriptions of the units of wide area are collected and are analyzed, due to which conclusion becomes more reliable,
2. It becomes easy to study and analyze small size of population and places,
3. In this method generally every person of family can be contacted.They become familiar to the reason for giving information. This raises people’s awareness,
4. In this method, information can be obtained from many units or people. It becomes easy for study.
Demerits
5. It takes long time to cover larger area,
6. More manpower, cost and means are needed in the method,
7. Sometimes manpower may not be available due to which collection of information may be difficult,
8. In certain situation this method cannot be used, for example, to examine or check skin or blood of sick persons etc.
Merits
1. It can be studied at less cost as small area is selected for sample,
2. It does not take long time and more labour for study,
3. Sample method becomes appropriate to study wide and big group. Generalizing with sample method, study of big size and group becomes possible.
4. Services of experts and specialists also can be taken as it can be done in shorter time.
5. Testing of description got from census method cannot be used that sample method.
6. To some context census method cannot be used but sample method can be used in all situations.
Demerits
1. As only samples are studies in this method, it becomes insufficient and the collected samples may not truly represent all the units. In such situation the conclusion may be wrong.
2. If method of sample collection could not be impartial or researcher is inexperienced and careless, the conclusion becomes wrong.
3. If there is no uniformity in all the units of the group or dissimilar characters are included in the unit, this method becomes useless.
1. Judgmental or deliberate sampling: this method depends on the researcher’s judgment. As samples are selected by the researcher and analyzed by himself, reality depends on the degree of efficiency, ability and impartially of their searcher. For example, 15 families technically called a sample can be selected to find out the economic condition of 100 families technically called a universe or population of a village. While selecting the 15 families, the researcher should be careful and use this prudence.
2. Random sampling: The universe is given numbers for the selection of sample. All items of the universe are numbered on separate slips or paper of identical size and shape, flooded in the same manner and are put in to an empty box. Numbers written in such papers cannot be known without unfolding the pieces of papers. All look same so any of the pikes can be selected. The certain number of pieces can be selected. The certain number of pieces can be selected. The certain number of pieces can be selected. The certain number of pieces required to constitute the desired to sample size are taken out blindly from the box.
3. Stratified random sampling: In this method. The total part of population to be studied is divided in to different classes and levels on the basis of certain types of their characteristics. The number of sample or percent to be taken from each class or level is fixed after determining this; samples are taken out from each class or levels.
4. Systematic sampling: in this method, list of population to be studied is prepared in an orderly manner. The size of sample is fixed on the basis of the list, and on the basis of the sample size, certain sampling interval is determined. The first item is selected at random from complete list and the rest other are selected on the basis of sampling interval. For the gap/difference formula K=N/n is used here K denotes determines the sampling interval, N denotes size of population, and n denotes sample size. This can be made clearer by the following example.
If 10 students are to be selected from 100 students, sampling interval is K=100/10. One of the intervals is selected by random method. Suppose 4 came out in selection. Then difference in each 10 are 4,14,24,34,44,54,64,74,84,94 which are taken as samples.
5. Multistage sampling: According to this method, sample selection process is used at different levels from a large member of population. As certain samples are selected from each stage, it is called multistage sampling for examples- if anything is to be studied taking 10,000 families from Nepal’s population, at first a few zones, after the districts are selected, then villages are selected from the districts. Then families are selected from the selected villages Random sampling method is applied at each stage.
6.3 Dispersion, Coefficient of Variance and Features of Variance:
Usually, the measure of central tendency shows the average of series only. But, the average along does not indicate what the actual structure of the series is. In such cases, one cannot say whether the information and the structure of the data given by the average is correct. For example, 15 is the average of 10, 15 and 20, and the average of 5, 10 and 30 is also 15. In the same cases, without the use other statically measures, it is difficult to take the average along as the analyzing factor of the statically data. The measurement of scatter or ups and downs of the items of a series from some central value is known as dispersion. It helps to analyses the structure or variability of statistical data.
The measures of dispersion have a great importance in these analyses of the data. Generally, the objectives of dispersion can be pointed out as follows:
1. To find out reliable an average is.
2. To act as a basis for analyzing factor of variability.
3. To compare the uniformity of different series.
4. To help in analyzing other statistical tools such as correlation, regression etc.
6.3.1 Features Qualities of dispersion
The measures such as range, quartile deviation, mean deviation, standard deviation etc. can be used for the measure of dispersion. According to the nature of a series and purpose of analysis, any measure can be used but a good measure of dispersion should posses the following properties.
1. It should be clear or easy to understand.
2. Its definition should be rigid, and should not be ambiguous.
3. It should be easy to compute.
4. It should be based on values of each and every items of a series.
5. It should be least affected by fluctuation of sampling.
6. It should be useful in other statistical analysis.
6.3.2 Absolute and related Dispersion
Measures of dispersion can be divided in to two parts. They are absolute and relative dispersion. If the measures of dispersion are expressed in units in which the original series are given, that are measure is called absolute measure. For example: If a series relating to the income of a group of persons is given in rupees, absolute measure of dispersion of the series should have rupees, as its unit. But sometimes, when two or more than two series having different units have to be compared, the absolute measure would erroneous and useless. In such cases, relative measure of dispersion is essential for comparison. When the absolute measure is divided by respective average, the obtained ratio (or the percentage obtained from the ratio) is called the relative dispersion. Relative measures of dispersion are pure numbers that are independent of unit of measurement. If the two given series have ‘KG’ and ton as their units respectively, these series cannot be compared. Relative measures are also known as coefficients of dispersion.
6.3.3 Methods of measuring Dispersion
The dispersion of a given series can be measured by different methods. The main methods are as follows:
1. Range
2. Quartile deviation or semi-interquartile Range
3. Mean Deviation
4. Lorenz Curve
6.3.3.1Range
Among the measures of dispersion, range is the simplest measure.
The ratio of the difference and sum of the maximum value and minimum value of a series is called coefficient of range.
Writing in formula,
L + S
Where, L= Largest value in the series
S= Smallest value in the series
Range of a series is an absolute measure of dispersion. But the coefficient of rage is a relative measure of dispersion. The methods of determining the rage of individual, discrete and continuous series are as follows:
a. Individual series: The following formula is used for determining the rage of individual series.
R=L-S
L + S
Here, R= range, l=value of largest item; and S= value of the smallest item.
Exercise 8.2(a)
1. What is meant by dispersion? What are its measures? What are qualities of ideal measure of dispersion?
2. What do you mean by absolute and relative measures of dispersion? Define range and coefficient of range what are its merits and demerits?
3. Calculate the range and coefficient of range of the following data.
70, 80, 53, 58, 51, 57, 55, 90, 65, 79, 74, 76
4. The following are the price of shares of company from Monday to Saturday. Calculate the range and coefficient of range.
|
Days |
Monday |
Tuesday |
Wednesday |
Thursday |
Friday |
Saturday |
|
Prices |
200 |
210 |
208 |
160 |
220 |
250 |
5. in a given series the coefficient of range is 0.29 and its maximum value is 64.5, find its minimum value.
6. Calculate range and coefficient of range from the following data.
Value: 4 -10 0 -2 -8 12 7 2 -5.
7. Calculate the rage and coefficient of range of the following data.
Size: 4.5 5.5 6.5 7.5 8.5 9.5 10.5
Frequency: 5 6 8 7 10 12 14
8. The weights of the 30 students of grade 12 are given in the following data. Calculate the rage of and coefficient of the range
|
Weight in kg |
52 |
54 |
57 |
60 |
54 |
63 |
60 |
|
NO. of Students |
4 |
4 |
5 |
8 |
6 |
2 |
1 |
9. Calculate the range
and coefficient of range of the following date.
|
Marks |
10-20 |
20-30 |
30-40 |
40-50 |
50-60 |
|
No. of students |
8 |
10 |
12 |
8 |
4 |
10. Find the range and coefficient of range of the following data.
Size: 5-10 10-15 15-20 20-25 25-30 30-35 35-40
Frequency: 3 8 7 12 9 8 6
6.3.4.2 Quartile Deviation or semi-Interquartile Range
The difference between third quartile or Q3 and first quartile or Q1 is calculated interquartile range. This is calculated by the following formula.
I.R.= Q3- Q1
Where, I.R= Interquartile large
Q3 = third quartile
Q1 = First quartile
One-half of the interquartile range is called semi-interquartile range or quartile deviation. It is calculated by the following formula.
2
Where, Q.D. –Quartile deviation
In any distribution, approximately 25% of observations are below (or lower than) Q1 so approximately 25% of observation are above (or greater than) Q3 so, approximately 50% of observation lie between Q1 and Q3. In a distribution small quartile derivation (Q.D). Shows the high uniformity or small variation of central 50% observations and a high quartile deviation (Q.D) shows the less uniformity or high variation of the center observation.
Coefficient of Quartile Deviation
The ratio of the difference and sum of third quartile deviation is an absolute measure of dispersion. It is calculated by the following formula.
- Q3 - Q1
Exercise 8.2 (b)
1. What is meant by ‘quartile deviation’? What are its merits and demerits?
2. Calculate the quartile deviation and its coefficient of the following data.
20, 28, 40, 12, 30, 15, 50
3. Calculate the quartile deviation and its coefficient
Marks: 10 20 30 40 50 80
No. of students: 4 7 15 8 7 2
4. From the following data, calculate the quartile deviation and its coefficient.
Height in cm: 150 151 152 153 154 155 156 157 158
No of student: 15 20 32 35 33 22 20 12 10
5. Calculate the quartile deviation and its coefficient
|
Wages Rs. |
0-10 |
10-20 |
20-30 |
30-40 |
40-50 |
50-60 |
60-70 |
70-80 |
|
No. of workers |
20 |
45 |
85 |
160 |
70 |
55 |
35 |
30 |
6. From the following series, find the quartile deviation and its coefficient.
Class-interval: 20-29 30-39 40-49 50-59 60-69 70-79
Frequency: 306 182 144 96 42 34
7. Find the quartile deviation and its coefficient from the following data.
Marks obtained No of students
‘More than’ 70 7
‘More than ‘60 18
‘More than ’50 40
‘More than’40 40
‘More than’30 63
‘More than’20 65
8. From the following data, find the suitable measure of dispersion.
|
Wages Rs. |
Below 30 |
30-40 |
40-50 |
50-60 |
60-70 |
Above 70 |
|
Frequency |
4 |
11 |
20 |
22 |
15 |
8 |
6.3.4.3 Mean Deviation
In a series, the arithmetic mean of deviations of items found from a measure of central tendency (like mean, median or model) is called mean deviation.
Calculate of Mean Deviation:
1. Individual Series: The mean deviation of individual series is found out by using the following formula.
N
N
n
Where deviation X-
x̄│, │ X - Md│&│X=
M0│ are the absolute values (ignoring minus sign)
of deviations taken from mean, median& mode respectively.
Merits and demerits of Mean deviation
1. It is easy to compute and simple to understand,
2. As it is easy to compute and simple to understand.
3. It is less affected by the value of extreme observation.
4. Since deviations are taken from a central value (like, mean, median, or mode), it is helpful for making comparative studies about formation of two or more than two frequency distributions.
Despite having the above mentioned merits, mean deviation has some demerits. They can be pointed out as follows.
1. The main defect of mean deviation is that the algebraic signs (plus and minus) of deviation are left. So, it becomes an unscientific measure and rarely used in statistical processes.
2. Since it can be calculated by using mean, median or mode, it is not accurate measure and not rigidly defined.
3. It is not suitable for other further algebraic treatment also.
Variance and coefficient of variation
1. Variance: The Square of
standard deviation is called variance. This term was first brought in use by
R.A. Fisher in 1913. It can be useful when the effect on several variables in a
series is to be studied separately. By removing the radical sign (√) from
the formulate for finding the standard deviation in various series, variance
can be obtained directly. If there is some letter without the radical sign in
the formula for finding standard deviation, the variance is obtained only by
removing the radical sign and squaring that letter also. The following formulae
are used in calculating the variance.
a. In
individual series
34
i. Variance = ΣX- x̄-^ x̄)2 [where, =x̄ actual
mean]
N
b. in discrete and continuous series
N N h
In case of a continuous series, x=
mid-point of class-interval.
2. Coefficient of variation: The quotient obtained by dividing the standard deviation by the mean of distribution is called coefficient of standard deviation.
Symbolically,
Coefficient of standard deviation
= 6
X
Where, 6 = standard deviation and x̄= actual mean.
Coefficient of standard deviation is a relative measure of dispersion. It is used in comparing two or more than the two series. Since its value comes in fraction, it is not considered to be useful for comparison. So, if the coefficient of standard deviation is multiplied by 100, the product obtained is called coefficient of variation (C.V.)
i.e. C.V. = = 6 *100
X
Merits and demerits of standard deviation:
Standard deviation has some merits and demerits. Its merits can be pointed out as follows.
1. It depends on the values of all items. So, it is the best measure of dispersion.
2. In its calculation, the median and mode are not used like in mean deviation is rigid and definite.
3. If all the measures of dispersion are calculated by various samples taken from a same data. Only standard deviation would have approximately the same value but other measures of dispersion (like, range, quartile deviation, mean deviation) would not. It means that standard deviation is less affected by the fluctuations of sampling than other measures of dispersion.
4. Algebraic sign are not ignored in its calculation. So, it is considered to be a more suitable measure mathematically.
5. The combined standard deviation of two or more groups can be calculated whereas it is not possible in any other measure of dispersion.
6. It is used in calculating the coefficient of variation which is more appropriate for comparing the variability of two or more series.
7. It can be used in many statistical work for example, it is used in calculating correlation, skewers etc.
Through the standard deviation has above mentioned merits, yet it is not free from drawbacks. Its demerits can be pointed out as follows.
1. In comparison to other measures of dispersion, its calculation takes greater time and labour and it is not easily understood.
2. It gives less weight to the items which are near the mean and greater weight to the extreme items.
3. It is highly affected by extreme items like arithmetic mean.]
Exercise:
1. Find the standard deviation from the following data.
X: 10 15 12 18 22 25 30
2. Find the standard derivation from the following data.
X: 37 39 41 43 45 47 49 51
F: 3 7 12 17 8 6 4 3
3. Find the standard deviation from the following income distribution data.
X: 45 55 65 75 85 95
F: 2 5 6 8 11 16
4. Find the standard deviation from the following data.
Wage: 10 20 30 40 50 60 70 80
Labour: 12 30 22 16 20 15 22 23
5. Calculate the standard deviation from the following data.
X: 6 7 8 9 10 11 12
F: 3 6 9 13 8 5 4
6. Marks obtained by 50 students are given below. Find the standard deviation.
Marks: 0-10 10-20 20-30 30-40 40-50
Number of students: 5 10 25 5 5
7. Calculate the standard deviation from the following data.
|
Class Interval |
Frequency |
|
5-10 10-15 15-20 20-25 25-30 30-35 35-40 40-45 |
6 5 15 10 5 4 3 2 |
8. Calculate the standard deviation from the following data.
|
Age in years |
No. of Workers |
|
50-55 45-50 40-45 35-40 30-35 25-30 20-25 |
25 30 40 45 80 110 170 |
9. Find out the variance of the following data.
Value: 10 11 12 13 14
Frequency: 3 12 18 12 2
10. Find the standard deviation from the following data.
Income (000): 37 39 41 43 45 47 49
Families: 3 7 10 13 6 6 7
11. Find the standard deviation from the following wage distribution data.
X: 45 50 60 65 78 80 85
F: 1 5 2 11 12 9 8
12. Find the standard deviation from the following data.
Wage: 100 200 300 400 500 600 700 800
Laborer: 2 4 5 3 7 2 2 3
13. Calculate the standard deviation from the following data.
X: 6 7 9 10 15 18
F: 3 6 7 8 5 7
14. Marks obtained by 40 students are given below. Find the standard deviation.
X: 0-10 10-20 20-30 30-40 40-50
F: 4 10 15 6 5
6.4 Index number
6.4.1 Introduction and Meaning
Index number is frequently used statistical device for measuring the change. Generally, it is used for measuring the change in price, production, income, wages which are changing in course of time, because of that some scholars have described it as the barometer of economic activities.
In general, index numbers’ is used for measuring the change. It has been defined by various scholars in different ways. Index number is a special type of average which provides a measurement of relative changes from time to time or from place to place. Index numbers are devices for measuring differences in the magnitude of a group of related variables. Another statistician Spiegel has defined index numbers are devices for measuring differences in the magnitude of a group of related variables with respect to time, geographical location or other characteristics such as income, profession etc.
F.Y. Edgeworth has defined index number giving focus on the change. According to Edgeworth. ‘Index number shows by its variation the change in a magnitude which is not susceptible either of accurate measurement in itself or of direct valuation in practice. Morris Humburg has explained it as the relative number or a relative which expressed the relationship between to figures, where one of the figures in used as a base (Morn’s Humberg). Berenson and Levine have defined index number as the measure of magnitude of change. Generally speaking, index numbers measure the size or magnitude of some object at a particular point in time as a percentage of some base of reference object in the past (Berenson and Levine).
From the above discussion it is clear that an index number is a specialized average designed to measure the change in a group of related variables over a period of time. In conclusion, following points can be mentioned regarding the index numbers.
1. Index number is specialized averages.
2. Index numbers measure the net change in a group of related variables.
3. Index numbers measure the effect of changes over a period of time.
6.4.2 Importance of index numbers:
The economic activities are changing over the time. Such changes affect social and economic life of people. Changes in price, production consumption, import, export affect the economic activities successively Because of that, various statistical devices. Index number is important in this regard. Following are the uses or importance of index numbers.
1. Index number helps in framing suitable policies: Index numbers help to measure the changes and magnitude of changes. By helping in measuring, index number provides the guidelines for economic and business policies, wages, price, income, production, etc. can be adjusted of index number. Index numbers are equally useful to other subjects and areas in addition to the economics. Sociologist, psychologist, health authorities etc. can use the index number to formulate the policies as per requirement.
2. Explanation of trend and tendencies: Index numbers are used for measuring the change in certain phenomenon over a period time. Income consumption production etc. and their changes are measured with the income help of this. Such measurement shows the tendencies of certain things, on the basis of which various forecasting activities can be carried out. By examining the trend of certain phenomenon, various important conclusions can be drawn for making policies.
3. Forecasting future economic activity: Index numbers are important in forecasting future economic activity. By using time series data, index numbers help to calculate the seasonal and cyclical variation of changes in various economic activities. On the basis of measuring such changes, producers and businessmen can make the plan for future regarding the certain targets.
4. Comparative study: Different point is considered for calculating the index numbers. It helps to compare the change of production, income, price, of different years. Such comparison is necessary for measuring the trends and situation of the economy.
5. Useful is deflating: Index numbers are considered useful in deflecting i.e. for adjusting data of previous years. It helps to adjust between price, wage, and salaries with actual cost of living. In the same way index numbers help to change the normal income in to real income and nominal sales into real sales through appropriate index numbers.
6.4.3 Types of index numbers:
1. Price index numbers: It helps to measure the changes in price between periods to another.
2. Quantity Index Numbers: Quantity index numbers help to measure the change in quantity such as production, sale, import etc.
3. Value index numbers: Value index numbers help to measure the changes in value of money over the period of time.
6.4.4 Construction of Index Numbers:
In spite of various index numbers,
price index number is important and common. Various formulae can be used for
measuring the price index. They are described as below:
|
Index numbers |
Simple Aggregative Weighted Aggregative Weighted average of relatives Simple average of relatives’ price Weighted Unweighted
1. Unweighted: Unweighted index numbers can be obtained as in the following two methods:
a. Simple aggregative
b. Simple average of price relatives
1. a. Simple aggregative method: This is simple method for calculating index numbers. In this method the total current year price for the various commodities in questions divided by the total of base year price and the quotient is multiplied by 100.
Σp0
P01 = Index number Σ =
summation
P1 = Current year price
P0
N
N= Number of goods
In the same way geometric mean can
be used for finding price relatives as
N
Here, p= p1/p0 * 100
2. Weighted Index Numbers: Weighted index numbers can be obtained as in the following two methods.
a. weighted Aggregative index numbers
b. Weighted Average of price relative method
2a. Weighted Aggregative index numbers: In this method relative importance of commodities is considered. Generally all goods are not equally important for human beings. Relatively important commodities get high weighted. This is various methods for this,
ΣP0q0
2. b. Weighted Average of price Relative method: In this method price and weighted are considered. Following formula is used for calculating the index number.
Σw
6.4.5 Factors Considering Constructing Index Numbers
Index number is important for economics and economic activities. Price, production, economic activities and their tendencies can be found with the help of index number. But proper calculation is necessary; following factors should be while calculating the index number in appropriate ways.
1. Objectives: Objectives of index number construction should be clear. If the objectives are not clear, the selection of base year and current year, may be reliable because of that the objective should be clear, measurable and purposeful.
Selection of commodities: certain commodities are necessary for calculating the index number; Commodities should be selected carefully considering their popularity in base year and current year.
a. Commodities should be according to interest, habit and tradition.
b. Commodities should be popular is all times i.e. in base year ad current year.
c. there should be the stability in consumption in all situations.
d. Selected commodities must have economic and social values.
E Commodities should be representative.
3. Selection of base year: Index number requires base year. Is should be selected carefully. Appropriate base year can provide the real change is variable for electing the appropriate year, following should be considered:
a. Base year should be normal. There would not be sharp fluctuation in production, economic and political activities.
b. Season of the year should be considered while selecting the base year generally normal season should be considered for collecting the prices of different commodities.
4. Price collection: Collection of prices is necessary for constructing the index number.
a. Prices should be collected from highly transacted areas
b. priority should be provided to published data.
c. Quality of sources of price information should be considered.
D. Wholesale or retail price should be collected according to objectives.
e. If the prices are different, average should be considered for using in index number calculation.
5. Determination of weight: Weight should be determined for constructing the weighted index number. Weight of the commodities should be according to the importance in human life.
6. Selection of method:
Index number should be calculated according 6to the objectives. There are various methods for calculating the index number. The methods should be according to the objectives of study.
6.4.6 Limitations of index numbers
Index number has been important device for measuring the change is different variables. Various activities of the economy such as production, income prices regarding application. According to them index number indicates only the trend rather than the real situation. They have shown various limitations of index number as below:
1: Index number provides the information of average may not be equally applicable and true in all situations.
2. Index number is based on sample and its reliability depends on the methods of sampling it cannot provide the real result, its sampling incurs some weaknesses.
3. Changes in quantity may have number of determinations. The changed value of index number cannot show the effect of various determinants and their share in total change.
4. Information regarding the index number may be different according o society, economic situation etc.
5. Index number requires reliability is selection of years, commodities and prices of difficult to select the base year, popular commodities and reliable prices for calculation, in this situation the index number cannot be representative.
6. Because of different methods, various results can be obtained; this creates the problems for finding the appropriate result.
7. Index number is calculated with certain objectives. It may be partial due to business feeling of the researcher
Exercise:
1. Calculating the index umber from the following data by simple aggregative method.
|
Goods |
Price per kg in 1982 |
Price per kg in 1983 |
|
Maize Rice Ghee Milk Potato |
20 24 4 6 10 |
30 24 4.8 7.2 14 |
2. Construct index number from the following data using (a) simple aggregative method (b) average of price relative method.
Goods: A B C D E
Price in 1989: 50 60 10 15 25
Price in 1995: 75 60 12 18 35
3. Construct index number by Fisher’s ideal formula.
|
goods |
Base year |
Current Year |
|||
|
Price |
Quantity |
price |
Quantity |
||
|
A B C D E |
6 2 4 8.5 8 |
10 2 6 12 16 |
10 2 6 12 16 |
56 120 61 24 22 |
|
4. Construct simple aggregative price index from the following data
Goods: A B C D E
Price in 19902: 5 4 28 10 3
Price in 1996: 6 5 33 12 4
5. Calculate the index number for 1996 using weighted average of price relative
Goods: A B C D
1991: 2 25 3 1
4.5 3.2 4.5 1.8
5 7 6 2
6.
|
goods |
Base year |
Current year |
|||||
|
price |
Quantity |
price |
Quantity |
||||
|
A B C D |
2 5 4 2 |
8 10 14 19 |
6 5 16 13 |
4 7 5 2 |
|||
7. Calculate the index number from the following data by simple aggressive method.
|
goods |
Price per kg in 2010 |
Price per kg in 2019 |
|
Millet Rice Wheat Barley maize |
20 24 22 24 18 |
30 24 33 28 24 |
8. Construct index number from the following data using (a) simple aggregative method (b) average of price relative method.
Goods: A B C D
PRICE IN 2000: 55 65 10 15 25 45
Price in 2010: 70 80 12 22 30 50
9.
|
goods |
Base year |
Current Year |
|||
|
Price |
Quantity |
price |
Quantity |
||
|
A B C D E |
6 2 4 8.5 8 |
10 2 6 12 16 |
10 2 6 12 16 |
56 120 61 24 22 |
|
10. Construct simple aggregative price index from the following data.
Goods A B C D E
Price in 2010: 5 4 8 10 13
Price in 2020: 9 9 12 13 20
11. Calculate the index number for 1996 using weighted average of price relative.
Goods: A B C D E
Price in 1995: 20 25 30 10 24
Price in 1999: 45 32 45 18 30
12. Construct the index numbers of prices from the following data by applying laspeyre’s method and paasche method.
|
Goods |
Base Year |
Current Year |
||||
|
Price |
Quantity |
Price |
Quantity |
|||
|
A B C D |
2 8 7 10 |
8 11 16 19 |
4 12 12 17 |
16 15 16 25 |
||
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