Measure of Dispersion | Business Statistics Notes | B.Com Notes Hons & Non Hons | CBCS Pattern

BUSINESS STATISTICS NOTES
B.COM 2ND AND 3RD SEM NEW SYLLABUS (CBCS PATTERN)
 MEASURE OF DISPERSION

Meaning of Dispersion

Average of a given distribution is a single data which represents the entire data. But the average alone cannot adequately describe a set of observations, unless all the observations are the same. It is necessary to describe the variability or dispersion of the observations. In two or more distributions the central value may be the same but still there can be wide disparities in the formation of the distribution. Measure of dispersion help up in studying this important characteristic of a distribution.

In the words of Brooks and Dick,” Dispersion is the degree of the scatter or variation of the variable about a central value.”

In the words Simpson and Kafka,” The measurement of the Scatterness of the mass of figures in a series about an average is called measure of variation or dispersion.”

It is clear from the above discussion is that Dispersion is the measure of variation of items. It measures the extent to which the items vary from central value. Dispersion is also known as average of the second order. Dispersion includes range, mean deviation, quartile deviation, and standard deviation. Mean, Median and Mode are the average of 1st order.

Purpose and Significance of Measure of Dispersion

Measures of Dispersion are needed for the following purposes:

(i)      To determine the reliability of an average. If dispersion is small, the average may me reliable and if dispersion is large data, the average may be unrealiable.

(ii)    To serve as a basis for the control of the variability by determining the nature and cause of variation.

(iii)   To compare two or more series with regard to their variability. A high degree of variation would mean little uniformity whereas a low degree of variation would mean great uniformity.

(iv)  To facilitate the use of other statistical measures such as correlation analysis, the statistical quality control, regression analysis etc.

Types of Measure of Dispersion

Measure of dispersion may be broadly classified into two types:-

a.       Absolute measures of dispersion: It is classified into

(i)    Range  

(ii)   Mean Deviation

(iii)  Standard Deviation

(iv) Quartile Deviation

(v)  The Lorenz Curve

b.      Relative measures of dispersion: It is classified into

(i)    Coefficient of Range      

(ii)   Coefficient of Mean Deviation

(iii)  Coefficient of Variation

(iv) Coefficient of Quartile Deviation.

Difference between absolute and relative measure of dispersion:

1.       Absolute measures are dependent on the unit of the variable under consideration whereas the relative measures of dispersion are unit free.

2.       For comparing two or more distributions, relative measures and not absolute measures of dispersion are considered.

3.       As compared to absolute measures of dispersion, relative measures of dispersion are difficult to compute and comprehend.

Desirable properties of a good measure of dispersion (Variation)

The following are the important properties which a good measure of dispersion should satisfy:

1.       It should be simple to understand and easy to compute.

2.       It should be simple to compute.

3.       It should be based on all the items.

4.       It should not be affected by extreme values.

5.       It should be rigidly defined.

6.       It should be capable of further algebraic treatment.

7.       It should have sampling stability.

Meaning of Range | Merits and Demerits of Range

Range:  Range is defined as the difference between the value of the smallest item and the value of the largest item included in the distribution. It is the simplest method of measuring dispersion. Symbolically,

Range= Largest value (L) – Smallest Value (S)

The relative measure corresponding to range, called the coefficient of range, is obtained by applying the following formula: Coefficient of Range= (L- S)/ (L + S)

Merits of Range:

(i)   It is simple to understand and easy to calculate.

(ii) It is less time consuming.

Demerits of Range:

(i)      It is not based on each and every item of the distribution.

(ii)    It is very much affected by the extreme values.

(iii)   The value of Range is affected more by sampling fluctuations

(iv)  Range cannot be computed in case of open-end distribution.

Meaning of Quartile Deviation (Q.D) OR Semi inter-quartile range | Merits and demerits of QD

The QD is half of the difference between the upper and lower quartiles. Symbolically, QD= ½ (Q3- Q1).

QD is an absolute measure of dispersion. The relative measure corresponding to QD, called the coefficient of QD, is obtained by applying the following formula: Coefficient of QD= (Q3 - Q1) / (Q3 + Q1). Coefficient of QD can be used to compare the degree of variation in different distributions.

Merits of QD:

(i)      It is based on 50% of the observations.

(ii)    It is not affected by the presence of extreme values.

(iii)   In case of open-end distribution, it can be computed.

Demerits of QD:

(i)      It is not based on each and every item of the distribution.

(ii)    It is not capable of further algebraic treatments.

(iii)   The value of Range is affected more by sampling fluctuations

Meaning of Mean Deviation (M.D) | Merits and demerits of MD

M.D: For a given set of observation, MD is defined as the arithmetic mean of the absolute deviation of the observations from an appropriate measure of central tendency. The formula for computing MD is:

MD= ∑│D│/ N

MD is an absolute measure of dispersion. The relative measure corresponding to MD, called the coefficient of MD, is obtained by dividing mean deviation by the particular average used in computing mean deviation. Thus, if MD has been computed from median, the coefficient of mean deviation shall be obtained by dividing MD by median. Coefficient of MD = MD/ (Mean or Median)              

Merits of MD:

(i)      It is simple to understand and easy to compute.

(ii)    It is based on each and every item of the data.

(iii)   MD is less affected by the values of extreme items than the Standard deviation.

Demerits of MD:

(i)      The greatest drawback of this method is that algebraic signs are ignored while taking the deviations of the items.

(ii)    It is not capable of further algebraic treatments.

(iii)   It is much less popular as compared to standard deviation.

Meaning of Standard Deviation (S.D) | Merits and Demerits of S.D.

S.D: The standard deviation commonly denoted by ‘σ’ (Sigma) is the most widely used measure of dispersion. It is the square root of the second moment of dispersion and is calculated from the arithmetic mean. In short, it may be defined as the root-mean-square deviation from the mean.

Merits of SD:

There are various advantages of Standard deviation due to which SD is regarded as the best measure of dispersion. Some of the advantages of standa

(i)      It is based on each and every item of the data and it is rigidly defined.

(ii)    It is capable of further algebraic treatment. Combined SD of two or more groups can be calculated.

(iii)   It is less affected by fluctuations of sampling than most other measures of dispersion.

(iv)  For comparing variability of two or more series, co-efficient of variation is considered as most appropriate and this is based on SD and Mean.

(v)    SD is most prominently used in further statistical work.

Demerits of SD:

(i)      It is not easy to calculate and to understand.

(ii)    It gives more weight to extreme items and less to those which are nearer to mean.

Distinguish between mean deviation and standard deviation

1. Algebraic signs are ignored while calculating mean deviations whereas standard deviation takes into account algebraic sign also.

2. Mean deviation can be computed either from median or mean whereas standard deviation is computed always from arithmetic mean.

Variance and Coefficient of variation

The term variance was used to describe the square of the standard deviation by Fisher in 1913. The concept of variance is highly important in advance work where it is possible to split the total into several parts, each attributable to one of the factors causing variation in their original series. Variance is calculated as: (SD)2.

Coefficient of variation: It is the ratio of Standard deviation to the mean expressed as percentage.  Coefficient of variation can be defined as the coefficient of standard deviation with respect to mean which is generally expressed in terms of percentage. The coefficient of variation is also known as coefficient of variability. Symbolically, Coefficient of Variation (C.V.) = (S.D / Mean)*100.

Purpose of Coefficient of variation:

Coefficient of variation is used to compare the variability of two or more series. A series having greater coefficient of variation is said to have more variable, i.e., less uniform, less stable or less consistent. Again a series, having coefficient of variation lesser is said to be less variable, i.e., more uniform, more stable or more consistent.

Difference between variance and coefficient of variation: Coefficient of variation is the percentage variation in the mean while Variation is the total variation in the mean.

Lorenz Curve

The Lorenz Curve, devised by Max O, Lorenz, a famous economics statistician, is a graphic method of studying dispersion. This curve was used by him for the first time to measure the distribution of wealth and income. Now the curve is also used to study the distribution of profit, wages, turnover, etc. However, still the most common use of this curve is in the study of the degree of inequality in the distribution of income and wealth between countries or between different periods of time. It is a cumulative percentage curve in which the percentage of items is combined with the percentage of other things as wealth, profits, turnover, etc.

Procedure in preparation of Lorenz curve:

While drawing the Lorenz Curve the following procedure is adopted:

1)      The size of items and frequencies are both cumulated and then percentages are obtained for these various cumulative values.

2)      On the X-axis start from 0 to 100 and take the percentage of cumulative frequencies.

3)      On the Y-axis start from 0 to 100 and take percentage of variable.

4)      Draw a diagonal line joining 0 to 100. This is known as line o equal distribution. Any point on this diagonal shows the same percentage of X as on Y.

5)      Plot the various points corresponding to X and Y and join them. The distribution so obtained, unless it is exactly equal, will always curve below the diagonal line. If two curves of distribution are shown on the same Lorenz presentation, the curve that is furthest from the diagonal line represents greater inequality. Clearly the line of actual distribution can never cross the line of equal distribution.

Relationship between Mean and Standard deviation

In a symmetrical distribution, the mean and standard deviation are related as follow:

Mean +1 sd covers 68.27% of the items.

 Mean +2 sd covers 95.45% of the items.

Mean +3 sd covers 99.73% of the items.

Relationship between Measures of Dispersion

In a normal distribution there is a fixed relationship between the three most commonly used measures of dispersion. The QD is smallest, the MD next and the SD is largest in the following percentage:

QD = 2/3sd or sd = 3/2 QD and MD = 4/5sd or sd = 5/4MD

Relationship between Mean and other measures of dispersion

In a normal distribution,

Mean +QD includes 50% of the items.

 Mean + MD includes 57.31% of the items.

Mean + SD includes 68.27% of the items or about 2/3 of items. 

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