ALLAMA IQBAL OPEN UNIVERSITY, ISLAMABAD

Quantitative Techniques (5564)Muhammad Umer Contact # 03005181376

Autumn - 2010

Q.1 (a)

Describe the major phases of statistics. Formulate a business problem and analyze it by applying these phases?

STATISTICS Statistics offers a range of methods for the collection, presentation and analysis of data. Underlying these methods is the framework of mathematics and mathematical modeling. Typically in statistical methods, a major role is played by the notion of uncertainty and the mathematical solution to deal with it, namely: probability, and closely linked the concept of random variation. This uncertainty arises when one realizes that an actual data set is just a specimen of a set of possible outcomes that one might have obtained in the given situation just as well.

PHASES OF STATISTICSy y y y y Data collection Organizing data Presentation of data Data analysis Data interpretation

DATA COLLECTION Data collection refers to the gathering of set of observations about variables and it is the starting point of research methods. Basically, there are two types of data which are: primary data and secondary data. Primary data is received from first hand sources such as: direct observation, interview, survey, and questionnaire etc. On the other hand, secondary data is received from secondary sources such as: printed material and published material etc. Here, we will only discuss the primary sources of data collection. ORGANIZING DATA After being collected and processed, data need to be organized to produce useful information. When organizing data, it helps to be familiar with some of the definitions.. PRESENTATION OF DATA Presentation of data in statistics, are very careful work that's done with lots of information. There are many different examples on how to present data in statistics, which are include mean, standard deviation, median, minimum, and maximum. To present data in statistics you will need to refer to table in body of paper. Make sure that all the information that you want to convey in a table or a graphic is understandable for the all reviewers. DATA ANALYSIS Data Analysis is a practice in which raw data is ordered and organized so that useful information can be extracted from it. The process of organizing and thinking about data is key to understanding what the data does and does not contain. There are a variety of ways in which people can approach data analysis, and it is notoriously easy to manipulate data during the analysis phase to push certain conclusions or agendas.

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For this reason, it is important to pay attention when data analysis is presented, and to think critically about the data and the conclusions which were drawn. Raw data can take a variety of forms, including measurements, survey responses, and observations. In its raw form, this information can be incredibly useful, but also overwhelming. For example, survey results may be tallied, so that people can see at a glance how many people answered the survey, and how people responded to specific questions. DATA INTERPRETATION Data Interpretation can be defined as "the application of statistical procedures to analyze specific observed or assumed facts from a particular study". Data interpretation is something that is pretty common in education circles. They come as questions in tests to understand how much a student has understood the subject at hand. In school, college, university and higher educational levels, data interpretation is common. In various entrance exams for colleges too, data interpretation is used as a means to understand a student's grasp of the subject

Business Problem And Analyze It

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Business Problem Anew water company Transparent Waters launches its business in Pakistan for this the company surveyors collect water samples from different rivers and well in order to know the quality of water available to people A Data Table is simply an organized way to display all of r water quality data, and will usually be included in the report about sampling results. Data tables may be hand-written or typed in a word processor, but are most useful when created using computer spreadsheet and database programs. Spreadsheet programs (Excel, Lotus, ) allow to print tables, perform calculations, and develop graphs with your data. Data tables may be organized in many ways, depending on what kind of problem you are looking at. One common approach is to create one table for each sampling location. The columns of the table would then be the various water quality parameters, and the rows would be the results for each sampling date: Example Table 1: Water Quality Data at River Sindh Dissolved Oxygen Water Temperature BOD Fecal Coliform mg/l degrees C mg/l #/100ml 10 10.5 9.2 8.5 6.1 4.3 6.4 11 13 12 15 20 20 19 1.1 0.5 0.8 1 3.1 3.1 2.2 120 150 70 30 20 20 25

Date 03/01/11 04/01/11 05/01/11 06/01/11 07/01/11 08/01/11 09/01/11

This kind of table is especially useful if you are trying to see how different parameters are related to each other.4

A second common method is to create one table for each water quality parameter. In this case the columns would be the various sampling locations, and the rows would be the results for each sampling date: Example Table 2: Dissolved Oxygen (mg/l) Dissolved Oxygen in mg/l River Jhelum River Swat River Chunab Mangla Dam Tarbella River Kabul 03/01/98 04/01/98 05/01/98 06/01/98 07/01/98 08/01/98 09/01/98 10.2 10.1 9.2 9.3 7.4 7.1 8.1 9.8 10.1 8.2 9.1 6.5 5.4 6.8 9.9 10.5 9.2 8.5 6.1 4.3 6.4 10.2 10.3 9.1 9.2 7.3 5.9 6.8 10.3 9.5 10.2 9.2 8.1 7.9 8.1 10.2 9.7 10 8.9 9.1 8.3 9.4

Date

This kind of table helps look at trends in data, such as how a parameter changes over time at one location, or how it changes as move downriver on a given sampling date. An important part of Quality Control is to make sure tables are transcribed accurately from your original water quality data. All tables should be carefully proofed and checked against original laboratory and field notes.

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Q. 1 (b) List at least two applications of statistics in each functional area of management?

INTRODUCTION TO THE USE OF STATISTICS IN MANAGEMENT Statistics may be defined as a systematic process of collection, classification, tabulation, analysis, interpretation and drawing valid inferences of numerical data in any field of human activity. In almost all the fields of human activity, the question that crops in is the variability of characteristics. The variability which is observed in nature is the sound footing of statistical analysis which tells with a certain degree of confidence, the relative and absolute risks involved. This helps in management in decision making and planning for the future. DECISION MAKING - USE OF STATISTICS IN MANAGEMENT Whether it is a factory or farm resources of men, machine and finance have to be coordinated against time and space constraints, to achieve the objectives in the most efficient manner. The common trend of all the managerial activity is the capability to evaluate the situation, the objectives, limitations and alternatives, obtain information and make decisions. With the ever-increasing growth in the size and competition, the business environment has become complex. Since the complexity of business environment makes the decision making process difficult, the decision maker can no longer rely entirely upon his judgment, experience or evaluation to make a decision. Instead he has to base his decisions upon data which show relationship, indicate trends and show rates of change in various variables or characteristics. USE OF STATISTICS IN EVERY AREA OF MANAGEMENT Application of statistics pervade virtually every area of management decision making whether it be production, finance, distribution, marketing or any other activity. In any organization, the management uses statistical techniques for making valid decisions on the basis of factual data on the current operations. These decisions are of so vital importance that they not only improve the present situation, but also effect the future operations and policies. Statistics plays an important role and is very much in use in production and inventory decisions, marketing decisions, investment and financial decisions and also in planning decisions. USE OF QUANTITATIVE TECHNIQUES IN BUSINESS AND MANAGMENT Due to increasing complexity in business and industry, decision making based on intuition has become highly questionable especially when the decision involves the choice among several courses of action each of which can achieve several management actions. So there is need for training the people who can manage a system efficiently and creatively. Quantitative Techniques now have a major role in effective decision making in various functional areas of management i.e. marketing, finance, production and personnel. These techniques are also widely used in planning, transportation, public health, communication, military, agriculture etc. Quantitative techniques are also used extensively as an aid in business decision making. Some of the areas where quantitative techniques can be used are,

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MANAGEMENT i). Marketing Analysis of marketing research information Statistical records for building and maintaining an extensive market Sales forecasting ii). Production Production planning, control and analysis Evaluation of machine performance Quality control requirement (to analyze the data/trends) Inventory control measures iii). Finance, Accounting and Investment Financial forecast, budget preparation Financial investment decisions Selection of securities Auditing function Credit policies, credit risk and delinquent accounts iv). Personnel Labour turn over rate Employment trends Performance appraisal Wage rates and incentive plans

Q3.(a) Solve using Guass-Jordan elimination. 2x1+4x2-10x3=-2 3x1+9x2-21x3=0 x1+5x2-12x3=1Solution:2 4 -10 3 9 -21 1 5 -12 = 2 0 1

1 -1 2 0 -6 2 1 5 -12

1 = 1 7 -3

1 -1 2 0 -6 15 0 6 -14

1 = -3 0

1 -1 2 0 1 -15/6 0 6 -14 =

1 -3/6 0

1 0 2-15/6 0 1 0 0 -15/6 -29 =

1-3/6 -3/6 3

1 0 0

0 -3/6 1 -15/6 0 1 =

3/6 -3/6 -3/29

1 0 0

0 1 0

0 0 1 =

3/6+(3/6)-3/29 -3/6+(15/6)(-3/29) -3/29

1 0 0

0 1 0

0 0 1 =

1/2+(1/2)(-3/29) -1/2+(5/2)+(-3/29) -3/29

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x = 1/2+(1/2)(-3/29) x = 1/2-3/58 x = 13/29 y = -1/2+(5/2)(-3/29) y = -1/2-15/58 y = -21/29 z = -3/29

Q3(b). Solve the following system of equation by Cramers rule. -x1+2*2=24 3x1-4*2=10Solution:Cramers rule gives the solution as follows: X = Dx/D , y = Dy/D

Where D,Dx,Dy are determinant defined by -1 2 = 24 10 -1 3 2 -4

3 -4 D=

= -1(-4)-(2)(3) D = 4-6= -2 Dx = 24 2

10 -4 = 24(-4)-(2)(10)

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= -96-20 Dx = -116 Dy = -1 3 = = 24 10

-1(10)-3(24) -10-72

Dy = -82 x = Dx/D = -116/-2 = 58 y = Dy/D = -82/-2 = 41

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Q.2 (a)

Determine the common ratio of the G .P. 49, 7, 1/7, 1/49

Find the sum to first 20 terms of G .P. Find the sum to infinity of the terms of G .P.

Solution:In mathematics, a geometric progression (also inaccurately known as a geometric series) is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. If the common ratio is:

y y y y y

Negative, the results will alternate between positive and negative. Greater than 1, there will be exponential growth towards infinity (positive). Less than -1, there will be exponential growth towards infinity (positive and negative). Between 1 and -1, there will be exponential decay towards zero. Zero, the results will remain at zero

The common ratio is = r r= ak/ak-1 r= 7/49= 1/7 (i) Find the sum to first 20 term of G.P a20 = a1(r)20-1 =49.(1/7)19 = 49/719 = 1/717 The 20th term of G.P is = 1/717 Series of the sequence is Sn = a(1-rn)/1-r S20 = 49(1-(1/7)20)/1-1/7 = 49(1-(2.857)/0.857

=49(-1.857)/0.857

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(ii)

find the sum to infinity term of G.P. Solution:S = k=1 , ark = a/1-r = 49/1-1/7 =49/6/7 = 1.1666

A woman deposits RS. 20,000 in a bank that pays 6% interest per year compounded annually. How much is in her account after 4 years. We assume that interest is added to her account and not withdrawn.Solution:P = deposit = 20,000 P is the principal amount r= profit 6% , n=4 , r is the annual rate n is the numbers of year

A is the amount of money accumulated after n year including interest A = P(1+r)n = 20,000(1+0.06)4 = 20,000(1.06)4 = 84800

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(a)

Distinguish between the census and sampling methods of data collection and compare their merits and demerits. Why is the sampling method unavoidable in certain situations? What are ogives? Point out the role. Discuss the method of constructing ogives with the help of an example. (20)

(b)

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CENSUS Census is a complete enumeration of an entire population of statistical units in a field of interest. It is also called complete enumeration survey. For example, population census canvases every household in a country to count for the number of permanent residents and other characteristics; census of manufacturing canvases all establishments engaging in manufacturing activities. Data from the census serve as base-year or benchmark data. Requirement: A complete and up-to-date register of all statistical units in the field of inquiry is required. Advantages: Census provides the most reliable statistics if done professionally and with integrity. Disadvantages: Very costly to enumerate and to process data. Timeliness is low: data is available for use only many months, even years after. Census is normally carried every five or ten years. SAMPLING METHOD When the investigator studies only a representative part of the total population and makes inferences about the population on the basis of that study. It is known as sampling method or Survey. In both methods, the investigator is interested in studying some characteristics of the population. Advantages: Provide more up-to-date statistics, which are reliable if scientifically designed and professionally implemented, less costly than census. Sampling errors can also be obtained. Surveys are normally carried out weekly, monthly, quarterly or annually. Disadvantages: Timeliness requires prompt data processing, thus less information may be asked. CENSUS AND SAMPLING Practically every country in the world conducts censuses and sampling surveys on a regular basis in order to get valuable data from and about their populations. This data is used by the federal and state governments in making numerous decisions with regard to various health care, housing, and educational issues, among others. While both these two data-gathering methods essentially serve the same purpose, they have a number of differences with regard to approach and methodology, as well as scope. These two methods may also differ in terms of the variance in the data gathered,. A census involves the gathering of information from every person in a certain group. This may include information on age, sex and language among others. A sample survey on the other hand commonly involves gathering data from only a certain section of a particular group. SAMPLING VARIANCE The main advantage of a census is a virtually zero sampling variance, mainly because the data used is drawn from the whole population. In addition, more precise detail can generally be gathered about smaller groups of the population. 13

As for sampling, there is a possibility of sampling variance, since the data used is drawn from only a small section of the population. This makes sampling a much less accurate form of data collection than a census. In addition, the sample may be too small to provide an accurate picture of the population. COST AND TIMETABLE A census can be quite expensive to conduct, particularly for large populations. In most cases, they are also a lot more time-consuming than sample surveys. Adding considerably to the timetable is the necessity of gathering data from every single member of the population. The huge scope of a census also makes it harder to maintain control of the quality of the data. For instance, anyone who does not complete a census form will be visited by a government representative whos only job to is to gather census data. A sample survey for its part costs quite a bit less than a census, since data is gathered from a much smaller group of people. In addition, sample surveys generally take a much shorter time to conduct, again given the smaller scope. This also means reduced requirements for respondents, which in turn leads to better data monitoring and quality control. CENSUS VS SAMPLING There are stark differences between Census and sampling though both serve the purpose of providing data and information about a population. Howsoever accurately a sample from a population may be generated there will always be margin for error, whereas in case of Census, entire population is taken into account and as such it is most accurate. Data obtained from both Census and sampling is extremely important for a government for various purposes such as planning developmental programs and policies for weaker sections of the society. It is obvious then that when whole population is taken into account, data collection is called Census Method, whereas when a small group that is representative of the entire population is used, it is called a Sample Method. SUMMARY Census refers to periodic collection of information about the populace from the entire population. Sampling is a method of collecting information from a sample that is representative of entire population. There are both advantages and disadvantages of both the methods. Whereas data from census is reliable and accurate, there is a margin of error in data obtained from sampling.

Census is very time consuming and expensive, whereas sampling is quick and inexpensive. However, if the next Census is far away, sampling is the most convenient method of obtaining data about the population.

SITUATIONS IN WHICH SAMPLING METHOD IS UNAVOIDABLE Sampling method is unavoidable in following situations 14

Unlimited population Distractive population nature Unapproachable population e.g. Mobilink users In quality control, such as finding the tensile strength of a steel specimen by stretching it till it breaks. Another example is in process checking in the manufacturing of pharmaceuticals where it is not possible to check the each and every tablet or injection. Secondly quality testing results in destruction of items itself

(b) What are ogives? Point out the role. Discuss the method of constructing ogives with the help of an example.OGIVE An Ogive is a specialized line graph which shows how many items there are which are below a certain value.

INTRODUCTION TO HOW TO CONSTRUCT AN OGIVE:Cumulative frequencies of a distribution can also be charted on a graph. The curve that results by plotting these is called the Ogive Curve. Since the cumulative frequencies can either be less than or more than type, there are two type of ogives called less than type and more than type ogive. The value of median and other partition values can be located from the ogives. The technique of drawing frequency curves and cumulative frequency curves is more or less the same. The only difference is that in case of simple frequency curves the frequency is plotted against the mid point of a class interval whereas in case of a cumulative frequency curve it is plotted at the upper or limit of a class interval depending upon the manner in which the series has been cumulated.

How to Construct an Ogive - TypesLess Than Ogive: - The less than cumulative frequencies are in ascending order. The cumulative frequency of each class is plotted against the upper limit of the class interval in this type of ogive and then various points are joined by straight line.

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More Than Ogive:- The cumulative frequencies in this type are in the descending order. The cumulative frequency of each class is plotted against the lower limit of the class interval.

Example on How to Construct an Ogive Example: Marks obtained by the students of a class in statistics test are : Marks 0 10 10 20 8 20 30 18 30 40 15 40 50 5

Number of students 4

Draw less than and more than ogives. Solutions. First, the less than and more than cumulative frequencies will be calculated and the ogives will be drawn on the basis of these cumulative frequencies. Calculation of Cumulative Frequencies

Marks 0 10 10 20 20 30 30 40 40 50

Frequency 4 8 18 15 5

Less than Cumulative More than Cumulative Frequency Frequency 4 50 12 30 45 50 46 38 20 5

The Two Ogives Are Shown In Below Figure

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Construct an Ogive - Uses From the standpoint of graphic presentation, the ogive is especially used for the following purposes:1. To determine as well as to portray the number of proportion of cases above or below a given value. 2. To compare two or more frequency distribution. Generally there is less overlapping when comparing several ogives on the same grid than when comparing several simple frequency curves in this manner.

3. Ogives are also drawn for determining certain values graphically such as median, quartiles,deciles, etc.

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Q.5 (a) Explain the terms Geometric Mean and Harmonic Mean. Point out some the public system applications of the concept? (b) What are statistical averages? What are the desirable properties for an average to posses? Mention different types of averages and state why the arithmetic means are the most commonly used among them?

GEOMETRIC MEANThe geometric mean is an average calculated by multiplying a set of numbers and taking the nth root, where n is the number of numbers. For example, the geometric mean of 4, 8, 16 is: (4 8 16)1/3 = 8

A common example of when the geometric mean is the correct choice average is when averaging growth rates, see compound annual growth rate. Compound annual growth rate (CAGR) is an average growth rate over a period of several years. It is a geometric average of annual growth rates:

CAGR = (ending value starting value)1/(number of years - 1

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If a company had sales of 10m in 2000 and 15m in 2005 then the CAGR of its sales is: (15 10)1/5 - 1 = .084 = 8.4% If percentage growth rates are used it is important to remember to add one to each of them before calculating the geometric average. For example, the CAGR over two years of 10% one year and 20% the next is (1.1 1.2)1/2 - 1

MERITS, DEMERITS AND USES OF GEOMETRIC MEANMERITS 1. It is a rigidly defined average. 2. It is based on all the observations. 3: It is capable of mathematical treatment. If any two out of the three values, i.e., (i) product of observations, (ii) GM of observations and (iii) number of observations, are known, the third can be calculated. 4. In contrast to AM, it is less affected by extreme observations. 5. It gives more weights to smaller observations and vice-versa.

DEMERITS1. It is not very easy to calculate and hence is not very popular. 2. Like AM, it may be a value which does not exist in the set of given observations. 3. It cannot be calculated if any observation is zero or negative.

USES 1. It is most suitable for averaging ratios and exponential rates of changes.2. It is used in the construction of index numbers. 3. It is often used to study certain social or economic phenomena.

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HARMONIC MEAN Definition

MERITS AND DEMERITS OF HARMONIC MEAN MERITS 1. It is a rigidly defined average. 2. It is based on all the observations. 3. It gives less weight to large items and vice-versa. 4. It is capable of further mathematical treatment. 5. It is suitable in computing average rate under certain conditions. DEMERITS 1. It is not easy to compute and is difficult to understand. 2. It may not be an actual item of the given observations. 3. It cannot be calculated if one or more observations are equal to zero. 4. It may not be representative of the data if small observations are given correspondingly small weights.

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Use of Harmonic MeanIn certain situations, especially many situations involving rates and ratios, the harmonic mean provides the truest average. For instance, if a vehicle travels a certain distance at a speed x (e.g. 60 kilometers per hour) and then the same distance again at a speed y (e.g. 40 kilometers per hour), then its average speed is the harmonic mean of x and y (48 kilometers per hour), and its total travel time is the same as if it had traveled the whole distance at that average speed.

In finance, the harmonic mean is used to calculate the average cost of shares purchased over a period of time. For example, an investor purchases $1000 worth of stock every month for three months and the prices paid per share each month were $8, $9, and $10, then the average price the investor paid is $8.926 per share. However, if the investor purchased 1000 shares per month, the arithmetic mean (which turns out to be $9.00) would be used. Source(s): http://en.wikipedia.org/wiki/Harmonic_me

b)

What are statistical averages? What are the desirable properties for an average to posses? Mention different types of averages and state why the arithmetic means are the most commonly used among them?

ARITHMETIC MEAN

The average of a distribution has been defined in various ways. Some of the important definitions are: (i) (ii) (iii) "An average is an attempt to find one single figure to describe the whole of figures". "Average is a value which is typical or representative of a set of data". "An average is a single value within the range of the data that is used to represent all the values in the series. Since an average is somewhere within the range of data it is sometimes called a measure of central value". - Croxton and Cowden

If n numbers are given, each number denoted by ai, where i = 1, ..., n, the arithmetic mean is the [sum] of the ai's divided by n The arithmetic mean, often simply called the mean, of two numbers, such as 2 and 8, is obtained by finding a value A such that 2 + 8 = A + A. One may find that A = (2 + 8)/2 = 5. Switching the order of 2 and 8 to read 8 and 2 does not change the resulting value obtained for A. The mean 5 is not less than the minimum 2 or greater than the maximum 8. If we increase the number of terms in the list for which we want an average, we get, for example, that the arithmetic mean of 2, 8, and 11 is found by solving for the value of A in the equation 2 + 8 + 11 = A + A + A. One finds that A = (2 + 8 + 11)/3 = 7. Changing the order of the three members of the list does not change the result: A = (8 + 11 + 2)/3 = 7 and that 7 is between 2 and 11. This summation method is easily generalized for lists with any number of 21

elements. However, the mean of a list of integers is not necessarily an integer. "The average family has 1.7 children" is a jarring way of making a statement that is more appropriately expressed by "the average number of children in the collection of families examined is 1.7". STATISTICAL MEAN In Statistics, the statistical mean, or statistical average, gives a very good idea about the central tendency of the data being collected. Statistical mean gives important information about the data set at hand, and as a single number, can provide a lot of insights into the experiment and nature of the data. EXAMPLES The concept of statistical mean has a very wide range of applicability in statistics for a number of different types of experimentation. For example, if a simple pendulum is being used to measure the acceleration due to gravity, it makes sense to take a set of values, and then average the final result. This eliminates the random errors in the experiment and usually gives a more accurate value than a single experiment carried out. The statistical mean also gives a good idea about interpreting the statistical data. For example, the mean life expectancy in Japan is higher than that of Brazil, which suggests that on an average, the people in Japan are likely to live longer. There may be many viable conclusions about this, such as that it is due to better healthcare facilities in Japan, but the truth is that we do not know this unless we measure it. Similarly, the mean height of people in Russia is higher than that of China, which means that on an average, you will find Russians to be taller than Chinese. Statistical mean is a measure of central tendency and gives us an idea about where the data seems to cluster around. For example, the mean marks obtained by students in a test are required to correctly gauge the performance of a student in that test. If the student scores a low percentage, but is well ahead of the mean, then it means the test is difficult and therefore his performance is good, something that simply a percentage will not be able to tell. FUNCTIONS AND CHARACTERSTICS OF AN AVERAGE 1. To present huge mass of data in a summarized form: It is very difficult for human mind to grasp a large body of numerical figures. A measure of average is used to summaries such data into a single figure which makes it easier to understand and remember. 2. To facilitate comparison: Different sets of data can be compared by comparing their averages. For example, the level of wages of workers in two factories can be compared by mean (or average) wages of workers in each of them.

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3. To help in decision-making: Most of the decisions to be taken in research, planning, etc., are based on the average value of certain variables. For example, if the average monthly sales of a company are falling, the sales manager may have to take certain decisions to improve it. CHARACTERISTICS OF A GOOD AVERAGE A good measure of average must posses the following characteristics: 1. It should be rigidly defined, preferably by an algebraic formula, so that different persons obtain the same value for a given set of data. 2. It should be easy to compute. 3. It should be easy to understand. 4. It should be based on all the observations. 5. It should be capable of further algebraic treatment. 6. It should not be unduly affected by extreme observations. . It should not be much affected by the fluctuations of sampling. DIFFERENT STATISTICAL MEANS

There are different kinds of statistical means or measures of central tendency for the data points. Each one has its own utility. The arithmetic mean, geometric mean, median and mode are some of the most commonly used measures of statistical mean. They make sense in different situations, and should be used according to the distribution and nature of the data. For example, the arithmetic mean is frequently used in scientific experimentation, the geometric mean is used in finance to calculate compounding quantities, the median is used as a robust mean in case of skewed data with many outliers and the mode is frequently used in determining the most frequently occurring data, like during an election. The arithmetic mean is by far the most common average. It is the simplest to compute and the easiest to understand. In fact, most people are only familiar with the arithmetic mean; however, this is often inaccurate and misleading. Some merits of arithmetic means are defined below MERITS ARITHMETIC MEAN MERITS Out of all averages arithmetic mean is the most popular average in statistics because of its merits given below: 1. Arithmetic mean is rigidly defined by an algebraic formula.

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2. Calculation of arithmetic mean requires simple knowledge of addition, multiplication and division of numbers and hence, is easy to calculate. It is also simple to understand the meaning of arithmetic mean, e.g., the value per item or per unit, etc. 3. Calculation of arithmetic mean is based on all the observations and hence, it can be regarded as representative of the given data. 4. It is capable of being treated mathematically and hence, is widely used in statistical analysis. 5. Arithmetic mean can be computed even if the detailed distribution is not known but sum of observations and number of observations is known. 6. It is least affected by the fluctuations of sampling. 7. It represents the centre of gravity of the distribution because it balances the magnitudes of observations which are greater and less than it. 8. It provides a good basis for the comparison of two or more distributions.

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