Notes on Types of Measures of dispersion,Range, Quartile Deviation, Mean Deviation | Grade 12 > Mathematics > Dispersion, correlation and regression | KULLABS.COM

Types of Measures of dispersion,Range, Quartile Deviation, Mean Deviation

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Types of Measures of Dispersion

Absolute Measure: Those measures of dispersion whose units are same as the units of the given series is known as the absolute measure of dispersion. These types of dispersions can be used only in the comparing the variability of the series or distribution having the same units. Comparison of two distributions with different units cannot be made with absolute measures.

Relative measures: On the other hand, the relative measures of dispersions are obtained as the ratio of the absolute measure of dispersion to the suitable average and are thus a pure number independent of units. Hence two distribution with different units can be compared with the help of relative measures of dispersion.

Methods of measuring dispersion:

The following are the methods of measuring dispersion:

  • Range
  • Quartile deviation or Semi-interquartile deviation
  • Mean deviation or average deviation
  • Standard deviation

Range

The simplest method of studying the variation in the distribution is the range. The range is defined as the difference between the largest item and the smallest item in the set of observations. So, in a set of observations if L is the largest item and S is the smallest item, then range is given by

Range = L – S

In a grouped frequency distribution, range is the difference between the upper limit of the largest class and lower limit of the smallest class.

The range is the absolute measure of dispersion.It cannot be used to compare two distributions with different units. So, the relative measures corresponding to the range known as the coefficient of range is defined by

$$coefficient\;of\;range\;=\frac{L-S}{L+S}$$

Merits and demerits of range

Merits:

  • It is rigidly defined.
  • Range is simple to understand and easy to calculate.
  • Only minimum time is required to know the variability with the help of range.

Demerits

  • It is not based on all observations.
  • Range is affected by the fluctuation of sampling.
  • Range is affected by extreme values.
  • Range cannot be calculated in case of open classes.
  • Range is not suitable for further mathematical treatment.

Semi-Interquartile range or Quartile deviation

The measure of dispersion depending upon the lower and upper quartiles is known as the quartile deviation. The difference between the upper and lower quartile is known as the Interquartile range. Half the interquartile range is known as Semi-interquartile range or quartile deviation.

$$\therefore\;Quartile\;deviation=\frac{Q_3-Q_1}{2}$$

The relative measure based on the lower and upper quartiles known as coefficient of quartile deviation is given by $$Coeffi,\;of\;Q.D\;=\frac{Q_3-Q_1}{Q_3+Q_1}$$

The variability of the items will be less or greater according to the value of the quartile deviation is less or greater. If the quartile deviation is small then the variability is less or the uniformity is great. In the same way, If the quartile deviation is greater then the variability is greater or the uniformity is less.

Merits and Demerits of Q.D.

Merits

  • It is rigidly defined.
  • It is simple to understand and easy to calculate.
  • It is the better measure of dispersion in comparison to range as it is based on 50% of central items.
  • It is not affected by extreme values.
  • It can be calculated even when end classes are open.

Demerits

  • It is not based on all observations.
  • QD. is affected by the fluctuation of sampling.
  • Q.D. is not suitable for further mathematical treatment.

Mean Deviation (Average Deviation)

Mean deviation is defined as the arithmetic mean of the deviations of the items from mean, median and mode when all deviations are considered positive.

$$IF\;\overline{x},\;M_d\;M_o\;be\;the\;arithmetic\;mean,\;median\;and\;mode\;of\;the\;set\;of\;variate\;values\;x,$$

$$\;then\;the\;mean\;deviation\;(M.D)\;are\;computed\;by\;the\;following\;formulae:\;$$

$$M.D.\;from\;mean\;=\frac{\Sigma\;/x-\overline{x}/}{n}=\frac{\Sigma\;/d/}{n}$$

$$Also\;\;\;M.D.\;from\;mean\;=\frac{\Sigma\;f/x-\overline{x}/}{N}=\frac{\Sigma\;f/d/}{N}$$

$$where\;/d/=/x-\overline{x}/\;and\;read\;as\;modulus\;óf\;d\;or\;x-\overline{x},\;n=number\;of\;observations\;and\;N=total\;frequency.$$

M.D. from median and mode can similarly be obtained by replacing mean by M­d­ and M­0­ ­respectively.

The relative measure of mean deviation is defined as follows:

$$coeff.\;of\;M.D.\;from\;mean\;=\frac{M.D.\;from\;mean}{mean}$$

$$coeff.\;of\;M.D.\;from\;median\;=\frac{M.D.\;from\;median}{median}$$

Taken reference from

( Basic mathematics Grade XII and A foundation of Mathematics Volume II and Wikipedia.com )



  • Absolute Measure: Those measures of dispersion whose units are same as the units of the given series is known as the absolute measure of dispersion. These types of dispersions can be used only in the comparing the variability of the series or distribution having the same units. Comparison of two distributions with different units cannot be made with absolute measures.

     

  • Relative measures: On the other hand, the relative measures of dispersions are obtained as the ratio of the absolute measure of dispersion to the suitable average and are thus a pure number independent of units. Hence two distribution with different units can be compared with the help of relative measures of dispersion.

  • The simplest method of studying the variation in the distribution is the range. The range is defined as the difference between the largest item and the smallest item in the set of observations. So, in a set of observations if L is the largest item and S is the smallest item, then range is given by

    Range = L – S

  • The measure of dispersion depending upon the lower and upper quartiles is known as the quartile deviation. The difference between the upper and lower quartile is known as the Interquartile range. Half the interquartile range is known as Semi-interquartile range or quartile deviation.

    $$\therefore\;Quartile\;deviation=\frac{Q_3-Q_1}{2}$$

    The relative measure based on the lower and upper quartiles known as coefficient of quartile deviation is given by $$Coeffi,\;of\;Q.D\;=\frac{Q_3-Q_1}{Q_3+Q_1}$$

  • Mean deviation is defined as the arithmetic mean of the deviations of the items from mean, median and mode when all deviations are considered positive.

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