Course sections

Introduction to Analytics, Lecture 4

Measures of Dispersion, Range, Mean Deviation, Standard Deviation

Measures of Dispersion

Dispersion is the measure of the extent to which the individual observation in a data set varies. It relates to those measures which capture the degree of heterogeneity of a set of statistical observation from a central value. Measuring heterogeneity involves construction of estimators, which pro-vide a standard or a representative value of the scatterings, as a function of all the sample observation. But, the heterogeneity of the data affects the efficiency of the estimator adversely, i.e. greater the dispersion in a data set lesser is the efficiency of the estimator. Therefore, to form an estimator of sufficient efficiency it is necessary to form an idea of the dispersion present in the data. The main classes of the measures of dispersion are:

  • Absolute measure of Dispersion
  • Relative measure of Dispersion

Absolute Measure of Dispersion
Absolute measures of dispersion refer to those measures of dispersion which depend on units of measurement. Hence, if the variability of two or more distributions with the same unit of measurement is to be compared then the absolute measures are helpful. The three main absolute measures of dispersion are:

  • Range
  • Mean Deviation
  • Standard Deviation

Range
The range of a set of statistical observations is defined as the highest and the lowest values in the set. This is the simplest method of measuring dispersion. Range is defined as: Range (X) = Xmax – Xmin where Xmax = Maximum value of the variable X, Xmin = Minimum value of the set X, X is a set containing observations x1, x2 …xn. Range can compare the variability of two or more distributions with the same units of measurement, but to compare the variability of the distribution given in the different units of measurement, the formula of range cannot be used.

Mean Deviation
Mean Deviation is defined as the arithmetic average of the deviations of various items from a measure of central tendency, may be mean, median or mode. Generally, mean deviation is calculated either from mean or median. Mean Deviation can also be calculated about any arbitrary average A.

Standard Deviation
Standard Deviation is considered to be an improvement over the mean deviation, since the former gets rid of signs, by taking instead of the absolute value of the deviation, the squares of the deviation of the variable about A. Standard Deviation is defined as: The positive square-root of the arithmetic mean of these quantities, i.e. it is the root-mean-squared deviation about A. The Standard Deviation is measured about the arithmetic mean of the data set since standard deviation is the least about mean. This is a striking feature of the measure of Standard deviation as a measure of dispersion.

Relative Measure of Dispersion
Relative Measures of Dispersion is defined as: Measures independent of the units of measurement and used for comparing dispersions of two or more distributions given in different units. Some of the most important measures of Relative dispersion are:

  • Co-efficient of Range
  • Co-efficient of Variation
  • Co-efficient of Mean Deviation

Co-efficient of Range
The compare the variability of a distribution with another, where the units of measurements are given in different units, it is not possible to use the absolute measure, range. The relative version of measuring the variability between the distributions is called the Coefficient of Range. Coefficient of Range is the ratio of difference between two extreme observations of their distribution to their sum.

Co-efficient of Variation
The Relative Measures of Dispersion based on Standard Deviation is called the Coefficient of Variation. This is a pure number independent of the units of measurement, and thus, it is suitable for comparing the variability, homogeneity or uniformity of two or more distributions. A distribution with smaller C.V. is said to be more homogeneous or less variable than the other, and a distribution with more C.V. is said to be more heterogeneous.

Co-efficient of Mean Deviation
The Coefficient of Mean Deviation is the relative measure associated with Mean-Deviation. It is de-fined as the ratio of the Mean Deviation and the Average about which it has been calculated.

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