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# MBA 1st Year Business Statistics Unit 1 Short Question Answer Study Notes

MBA  1st Year Business Statistics Unit 1 Short Question Answer Study Notes Study Material Unit wise Chapter Wise Notes Syllabus Scope, function and limitations of statistics, Measures of Central Tendency-Mean, Median, Mode, Percentiles, Quartiles, Measures of Dispersion-Range, Interquartile range, Mean Deviation, Mean absolute deviation, Standard Deviation, Variance, Coefficient of Variation. Measures of Shape and Relative location; Skewness and Kurtosis; Chebyshev’s Theorem. Topic Wise Syllabus Of The Content Study Notes.

Section B

Q.1. Discuss the application of statistics in managerial decision.(2008-09)

Or Define statistics and discuss its application in managerial decision-making.

Or What is statistics? What are the various uses of statistics in the management of an organisation?

(2011-12)

Or Discuss briefly the role of statistics in the successful management of business enterprise.

(2014-15)

Ans. Statistics: Statistics is considered as analysis of figures for forecasting or drawing inferences. Statistical methods are becoming very useful in every sphere of life.

“The citizen today is indulged with white papers, economic surveys and a multitude of reports not from the government but from banks, insurance companies and the individual firms, all of which present and argue from the mass of statistical data.’

-Moore ‘Statistics born of the practical needs of the state is finding increasing applications in everyday life.’

-P.Maslor

Uses and Role of Statistics in Management

Uses and role of statistics in management is described as under:

1. Statistics is a branch of applied mathematics which specialises in data and its analysis is helpful in marketing.

2. Statistics is widely used in economics study and research and is concerned with production

and distribution of wealth, savings and investments.

3. Statistical techniques are proved extremely useful in the study of samples for statistical quality control.

4. Statistics is widely used in entrepreneurship and is necessary for the formulation of policies

5. Statistics is essential in research work and experiments are conducted with the help of

statistical methods to gather and analyse data.

6. Statistics are lifeblood of successful commerce and is indispensable in business and commerce.

7. Statistics is vital in making a sound investment whether it is in buying or selling of stocks and

securities or real estate.

Applications of Statistics in Managerial Decision-making

Applications of statistics in managerial deicision-making are as follows:

1. Product selection and competence strategies in marketing and sales.

2. Product mix and product positioning in production management.

3. Buying policy, material planning and lead time in case of material management.

4. Optimum organisation level, job evaluation in personnel management.

5. Project selection in research and development.

Q.2. Write down the functions of statistics.(2012-13)

Ans. There are many functions of statistics among which the most important are as follows:

1. Condensation: It means to reduce or to condense which is mainly applied at embracing the understanding of huge mass of data providing new observations thus reducing the complexity of data.

2. Comparison: Collected data is compared using the methods of classification and tabulation. As statistics is an aggregate of facts and figures, comparison is possible and helps to understand it.

3. Forecasting: It means to predict or to estimate before hand. It plays a dominant production sales, time series analysis, etc.

4. Estimation: Unknown value of the population parameter is estimated on the basis of observations.

5. Hypothesis Testing: Statistical methods are useful in formulation and hypothesis testing population is characterised on the basis of available information from sample observations,

Q.3. “Statistics are numerical statements of facts in any department of enquiry and placed in relay to each other’. Comment and discuss the characteristics of statistics.

Ans. Statistics is defined by using two concepts by the statisticians:

1. Statistics as a Statistical Data: It refers to collection of numerical data. Statistics are numeri statement of facts in any department of enquiry placed in relation to each other. This gives importan to numerical aspects and provides comparative study of figures.

2. Statistics as Statistical Methods: This is based on the concept that statistics is what it dope what statisticians do. Statistics may be called the science of counting in which data is collected hue making estimates. It is a science of averages.

‘Statistics are numerical statements of facts in any department of enquiry that are related to each other. This does not include the statistical methods.’-A.L. Bowley Statistics are expressed in numbers and all statistical statements are facts but all numerical statements of facts are not statistics.

Characteristics of Statistics

Following are the characteristics of statistics:

1. Statistics are the aggregation of facts. Single or isolated figures are not stated as statistics as they cannot perform some of the tasks of statistics. For example, single death or birth do not form any statistics but when these figures show birth rate or death rate, they become part of statistics.

2. Statistics are expressed in numbers. Qualitative statements do not show accurate interpretations and hence are not statistics.

3. Statistics can be compared with other subjects and statistical data can also be compared to each other. For example, the performance appraisal of two departments can be used to compare the efficiency of the two departments.

4. Statistics are affected by multiplicity of causes and a number of factors. So, only one cause is not responsible to given data. Statistics are collected in a systematic manner for a predetermined purpose. So, statistics are

collected for a purpose and under a plan.

Q.4. Discuss the limitations of statistics. Give its uses also.

Ans. Limitations of Statistics: Statistics is applicable in all sciences where quantitative measurement of phenomenon is possible, but it is not without limitation. Therefore, for the proper application statistics, it is also necessary to know the limitations and misuses of statistics.

The following are the limitations of statistics:

1. Statistics Deals with Aggregates of Items and not with Individual: Statistics is the study of mass data and deal with aggregates or group. In fact over data on an item considered individual does not constitute statistical data.

2. Statistics Deals Only with Quantitative Data: If the study yield qualitative data which cann be meaningfully converted to quantitative data, valid conclusion cannot be drawn from such stue using statistical analysis. Qualitative phenomena like honesty, intelligence, poverty, etc. cannot analysed statistically unless these attributes are assigned suitable quantitative measures.

3. Statistical Laws are True Only on an Average: Law of statistics are not universally applicable as the law of Physics, Chemistry and Mathematics. These may not be true for a particular individual. If it is statistically established that a particular food results in an increase in weight, the statement will be true on an average and may not be true for an individual.

4. Statistics is Only One of the Method of Studying a Phenomenon: Statistical methods do not always provide us best v Statstics Vo possible solution to a given problem. In varying cultural and is what statistics does religious situations, it may fail to reveal and pin point the or statistics is what underlying factors responsible for the variation in a phenomenon under study. Thus, statistical conclusion need to be supplemented by statisticians do. the other variations.

5. Statistics can be Misused and Misinterpreted: The greatest limitation of statistics is that it is likely to be misused. The misuse may arise due to several reasons, e.g. when conclusions are based on incomplete information or are drawn by unskilled investigators.

Inadequate and faulty procedure of data collection and inappropriate comparison may arrive at fallacious conclusion.

Uses of Statistics

Statistics is a numerical statement of facts and is a body of methods for making decisions in the face of uncertainty. It has various uses:

1. Statistics helps in giving a better understanding and correct explanation of natural phenomenon.

2. Statistics helps in assembling appropriate quantitative data.

3. Statistics helps in suitable and efficient planning of statistical investigation in any area of study.

4. Statistics is helpful in drawing valid inferences by calculating consistent parameter regarding population through the model data.

5. Statistics helps in presenting complex data in tabular, diagrammatic and graphical form for its

clear representation.

Q.5. What is meant by measure of central tendency? What are the characteristics of a good measure of central tendency?(2007-08)

Ans. Measures of Central Tendency: A central tendency is a single value which is used to represent an entire set of data. It is basically such type of typical value around which most of other values cluster. It can be said that the tendency of the observations to concentrate around a control point is known as central tendency.

Statistical measures indicate the location or position of a control value to describe the central tendency of the entire data and is called the measure of central tendency. There are a lot of measures of central tendency, some of which are broadly classified as mathematical averages and positional averages.

Characteristics

An ideal measure of central tendency should possess the following characteristics:

1. It should be clearly defined and not admit of misconstruction.

2. It should be representative of the whole group under consideration.

3. It should be precisely expressed in a single figure.

4. It should be chosen from a large amount of data so as to nullify the effect of abnormalities and to avoid undue fluctuations where additional matter is introduced.

5. It should be stable, i.e. it should not materially change if some more units of same group are

included at random.

6. It should be easy to understand specially.

7. It should admit of mathematical treatment.

Q.6. Differentiate between parameter and statistics.

Ans. Parameter and Statistics: A parameter is a numerical value that is equivalent to an entir population. The value of parameter is a fixed number and it doesn’t depend on the sample. It can seen in the case of population mean or mode.

A statistics is a numerical value that states something about a sample. It finds ever increasin application in everyday life and is born of the practical needs of the state to register its population. Th value of statistics can vary from sample to sample and it is said that it is dependent on the sample. The best example is seen in sample mean.

Q.7. What do you understand by dispersion? What is the need of studying dispersion? (2006-07)

Ans. Meaning of Dispersion: The term dispersion refers to the variability in the size of items cates that the size of items in a series is not uniform. The values of various items differ from other. If the variation is substantial, dispersion is said to be considerable and if the variation is little dispersion is insignificant. This is rather a general sense in which this term is used.

however, the term, dispersion not only gives a general impression about the variability of a serie but also a precise measure of this variation. Usually in a precise study of dispersion, the deviations of Size of items from a measure of central tendency, are found out and then these deviations are averaged to give a single figure representing the dispersion of the series.

‘Dispersion is the measure of the variation of the items.’

-A.L. Bowley

‘Dispersion is a measure of the extent to which the individual vary.’

Need of Studying Dispersion/Objectives of Dispersion

Need of studying dispersion is described as under:

1. To Judge the Reliability of Measures of Central Tendency: Measure of dispersion is the only means to test the representative character of an average. If the extent of the scatter is less, it indicates a greater degree of uniformity in the value of items and as such the average may be regarded as representative. On the other hand, if the scatter is large, average is bound to be less reliable since it shows a lower degree of uniformity in the value of observations. It is observed that when dispersion is small, the average is a typical value which closely represents the individual values.

2. To Control the Variability Itself: Measures of dispersion are indispensable in analysing the nature and locating the causes of variation.

‘In matters of health variations in body temperature, pulse, beat and blood pressure are basic guides to diagnosis. Prescribed treatment is designed to control their variation. In industrial production, efficient operation requires control of quality variation, the causes of which are sought through inspection and quality control programmes.’–Spurr and Bonini

In social sciences, the measurement of inequality in the distribution of income and wealth requires the measures of variation.

3. To Compare Two or More Series with Regard to their Variability: When two series are to be compared, due consideration has to be given to their dispersion, i.e. the extent to which the items are spread around their respective averages. Using measures of dispersion, one can find out degree of uniformity or consistency in two or more sets of data. For example, if it is desired to make a comparison between the prices of equity shares of two or more companies over a period of time, a measure of dispersion for each such series of prices would be very helpful.

4. To Facilitate the Use of other Statistical Measure: Many other important statistical techniques like correlation, regression, test of hypothesis, analysis of fluctuations in a time series have roots in the measures of variation of one kind or the other.

Q.8. What do you mean by dispersion? Enlist the important measures of dispersion. (2005-06)

Or Define dispersion. Why standard deviation is most widely used measure of dispersion? (2014-15

Ans. Dispersion and Measures of Dispersion: Also known as spread or variation, dispersion 15 defined as measure of extent to which the items vary from some central value. The degree to wn numerical data tends to spread about value is called measure of dispersion of the data. It measures on the degree but not the direction of the variation. Different sets of data may have the same measure central tendency but differ greatly in terms of variation. A good measure of dispersion should possess the same properties as that of a good measure of central tendency.

Some Important Measures of Dispersion

Some well known measures of dispersion which provide a numerical index of the variability of the given data are:

1. Range: It is the simplest method of studying dispersion which represents the difference between the value of the extremes-the largest value and the smallest value.

R=L-S   [L = Largest value, S = Smallest value]

2. Mean Deviation: It is a more sensitive measure of dispersion that has the advantage of giving equal weight to the deviation of every value from the mean or median.

M.D. – Ef|xi – x| /n

[x = Mean, N = No. of observations]

3. Quartile Deviation: It is easy to calculate and understand and is unreliable if there are gaps in the data around the quartiles. It is superior to the range as a rough measure of dispersion.

Q.D. – Q3 – Q1/2

[Q3 = 3rd quartile,Q, = 1st quartile]

4. Standard Deviation (6): This is the most widely used and important measure of dispersion, where in computing the average deviation, the signs are ignored. S.D. overcomes this problem by squaring the deviations which makes them all positive.

Q.D. = E(x – x)2/2

[X = Mean, N= No. of observations]

Q.9. Distinguish between measures of central tendency and dispersion. Illustrate with the help of examples.(2006-07)

Ans. Measures of central tendency refers to a single value that attempts to describe a set of data by identifying the central position within that set of data. It is a typical value around which other figures congregate. For example, the monthly income of 5 persons is 132, 140, 144, 136 and 148.

Arithmetic mean can be calculated as,

A.M.(x) =  =  = 140

Thus, statistical measures indicate the location or position of a control value to describe the central tendency of the entire data.

Measures of dispersion can be defined as the degree to which observations are clustered around a line of best fit. It gives the variability of data and depicts the spread of data. These measures are independent of units of measurement. The measure of dispersion include various methods like range, mean deviation, standard deviation, etc. and can be explained as under:

For example, the net profit of a business concern in thousands of rupees is given below:

Range R=L-S= 30-10= 20 thousand

Coefficient of range =  =  = a=0.5 or 50%

Q.10. ‘Measures of central tendency, dispersion and skewness are complementary to one another in understanding a frequency distribution.’ Elucidate.

(2005-06)

Ans. Measures of central tendency or averages, represent a whole series and its value always lies in between the minimum and maximum values and generally it is located in the centre or middle of the Frequency distribution containing its major characteristics. Condensation of data is necessary in statistical analysis because a large number of big figures are not only confusing for the mind but are also lifficult to analyse also.

Dispersion or spread is the degree of the scatter or variation of the variable about a central va Dispersion not only gives a general impression about the variability of a series, but also a precise meas of this variation. In a precise study of dispersion, the deviations of size of items from a measure of cent tendency are found out and then these deviations are found out and after that these deviations averaged to give a single figure representing the dispersion of the series.

The empirical relations between various averages/measures of central tendency and measures dispersion hold good only in asymmetrical distribution. A normal curve is a bell shaped frequency cury in which the values on either side of a measure of central tendency are symmetrical. Skewness is opposite of symmetry and its presence tells us that a particular distribution is not symmetrical or in other words it is skewed. Thus, measures of central tendency tell us about the concentration of items around the central value and measures of skewness tell us whether the dispersal of items from average is symmetrical.

Fig. Distribution of data.

Taken together, all the three aspects will give us a more comprehensive idea about the given series.

Skewness = Mean – Mode

Coefficient of skewness =

Thus, from the above discussion we can say that ‘Measures of central tendency, dispersion and skewness are complementary to one another in understanding a frequency distribution.’

Q.11. Define kurtosis.

Ans. Kurtosis: The peakedness of the frequency distribution is another characteristic which might be measured. The arithmetic mean of two or more series may be the same but dispersion of their items may be different. Kurtosis defines whether a distribution is more flat or more peaked than the normal distribution, it studies the concentration of items about the central part of the series. If the item concentrate too much at the centre, the curve becomes leptokurtic and if the concentration in the centre is comparatively little, the curve becomes platykurtic.

The coefficientß, is used as kurtosis. The measures of kurtosis is given as:

Kurtosis = B2 – 3

Where the results are:

1. If kurtosis is zero, the curve is mesokurtic, i.e. normal curve.

2. If kurtosis is a positive value, i.e. B,>3 the curve is

leptokurtic and curve is more peaked.

3. If kurtosis is a negative value, i.e. B, <3, the curve is platykurtic. Kurtosis is not much used so far as sociological studies are concerned. It is mainly used in biological studies.

Q.12. Distinguish between skewness and kurtosis.(2015-16)

Or What is meant by skewness? How is it measured?(2007-08, 11-12)

Or Write the differences between skewness and kurtosis.(2012-13)

Ans. Skewness: The measures of central tendency and variation do not reveal all the characteristics of a given set of data. For example, two distributions may have the same mean and standard deviation but may differ widely in the shape of their distribution. Either the distribution of data is symmetrical or it is not. If the distribution of data is not symmetrical, it is called asymmetrical or skewed. Thus, skewness refers to the lack of symmetry in distribution.

Measuring Skewness: The measure of skewness tells us the direction of dispersion about the centre of the distribution. tstanding the tendency for Skewness refers to the lack of symmetry in distribution.

A simple method of finding the direction of skewness is to points into a certain consider the tails of a frequency polygon.The concept of skewness is clear from the figures showing symmetrical, positively skewed and negatively skewed distributions,

Fig. Distribution of data.

It is clear that data are symmetrical when the spread of the frequencies is same on both sides of the middle point of frequency polygon.

Thus, Mean = Median = Mode

When the distribution is not symmetrical, it is said to be skewed-positively skewed or negatively skewed. In such a situation either,

Mean > Median > Mode or Mean < Median < Mode

Skewness is measured using the formula:

Skewness = (3 * (mean – median))/Standard deviation. Where standard deviation shows how far the numbers spread out from the mean and median.

Kurtosis: Refer to Sec-B, Q.11.

Q.13. Explain variance and coefficient of variation.(2013-14)

Ans. The square of the standard deviation is called variance. Therefore, Variance = 02. It is comparable with standard deviations and greater the standard deviation, greater the variability. For grouped data,

Ef(x – X)

Standard deviation

A frequently used relative measure of variation is the coefficient of variation denoted by C.V. This measure is simply the ratio of the standard deviation to mean expressed as the percentage. So, it is the percentage variation in mean, standard deviation being considered as the total variation in the mean. Thus, it shows relationship between the standard deviation and the arithmetic mean.

Coefficient of variation C.V.= x 100

When the coefficient of variation is less in the data, it is said to be less variable or more consistent.

Thus, coefficient of variation is useful because the standard deviation of data must al understood in context of the mean of the data. Instead, the actual value of the C.V. is independe unit in which the measurement has been taken, so it can be said as a dimensionless num compare the data sets with different units or widely different means, coefficient of variation instead of standard deviation.

Q.14. Write a note on Chebyshev’s theorem.

Ans. Chebyshev’s Theorem: This theorem states that the fraction of any set of numbers within k standard deviations of those numbers of the mean of those numbers is at least

1.

K =

Which must be greater than 1.

Using Chebyshev’s Theorem

Let the percent of the values used in this theorem (at least) will fall between 123 and 179 for a data set with mean of 151 and standard deviation of 14.

1. We subtract 151-123 and get 28, which tells us that 123 is 28 units below the mean.

2. We subtract 179-151 and also get 28, which tells us that 151 is 28 units above the mean.

3. Those two together tell us that the values between 123 and 179 are all within 28 units of the mean. Therefore the ‘Within number’ is 28.

4. So, we find the number of standard deviations, k which the ‘Within number’, 28, amounts to by dividing it by the standard deviation:

K =  = 2

So, now we know that the values between 123 and 179 are all within 28 units of the mean, which is the same as within k= 2 standard deviations of the mean.

Now, since k> 1 we can use Chebyshev’s formula to find the fraction of the data that are within k=2 standard deviations of the mean. Substituting k= 2, we have

So, of the data lie between 123 and 179. And since – = 75% that implies that 75% of the data values are between 123 and 179. –

Q.15. Following is the distribution of marks of 50 students in a class:

Calculate the median.

Sol.                                                     Computation of Median

 Value Frequency c.f. 0-10 4 50 10-20 6 46 20-30 20 40 30-40 10 20 40-50 7 10 50-60 3 3

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