32 Measures of Dispersion – I

1. Introduction

 

In a series, all the items are not equal. There is difference of variation among the values. The degree of variation is evaluated by various measures of dispersion. “The degree to which numerical data tend to spread about an average value is called the

 

“Variation or dispersion of the data”. It means that the deviation of each individual values around the measures of central values. If the value of dispersion is less it implies more reliability of the averages. If the value of dispersion is higher, it will not represent the averages. It is less reliable. Otherwise the measure of central value will not represent the individual values. In this module we are going to discuss range, mean deviation and quartile deviation alone.

 

2. Objectives

 

In this module, the following aspects of measures of dispersion are to be discussed along with percentile as it is directly used to calculate Kelly’s co efficient of skewness.

  1. Percentile
  2. Importance or significance of measures of dispersion
  3. Criteria or characteristics or desirable properties of a measure of dispersion
  4. Absolute and Relative Measures
  5. Range
  6. Quartile Deviation

3.Percentiles What is percentile?

 

P1, P2, P3….. P99  are the ninety nine percentiles. They divide a series into 100 equal parts.

 

Formulae: There are ninety nine percentiles. Instead of considering the formula for each percentile, the method and the formula for kth percentile are considered. The required formula can be obtained from it by substituting k = 1,2,3,….99.

 

In  Individual  observations  and  Discrete  Series,  Pk  is  the  value  of  an  item  at  k(N+1)th 41(N+1)th position. For example, P41 is the value of the item at position when 100  100 all the items are in ascending order. In Continuous Series, the continuous class interval in which kN/100th item is included and identified by considering the class intervals in ascending order. After identifying the percentile class, the following formula is used to calculate median.

   ik( kN −cfk)

Pk = Lk + [   100  ]

fk

 

The subscript k may not be necessary if L, cf, f and I are identified properly for each particular Pk while calculating more than Pk.

4. Importance or Significance of Measures of Dispersion Dispersion is measured for the following purposes:

 

  1. Firstly,the reliability of a measure of central tendency is known through measure of dispersion.
  2. Secondly,measures of Dispersion provide a basis for the control of variability.
  3. Thirdly, they help to compare two or more sets of data with regard to their variability.
  4. Fourthly, they enhance the utility and scope of statistical techniques.
  5. Criteria or desirable properties of a measure of dispersion
  •  It should be rigidly defined
  • It should be based on all the items
  • It should not be unduly affected by extreme items
  • It should lend itself for algebraic manipulation
  • It should be simple to understand and easy to calculate
  • It should have sampling stability.

The above are the properties. Next we are going to discuss the absolute and relative measures of dispersion.

 

There are two kinds of measures of dispersion, viz., absolute measures of dispersion and relative measures of dispersion. Absolute measures indicate the amount of variation in a set of values. They are quoted in terms of the units of observation. For example, when rainfall on different days are available in cm., any absolute measure of dispersion gives the variation in rainfall in cm. Relative measures are used to compare the variation in two or more sets. They are free from the units of measurements of the observations. They are pure numbers. For example, when rainfall on different days are given in cm., a relative measure such as coefficient of variation does not give the variation in cm. Consequently rainfalls in two places, say, one in cm and the other in inch, can be compared using coefficient of variation. Further, the set which has less variation is said to be less variable or more stable or more consistent or more uniform or more homogeneous, etc. The various absolute and relative measures of dispersion are listed below: Absolute measures include range, mean deviation around mean, median, mode, standard deviation and variance. Relative measures of dispersion include co efficient of range, coefficient of quartile deviation, co efficient of mean deviation and co efficient of variation.

 

In this module we are going to discuss range, quartile deviation and mean deviation one by one. First we are going to DISCUSS about range

 

What is range?

 

Range is the difference between the greatest (largest) and the smallest of the values.

 

In symbols, Range = L – S

 

L – Largest Value

 

S – Smallest Value

 

Let us to calculate range for individual series

 

For example Find the value of range and its coefficient for the following data

 

8             10           5             9             12           11

Let us discuss range for discrete series

 

In the discrete series, the frequency of variable is given

 

The following example will clear the calculation of range for discrete series.

 

Example Calculate range and its coefficient from the following distribution.

 

Size 60-62 63-65 66-68 69-71 72-74
Number 5 18 42 27 8

 

Solution:

 

After rewriting the class intervals continuously, the lower boundary of the lowest class, S=59.5 and the upper boundary of the highest class, L=74.5.

10. Uses of Deviation:

 

Mean deviation provides an opportunity to calculate deviation, absolute deviation, total deviation and average of the deviations. Standard deviation is the most important absolute measure of dispersion. Knowledge of the principle of mean deviation facilities understanding the concept of standard deviation. Standard deviation is a part of almost all the theories of Statistics, viz., skewness, kurtosis, correlation, regression, sampling estimation, inference, S.Q.C., etc. It is found to be much useful in forecasting business cycles in a few other statistical activities connected with business, economics and sociology.

 

Merits:

  1. Mean deviations are rigidly defined
  2. They are based on all the items
  3. They are affected less by extreme items than standard deviation.
  4. They are simple to understand and not difficult to calculate.
  5. They do not vary much from sample to sample.
  6. They provide choice. Among the three mean deviations, the one that is suitable to a particular situation can be used.
  7. Formation of different distributions can be compared on the basis of a mean deviation.

Demerits:

 

  1. Omission of negative sign of deviations makes them non-algebraic. It is pointed out as a great drawback.
  2. They could not be manipulated. Combined mean deviation could not be found.

It is not widely used in business or economics.

 

In the above module, we discussed measures of dispersion such as range, quartile deviation and mean deviation along with percentiles. Standard deviation and co efficient of variation are not discussed here. It will be discussed in the other module. Range is used in the assessment of temperature, stock markets etc. Mean deviation and quartile deviation are rarely used in practice. Various measures of dispersion are calculated with examples. It may give some idea to calculate various measures of dispersion. The use of various types of dispersion depends on the purpose of calculation and type of data. Try to calculate measures of dispersion for some more examples from text books which will give you more practice.