# BBA Business Statistics Most Important Long Question Answer

**BBA Business Statistics Most Important Long Question Answer :- in this post BBA 2nd semester Business statistics note and question paper, model paper, solve paper, and most important question with answer with example.**

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**Q.1. Define harmonic mean and discuss its merits and limitations. For what special
calculation it is commonly used? **

**Ans. Harmonic Mean :**Harmonic mean of a series is the reciprocal of the arithmetic average of

the reciprocals of the value of its various items.

**Calculation of Reciprocals :** The reciprocal of a number is that value which is obtained by dividing 1 by the value. It means that the reciprocal of 25 will be 1 / 25 and reciprocal of 12 will be 1 / 12 The reciprocal can easily be obtained by the use of reciprocal tables. In this table, the numbers from 1 to 9.9 are given in the first column. After this, there are ten columns ranging from 0 to 9 and in the last there are columns for mean differences ranging from 1 to 9. To consult the table, the first two significant digits should be found in the first column. The third digit is seen in the top horizontal column, while the fourth digit in the column of mean differences. The value of reciprocal is taken from the column of the third digit in the row of first two digits given in the first column. The value of mean difference for the fourth digit is deducted from the reciprocal value. It must be remembered that if the decimal point moves by one-digit to the right in the given number it moves by one digit to the left in the reciprocal.

**Harmonic Mean in Individual Series :**In this series, reciprocal of each value (X) is found out. The sum of reciprocals (E Reci. X) is divided by the number of items (N) and again the reciprocal of this quotient is seen which becomes the value of H.M.

**Harmonic Mean in Discrete Series :**In a discrete series, we first find out the reciprocals of each value and multiply it by the corresponding frequency. Then, the sum of this multiplication (y f Reci. X) is divided by total number of frequencies (N) and now find out the reciprocal of this

quotient.

**Harmonic Mean in Continuous Series :**In a continuous series each class interval is

converted in mid-points and then the procedure is the same as applied to discrete series.

**Weighted Harmonic Mean :** If the relative importance of different values in a series is different,

then weighted harmonic mean is calculated. Its process is as follows :

- Firstly, we take reciprocals of different values.
- Each reciprocal is multiplied by the concerned weight and the total of such multiplication ( Y W Reci. X) is obtained.
- The total obtained in step (2) is divided by the total of weight and the reciprocal of the resultant figure is found out. Thus,

**Merits of Harmonic Mean**

**Rigidly Defined :**M. satisfies the test of rigid definition because its value is always definite.**Accuracy :**It is based on all the observations of the series. So, it satisfies the test of accuracy also.**Less Importance to Big Values**: Harmonic mean gives greater importance to small values and less importance to big values. Hence, some big values cannot influence the answer much.**Less Effect Of Sampling :**Harmonic mean is also not affected very much by fluctuations of sampling.**Algebraic Analysis :**It is capable of further algebraic treatment.**Suitability :**This average is useful in study of time, rate, speed, velocity etc.

**Limitations of Harmonic Mean**

1. Difficult to Compute : Its process of computation is difficult. It is based on reciprocals, which require proper care even if they are seen in reciprocal table.

* 2. All Items Necessary : *Harmonic mean can be computed only when all the terms of the series are known.

* 3. Problem of Representative Value : *Harmonic mean of a series may be a value, which does not exist in a series. In such a case, this average may not work as representative value of the series.

Q. 2. Discuss objects and importance of dispersions What are the characteristics for a

satisfactory measure of dispersion?

Ans. Dispersion : Dispersion literally means scatteredness. It is the spread or scatter of values

from a measure of central tendency. It is studied to have an idea of the homogeneity or heterogeneity the frequency distribution.

Definitions of Dispersion

In the words of* D.C. Brooks and W.F.L. Dick, *“Dispersion or spread is the degree of the scatter or variations of variable about a central value.’

*In the Words of*

*“Dispersion is the measure of the variations of the items.” In the words*

**Bowley,**of Spiegal, “

The degree to which numerical data tend to spread about an average value is called the variation or dispersion of data.” In short, variation in the data is called dispersion and this variation is measured by statistical tools known as measure of dispersion or measures of variations or measure of variability. Measure of dispersion are called averages of the second order because they are based on the deviations of the different values from the mean or other measure of central tendency, which are called averages of the first order. Objects and importance of Dispersion Measure of dispersion are computed to serve the following objectives :

**1. To Judge the Reliability of the Average :** By measure of dispersion we can judge the reliability of the average i.e., we can see that average represent a distribution completely or not. For example, if dispersion in a series is small it means average represents the distribution in a proper way. On the other hand, if dispersion is large, we say that average is not representing the distribution in a proper way. 2. To know the Range of Values :Calculation of dispersion is necessary to know the difference between the highest and the lowest values. 3. Make a Comparative Study of the Variability of two or more dispersion are Useful in comparing two or more series with regard the disparities. A greater degree of variability means lacks of consistency or uniformity of the data. On the other hand, small degree of variation means greater consistency or uniformity. 4. To Control Variability : Another purpose of measuring Variations is to be determine nature cause of variations in order to the variation itself. For example, in matters of health, variations in blood pressure is the basic guide to diagnosis. Prescribed treatment is designed to control the variation. 5. Basis Of Other Statistical Measurement : Measure of dispersion are very useful for finding other measure like correlation, regression, skewness, quality control and Kurtis etc.

### Characteristics for a Satisfactory Measure of Dispersion

The following are the essential requisites for a satisfactory measure of dispersion :

(a) It should be rigidly defined.

(b) It should be based on all observations,

(c) It should be readily comprehensive.

(d) It should be simple to understand and easy to calculate.

(e) It should be capable of further algebraic treatment.

**Q. 3. What is range? Give its coefficient. Also give its merits, demerits and uses.**

* Ans. *Range is the simplest measure of dispersion. For a given set of observations, the range is the difference between the largest and the smallest observation. Thus

Range =R=L-S

Here, L= the largest observation.

S=the smallest observation.

R=the Range.

In case of grouped data, the range is defined as the difference between the upper limit of the highest class and the lower limit of the smallest class.

Coefficient of Range Range is an absolute measure of dispersion which is unsuitable for comparing variation in two

or more distributions expressed in different units. To over come this difficulty, a relative measure of dispersion called the coefficient of range is calculated by the following formula :

Coefficient of Range =Merits of Range

1. It is easy to calculate and provides a broad picture of the scatteredness in data quickly.

2. Its composition is not based on all the observations.

### Demerits of Range

- It is affected by extreme items. A single extreme value, either the maximum or minimum, can affect the value of range.
- Range is a crude measure of dispersion. It does not tell us about the variation in the observations relative to the average.
- It is very much influenced by sampling fluctuations.
- Its composition is not based on all the observations.
- Range cannot be calculated for open-end distributions. In view of these limitations, range is not a good measure of dispersion and should be used With great care and caution.

### Uses of Range

Despite various limitations, the range gives us a quick and simple measure and is useful in the

following cases :

**(a) In Quality Control :** Range of the observations is used for analysing the variations in the

quality of the product produced by a manufacturing concern. Thus, range has an important place in quality control.

**(b) In the Measurement of Fluctuations :** It is a very useful measure to study fluctuations of

observations in certain cases, specially in metereological department, where the variations in the differences between maximum and minimum temperature are used for weather forecasts.

Variations in the prices of share and other commodities can be analysed with the help of range.

**Q. 27. Find the range and coefficient of range from the following data :**

**Q. 5. What is meant by mode? Discuss its relative merits and demerits as measures of**

** central tendency.**

**Ans. Mode :** The word ‘mode’ has its origin in French word ‘La-Mode’ which means fashion or the most. popular phenomenon. In this context it is said that “Mode means most fashionable item.”

### Definition of Mode

From statistical point of view “The mode of a set of observation is that which occurs most often or with the greatest frequency.”

Some of its definitions are as follows :

According to **Kenney and Keeping,** “The value of variable which occurs most frequently in a distribution is called mode.”

In the **words of Croxton and Cowden,** “The mode of a distribution is the value at the pointaround which the items tend to be most heavily concentrated. It may be regarded as the mosttypical value of a series.”

According to **Boddington,** “Mode may be defined as the predominant kind, type or size oritem or the position of greatest tendency.”

Example : If maximum number of shoe sold in a shop is the size number 7, then mode is 7. Similarly, if the maximum number of readymade shirts sold is of size number 34, then mode size is 34.

### Calculation of Mode

The rules and process for the calculation of mode in different series are as follows :

**1. Calculation of Mode in Individual Series :** In an individual series, mode is located generally by observation and the value occurring maximum number of items is the modal value. However, for the convenience of counting data may be placed in an array or may be converted into a discrete

**2. Calculation of Mode in Discrete Series :**

**(a) Inspection Method :** If there is regularity and homogeneity in the series, mode can be

located by inspection of the series. The size or value having the highest frequency will be identified as mode.

**(b) Grouping Method :** Under following circumstances mode is located by grouping method :

I. When the maximum frequency is repeated or approximately equal concentration is found in two or more neigh bouring values.

II. When the maximum frequency occurs either in the very beginning or at the end of the distribution.

III. When there are irregularities in the distribution i.e., the frequencies of the variable increase or decrease in a haphazard way.

In grouping method six columns are drawn in addition to the column of value (X) and frequencies are grouped in the following order .

• **First column :** The frequencies given in the question are shown in this column.

• **Second column :** In this column, frequencies are grouped into twos, starting from the top.

• **Third column :** In this column, frequencies are again grouped into twos, but the first frequency is left out i.e., in this column grouping starts from the second frequency.

• **Fourth column :** In this column, frequencies are grouped into threes, starting from the top.

• **Fifth column :** In this column, frequencies are grouped into threes but this grouping starts from the second frequency.

• **Sixth column :** In this column, frequencies are grouped again in threes but this grouping starts from the third frequency.

After preparing the grouping table, tallies are marked against the values having highest frequency in first column and highest total in each of the other column. Finally, the value securing maximum tallies will be modal value.

**3. Calculation of Mode in Continuous Series :** In continuous series it should be checked before the calculation of mode that each class interval should be equal. If they are not equal, they should be equalised. After it there are two steps in the computation of mode :

**(a) Determination of modal value :** It can be found out by observation or in case of irregular

distribution by grouping method.

**(b) Calculation of modal value :** It is calculated by applying the following formula in modal class :

**Z=L _{1}+**

**Z=Value of the mode**

**L _{1}=Lower limit of the modal group**

**F _{1}=Frequency of the modal class**

**F _{0}=Frequency of the pre-modal crass**

**F _{2}=Frequency of the post-modal class**

**i=Class interval **

**Note:** It is important that the value of mode would always exist within the class limits of modal class. If the answer of mode is beyond these limits, the following alternative formula should be used :

**Z=L _{1}+**

Empirical Mode

Mode may also be calculated on the basis of its relationship with mean and median by applying the following formula :

**Mode = 3 Median—2 Mean or X = 3M —2 X**

Such mode is called as ‘Empirical mode’.

Some Specific Problems Relating to Mode

Some specific problems arising in the computation of mode and the examples based on these problems are as follows :

1. **No Mode or More than One Mode in Individual Series :** There may be a situation in an individual series when mode is ill-defined. It happens

a. when there is no repetition of any value or b. when there is equal repetition of two or more values. In such cases, determination of modal value has been clarified.

**2. Unequal Intervals :** If in a continuous series class intervals are not equal, then we must make them equal before we start the computation of mode. This process may be either by combining or by dividing the class intervals, as the case may be. It is also important that frequencies are also adjusted according to the adjustments in class intervals.

**3. Inclusive Series :** If mode is to be calculated in inclusive series, it will have to be converted into exclusive type classes.

**4. Series Based on Mid-values :** In such a case, mid-values will be converted into class intervals and then mode will be computed.

**5. Series in a Descending Order :** If class intervals are in descending order, there are two options for solving such questions :

a. Class intervals and corresponding frequencies are arranged in ascending order and the normal formula is applied. b. The question is solved in descending order and the changed formula, as given below, is applied :

**6. Determination of Unknown Frequency :** If in a continuous series modal value is known but the frequency of modal class or class preceding or succeeding to it is missing the unknown frequency can be determined as explained below :

**7. All techniques in a Continuous Series :** In computation of mode in a continuous series following problems may arise :

a. Intervals are unequal and they have to be equalised.

b. Class intervals are inclusive and they have to be converted into exclusive type of intervals.

c. Modal class is not clear by inspection and grouping is required.

d. Modal class is not clear by grouping also and density method is required.

e. The answer from the original formula crosses the limits of modal class and alternate formula is required. A problem including all these techniques has been solved**. **

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