Statistice for Management MB0040 B1129 1

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MB0040Statistics for Management.

(Book ID: B1129)

Set1

1. (a) Statistics is the backbone of decision-making. Comment.

(b) Statistics is as good as the user. Comment.

Answer:

(a)Statistics is the back bone of decision-making. Comment.

Due to advanced communication network, rapid changes in consumer behaviour, variedexpectations of variety of consumers and new market openings, modern managers have adifficult task of making quick and appropriate decisions. Therefore, there is a need for them

to depend more upon quantitative techniques like mathematical models, statistics,

operations research and econometrics.

Decision making is a key part of our day-to-day life. Even when we wish to purchase a

television, we like to know the price, quality, durability, and maintainability of variousbrands and models before buying one. As you can see, in this scenario we are collecting

data and making an optimum decision. In other words, we are using Statistics.

Again, suppose a company wishes to introduce a new product, it has to collect data onmarket potential, consumer likings, availability of raw materials, feasibility of producing

the product. Hence, data collection is the back-bone of any decision making process.

Many organisations find themselves data-rich but poor in drawing information from it.

Therefore, it is important to develop the ability to extract meaningful information from rawdata to make better decisions. Statistics play an important role in this aspect.

Statistics is broadly divided into two main categories. The two categories of Statistics are

descriptive statistics and inferential statistics.

Descriptive Statistics: Descriptive statistics is used to present the general description ofdata which is summarised quantitatively. This is mostly useful in clinical research, when

communicating the results of experiments.

Inferential Statistics: Inferential statistics is used to make valid inferences from the data

which are helpful in effective decision making for managers or professionals.

Statistical methods such as estimation, prediction and hypothesis testing belong toinferential statistics. The researchers make deductions or conclusions from the collected

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data samples regarding the characteristics of large population from which the samples are

taken. So, we can say Statistics is the backbone of decision-making.

(b) Statistics is as good as the user. Comment.

Statistics is used for various purposes. It is used to simplify mass data and to make comparisons

easier. It is also used to bring out trends and tendencies in the data as well as the hidden relations

between variables. All this helps to make decision making much easier. Let us look at each

function of Statistics in detail.

1. Statistics simplifies mass data

The use of statistical concepts helps in simplification of complex data. Using statistical

concepts, the managers can make decisions more easily. The statistical methods help in reducing

the complexity of the data and consequently in the understanding of any huge mass of data.

2. Statistics makes comparison easier

Without using statistical methods and concepts, collection of data and comparison cannot be

done easily. Statistics helps us to compare data collected from different sources. Grand totals,

measures of central tendency, measures of dispersion, graphs and diagrams, coefficient of

correlation all provide ample scopes for comparison.

3. Statistics brings out trends and tendencies in the data

After data is collected, it is easy to analyse the trend and tendencies in the data by using thevarious concepts of Statistics.

4. Statistics brings out the hidden relations between variables

Statistical analysis helps in drawing inferences on data. Statistical analysis brings out the hidden

relations between variables.

5. Decision making power becomes easier

With the proper application of Statistics and statistical software packages on the collected data,

managers can take effective decisions, which can increase the profits in a business. Seeing allthese functionality we can say Statistics is as good as the user

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Division in Statistics

Descriptive Statistics: Descriptive statistics is used to present the general description of data

which is summarized quantitatively. This is mostly useful in clinical r esearch, when

communicating the results of experiments.

Inferential Statistics: Inferential statistics is used to make valid inferences from the data which

are helpful in effective decision making for managers or professionals. Statistical methodssuch as estimation, prediction and hypothesis testing belong to inferential statistics.

The researchers make deductions or conclusions from the collected data samples regarding the

characteristics of large population from which the samples are taken. So, we can say Statistics is

the backbone of decision-making.

2. Distinguish between the following with example.

(a)Inclusive and Exclusive limits.

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(b)Continuous and discrete data.

(c)Qualitative and Quantitative data.

(d)Class limits and class intervals.

Answer:

a)

Inclusive and Exclusive limits.Inclusive and exclusive limits are relevant from data tabulation and class intervals point of

view.

Inclusive series is the one which doesn't consider the upper limit, for example,00-10

10-20

20-3030-40

40-50

In the first one (00-10), we will consider numbers from 00 to 9.99 only. And 10 will be

considered in 10-20. So this is known as inclusive series.Exclusive series is the one which has both the limits included, for example,

00-0910-19

20-29

30-3940-49

Here, both 00 and 09 will come under the first one (00-09). And 10 will come under the

next one.

b) Continuous and discrete data.All data that are the result of counting are called quantitative discrete data. These data

take on only certain numerical values. If you count the number of phone calls you receivefor each day of the week, you might get 0, 1, 2, 3, etc.

All data that are the result of measuring are quantitative continuous data assuming that

we can measure accurately. Measuring angles in radians might result in the numbers /6,/3, /2, /, 3/4, etc. If you and your friends carry backpacks with books in them to

school, the numbers of books in the backpacks are discrete data and the weights of thebackpacks are continuous data.

c) Qualitative and Quantitative dataData may come from a population or from a sample. Small letters like x or y generally

are used to represent data values. Most data can be put into the following categories:

Qualitative Quantitative

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Qualitative data

Qualitative data are the result of categorizing or describing attributes of a population. Haircolor, blood type, ethnic group, the car a person drives, and the street a person lives on are

examples of qualitative data. Qualitative data are generally described by words or letters. For

instance, hair color might be black, dark brown, light brown, blonde, gray, or red. Blood type

might be AB+, O-, or B+. Qualitative data are not as widely used as quantitative data becausemany numerical techniques do not apply to the qualitative data. For example, it does not

make sense to find an average hair color or blood type.

Quantitative dataQuantitative data are always numbers and are usually the data of choice because there are

many methods available for analyzing the data. Quantitative data are the result of counting ormeasuring attributes of a population. Amount of money, pulse rate, weight, number of people

living in your town, and the number of students who take statistics are examples of

quantitative data. Quantitative data may be either discrete or continuous.

All data that are the result of counting are called quantitative discrete data. These data takeon only certain numerical values. If you count the number of phone calls you receive for each

day of the week, you might get 0, 1, 2, 3, etc.

Example 2: Data Sample of Quantitative Continuous Data

The data are the weights of the backpacks with the books in it. You sample the same five

students. The weights (in pounds) of their backpacks are 6.2, 7, 6.8, 9.1, 4.3. Notice that

backpacks carrying three books can have different weights. Weights are quantitative

continuous data because weights are measured.

Example 3: Data Sample of Qualitative Data

The data are the colors of backpacks. Again, you sample the same five students. One studenthas a red backpack, two students have black backpacks, one student has a green backpack,

and one student has a gray backpack. The colors red, black, black, green, and gray are

qualitative data.

d) Inclusive and Exclusive Class IntervalsInclusive Class Interval:When the lower and the upper class limit is included, then it is an inclusive class interval.

For example - 220 - 234, 235 - 249..... etc. are inclusive type of class intervals. Usually in

the case of discrete variate, inclusive types of class intervals are used.

Exclusive Class Interval:

When the lower limit is included, but the upper limit is excluded, then it is an exclusive

class interval. For example - 150 - 153, 153 - 156.....etc are exclusive type of class

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intervals. In the class interval 150 - 153, 150 is included but 153 is excluded.

Usually in the case of continuous variate, exclusive types of class intervals are used.

Consider the frequency table shown below

Note: While analyzing a frequency distribution, if there are inclusive type of class intervals

they must be converted into exclusive type.This can be done by extending the class intervals from both the ends. Thus the class intervals 220

- 234, 235 - 249, ....... should be converted into exclusive type 219.5 - 234.5, 234.5 - 249.5.... etc.

After the conversion the frequency table would look like this

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3. In a management class of 100 students three languages are offered as an additional

subject viz. Hindi, English and Kannada. There are 28 students taking Hindi, 26 taking

Hindi and 16 taking English. There are 12 students taking both Hindi and English, 4 taking

Hindi and English and 6 that are taking English and Kannada. In addition, we know that 2

students are taking all the three languages.

i) If a student is chosen randomly, what is the probability that he/she is not taking any of

these three languages?

ii) If a student is chosen randomly, what is the probability that he/ she is taking exactly one

language?

Answer:

Let students taking Kannada as language be S (K) = 28

Let students taking Hindi as language be S (H) = 28

Let students taking English as language be S (E) = 28

Let students taking Kannada and English be S (K E) = 12

Let students taking Hindi and English be S (H E) = 4

Let students taking Hindi and Kannada be S (H K) = 6

Let students taking all the three subjects be S (K H E) = 2

If a student is chosen randomly, probability that he/she is not taking any of these three languagesis P (not taking any language)

= [ 1{ S(K)+ S(H)+S(E)+ S (K E )+ S ( H E )+ S ( H K )+ S (K H E} / 100 ]

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= [1{28+26+26+6+4+12+2} / 100]

= 1(94 / 100)= 10.94 = 0.06

If a student is chosen randomly, probability that he/ she is taking exactly one language is

P (taking exactly one language) = [ { S(K)+ S(H)+S(E) } ] / 100

= [28+26+16] = 70= 70

= {70/100}

= 0.7

4. List down various measures of central tendency and explain the difference between

them?

Answer:

Measures of Central Tendency

Several different measures of central tendency are defined below.

1 Arithmetic Mean

The arithmetic mean is the most common measure of central tendency. It simply the sum of the

numbers divided by the number of numbers. The symbol m is used for the mean of a

population. The symbol M is used for the mean of a sample. The formula for m is shown

below:

Where X is the sum of all the numbers in the numbers in the sample and N is the number of

numbers in the sample. As an example, the mean of the numbers 1 + 2 + 3 + 6 + 8 = 20/5 = 4

regardless of whether the numbers constitute the entire population or just a sample from the

population.

The table, Number of touchdown passes (Table 1: Number of touchdown passes), shows the

number of touchdown (TD) passes thrown by each of the 31 teams in the National Football

League in the 2000 season.

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The mean number of touchdown passes thrown is 20.4516 as shown below.

Number of touchdown passes

Although the arithmetic mean is not the only "mean" (there is also a geometric mean), it is by

far the most commonly used. Therefore, if the term "mean" is used without specifying whether

it is the arithmetic mean, the geometric mean, or some other mean, it is assumed to refer to the

arithmetic mean.

2 Median

The median is also a frequently used measure of central tendency. The median is the midpoint

of a distribution: the same number of scores are above the median as below it. For the data inthe table, Number of touchdown passes (Table 1: Number of touchdown passes), there are 31

scores. The 16th highest score (which equals 20) is the median because there are 15 scores

below the 16th score and 15 scores above the 16th score. The median can also be thought of as

the 50th percentile3. Let's return to the made up example of the quiz on which you made a

three discussed previously in the module Introduction to Central Tendency4 and shown in

Table 2: Three possible datasets for the 5-point make-up quiz.

Three possible datasets for the 5-point make-up quiz

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For Dataset 1, the median is three, the same as your score. For Dataset 2, the median is 4.

Therefore, your score is below the median. This means you are in the lower half of the class.

Finally for Dataset 3, the median is 2. For this dataset, your score is above the median and

therefore in the upper half of the distribution.

Computation of the Median: When there is an odd number of numbers, the median is simplythe middle number. For example, the median of 2, 4, and 7 is 4. When there is an even number

of numbers, the median is the mean of the two middle numbers. Thus, the median of the

numbers 2, 4, 7, 12 is 4+7/2 = 5:5.

3 Mode

The mode is the most frequently occuring value. For the data in the table, Number of

touchdown passes (Table 1: Number of touchdown passes), the mode is 18 since more teams

(4) had 18 touchdown passes than any other number of touchdown passes. With continuousdata such as response time measured to many decimals, the frequency of each value is one

since no two scores will be exactly the same (see discussion of continuous variables5).

Therefore the mode of continuous data is normally computed from a grouped frequency

distribution. The Grouped frequency distribution (Table 3: Grouped frequency distribution)

table shows a grouped frequency distribution for the target response time data. Since the

interval with the highest frequency is 600-700, the mode is the middle of that interval (650).

Grouped frequency distribution

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Proportions and Percentages

When the focus is on the degree to which a population possesses a particular attribute, the

measure of interest is a percentage or a proportion.

A proportion refers to the fraction of the total that possesses a certain attribute. Forexample, we might ask what proportion of women in our sample weigh less than 135

pounds. Since 3 women weigh less than 135 pounds, the proportion would be 3/5 or 0.60.

A percentage is another way of expressing a proportion. A percentage is equal to theproportion times 100. In our example of the five women, the percent of the total who

weigh less than 135 pounds would be 100 * (3/5) or 60 percent.

Notation

Of the various measures, the mean and the proportion are most important. The notation used todescribe these measures appears below:

X: Refers to a population mean. x: Refers to a sample mean. P: The proportion of elements in the population that has a particular attribute. p: The proportion of elements in the sample that has a particular attribute. Q: The proportion of elements in the population that does not have a specified attribute.

Note that Q = 1 - P.

q: The proportion of elements in the sample that does not have a specified attribute. Notethat q = 1 - p.

5. Define population and sampling unit for selecting a random sample in each of the

following cases.

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a) Hundred voters from a constituency

b) Twenty stocks of National Stock Exchange

c) Fifty account holders of State Bank of India

d) Twenty employees of Tata motors.

Answer:

Statistical survey or enquiries deal with studying various characteristics of unit belonging to a

group. The group consisting of all the units is called Universe or Population

Sample is a finite subset of a population. A sample is drawn from a population to estimate the

characteristics of the population. Sampling is a tool which enables us to draw conclusions aboutthe characteristics of the population.

In sampling there are two types namely discrete and the other is the continuous. Discrete

sampling is that the data given are of the finite and their calculations are made easy. Continuous

sampling is one where the data are of infinite form. Its intervals are indicated by , greaterthan but lesser than, lesser than and greater than..

The finite number of items in a sample is size. In practice samples greater than 30 are large

samples and if less it is small samples.

A measure associated with the entire population is called as population parameter. Or just an

parameter.

Given a population, suppose we consider all possible samples of a certain size N that can bedrawn from the population. For each sample suppose we compute a statistic such as mean,

standard deviation etc. These sample vary from sample to sample. We group these different

statistics according to their frequencies which is called as frequency distribution to formed so

called as sampling distribution., standard deviation of a sampling distribution is called itsstandard error.

Suppose we draw all possible samples of a certain size N from a population and find the mean ofX bar of each of these samples. Frequency distribution of these means is called as sampling

distribution of means. If the population is infinite, then , , be the standard deviation and

mean respectively then the standard deviation denoted by is given by

= / sqrt of N

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is used to calculate the standard normal variate for the population where its size is more than 30.

6. What is a confidence interval and why it is useful? What is a confidence level?

Answer:

Under a given hypothesis H the sampling distribution of a statistic S is a normal distribution with

the mean and the standard deviation then Z = is the standard normal

variate associated with S so that for the distribution of z the mean is zero and the standard

deviation is 1. Accordingly for z the Z% confidence level is (-z c, zc) this means that we can be

Z% confident that if the hypothesis H is true than the value of z lie between zc and zc. This is

equivalent saying that when H is true there is (100 Z) %chance that the value of z lies outside

the interval (-zc. zc) if we reject a true hypothesis H on the grounds that the value of z liesoutside the interval (-z, zc) we would be making a type 1 error and the probability of making this

error is (100-Z) % the level of significance.

Confidence level is very much useful as we can predict any assumptions can be made so that it

will not lead us to the wrong way even if it doesnt be so great. As explained the confidence level

is betweenzc to z and the peak is at 100% which is the best.

In some cases we predict but do not consider it, and sometimes we will not predict but

hypothesis need it so this is called as the TYPE 1 errors and TYPE 2 errors.

According to the levels of the Z the confidence is assured... in the above the field shaded portion

is the critical region.

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MB0040Statistics for Management.

(Book ID: B1129)

Set2

1. What are the characteristics of a good measure of central tendency?

(b) What are the uses of averages?

Answer:

(a): The characteristics of a good measure of central tendency are:

Present mass data in a concise form: The mass data is condensed to make the data readable and

to use it for further analysis.

Facilitate comparison: It is difficult to compare two different sets of mass data. But we can

compare those two after computing the averages of individual data sets.

While comparing, the same measure of average should be used. It leads to incorrect conclusions

when the mean salary of employees is compared with the median salary of the employees.

Establish relationship between data sets: The average can be used to draw inferences about the

unknown relationships between the data sets. Computing the averages of the data sets is helpful

for estimating the average of population.

Provide basis for decision-making: In many fields, such as business, finance, insurance and

other sectors, managers compute the averages and draw useful inferences or conclusions fortaking effective decisions.

The following are the requisites of a measure of central tendency:

It should be simple to calculate and easy to understand It should be based on all values It should not be affected by extreme values It should not be affected by sampling fluctuation It should be rigidly defined It should be capable of further algebraic treatment

(b): Appropriate Situations for the use of Various Averages

1. Arithmetic mean is used when:

In depth study of the variable is needed1


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The variable is continuous and additive in nature The data are in the interval or ratio scale When the distribution is symmetrical

2. Median is used when:

The variable is discrete There exists abnormal values The distribution is skewed The extreme values are missing The characteristics studied are qualitative The data are on the ordinal scale

3. Mode is used when:

The variable is discrete There exists abnormal values The distribution is skewed The extreme values are missing The characteristics studied are qualitative

4. Geometric mean is used when:

The rate of growth, ratios and percentages are to be studied The variable is of multiplicative nature.

5. Harmonic mean is used when:

The study is related to speed, time Average of rates which produce equal effects has to be found.

2. Your company has launched a new product .Your company is a reputed company with

50% market share of similar range of products. Your competitors also enter with their new

products equivalent to your new product. Based on your earlier experience, you initially

estimated that, your market share of the new product would be 50%. You carry out

random sampling of 25 customers who have purchased the new product ad realize that

only eight of them have actually purchased your product. Plan a hypothesis test to check

whether you are likely to have a half of market share.

Answer:

Null hypothesis H0: P=PsAlternative hypothesis P < Ps

Let the level of significance be 2%

2% = Z tab = 2.05

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Z = |PPs|vPQ/n

P = Population proportion = 50% = 0.5

Ps = Sample Population = 8/25 = 0.32Q = 1-P=10.5 = 0.5

N = 25

vPQ/n

= 8/25*100= 32%.

3. The upper and the lower quartile income of a group of workers are Rs 8 and Rs 3 per

day respectively. Calculate the Quartile deviations and its coefficient?

Answer:

Unlike range, quartile deviation does not involve the extreme values. It is defined as:

Q.D. = (Q3 - Q1)/2

Q.D = (83) /2

= 5 /2

= 2.5

Coefficient of Q.D = Q3 - Q1 / Q3 + Q1

= 8 -2 / 8 + 2

= 6 / 1

4. The cost of living index number on a certain data was 200. From the base period, the

percentage increases in prices wereRent Rs 60, clothing Rs 250, Fuel and Light Rs 150

and Miscellaneous Rs 120. The weights for different groups were food 60, Rent 16, clothing

12, Fuel and Light 8 and Miscellaneous 4.

Answer:

Cost of living Index X =

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GroupGroup

index (P)

Group weight

(w)wP

Food 0 60 60

House Rent 60 16 960

Clothing 250 12 3000

Fuel &Lighting

150 8 1200

Miscellaneous 120 4 480

Total 100 5640

P01 = Wp / w

= 5700 / 100

= 57

5. Education seems to be a difficult field in which to use quality techniques. One possible

outcome measures for colleges is the graduation rate (the percentage of the students

matriculating who graduate on time). Would you recommend using P or R charts to

examine graduation rates at a school? Would this be a good measure of Quality?

Answer:

The four different components of Statistics as per Croxton and Cowden (shown in figure)

can be used to analyse and measure graduation rate in college. As per Croxton and Cowden

analysis we need to use P-Chart.

Basic components of Statistics according to Croxton and Cowden

1. Collection of Data

Careful planning is needed while collecting data. The different methods used for collecting data

such as census method, sampling method and so on. The investigator has to take care while

selecting appropriate collection methods.

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In the census method, every unit or object of the population is included in the investigation. For

example, if we want to study the average annual income of all the families in a given area whichhas 500 families, we must study the income of all 500 families. When the population is large,

census method would be difficult.

A sample of units or objects is taken from the population to describe the overall characteristics ofthe population from which the sample was drawn. This method of collecting data is called

sampling. This method is helpful when size of the population is large or when the results are

needed in short time.

2. Presentation of DataThe collected data is usually presented for further analysis in a tabular, diagrammatic or graphic

form. The collected data is condensed, summarised and visually represented in a tabular or

graphical form.

Tabulation is a systematic arrangement of classified data in rows and columns. For the

representation of data in diagrams, we use different types of diagrams such as one-dimensional,

two-dimensional and three-dimensional diagrams.

Line diagrams, bar diagrams are one-dimensional diagrams. (Refer to figure 1.3 and

figure 1.4 for the illustrations of line diagram and bar diagram respectively)

Pie-charts are the two-dimensional diagrams which are in the form of circle. In pie-chart, total

and component parts are shown in circular shape.

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3. Analysis of DataThe data presented has to be carefully analysed to make any inference from it. The inferences

can be of various types, for example, as measures of central tendencies, dispersion, correlation,

regression.

Measures of central tendency will quantify the middle of the distribution. The measures in case

of population are the parameters and in case of sample, the measures are statistics that areestimates of population parameters. The three most common ways of measuring the centre of

distribution is the mean, mode and median.

In case of population, the measures of dispersion are used to quantify the spread of thedistribution. Range, interquartile range, mean absolute deviation and standard deviation are four

measures to calculate the dispersion.

4. Interpretation of DataThe final step is to draw conclusions from the analysed data. Interpretation requires high degree

of skill and experience. We can interpret the data easily from pie-charts.

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Thus, Statistics contains the tools and techniques required for the collection, presentation,

analysis and interpretation of data. Thus, we see that this definition is precise andcomprehensive.

6. (a) Why do we use a chi-square test?

(b) Why do we use analysis of variance?

Answer:

(a) Why do we use a chi-square test?

A Chi-Square is a non parametric test which can be applied on categorical data or

qualitative data. This test can be applied when we have few or no assumptions about the

population.

Actually, Chi-Square tests allow us to do a lot more than just test for the quality of several

proportions. If we classify a population into several categories with respect to two attributes

(such as age and job performance), we can then use a Chi-Square test to determine whether thetwo attributes are independent of each other. So, Chi-Square tests can be applied on contingency

table.

The c2 test is used broadly to:

Test goodness of fit for one way classification or for one variable only. 7


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Test independence or interaction for more than one row or column in the form of acontingency table concerning several attributes.

Test population variance s2 through confidence intervals suggested by c2 test.(b) Why do we use analysis of variance?

Analysis of variance is useful in such situations as comparing the mileage achieved by

five different brands of gasoline, testing which of four different training methods produce thefastest learning record, or comparing the first-year earnings of the graduates of half a dozen

different business schools. In each of these cases, we would compare the means of more than two

samples. Hence, in most of the fields, such as agriculture, medical, finance, banking, insurance,education, the concept of Analysis Of Variance (ANOVA) is used.

In statistical terms, the difference between two statistical data is known as variance. When two

data are compared for any practical purpose, their difference is studied through the techniques ofANOVA. With the analysis of variance technique, we can test the null hypothesis and the

alternative hypothesis.

Null hypothesis, H0: All sample means are equal.

Alternate Hypothesis, HA: all sample means are not equal or at least one of sample meansdiffer.

Initially the technique was applied in the field of Zoology and Agriculture, but in a later stage, it

was applied to other fields also. In analysis of variance, the degree of variance between two ormore data as well as the factors contributing towards the variance is studied.

In fact, Analysis of Variance is the classification and cross-classification of statistical data withthe view of testing whether the means of specific classification differ significantly or whetherthey are homogeneous.

The Analysis of Variance is a method of splitting the total variation of data into constituent parts

which measure different sources of variations.

The total variation is split up into the following two-components.

Variance within the subgroups of samples Variation between the subgroups of the samples

Hence, the total variance is the sum of variance between the samples and the variance within the

samples. After obtaining the above two variations, these are tested for their significance by F-test

which is also known as variance ratio test.

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Statistice for Management MB0040 B1129 1