MEASURES OF DISPERSION

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A.G.Tapashetti, A.S.P.College of Commerce, Bijapur

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A.G.Tapashetti, A.S.P.College of Commerce, Bijapur Dispersion indicates the measure of the extent to which individual items differ. It indicates lack of uniformity in the size of items. “Dispersion or spread is the degree of the scatter or variation of the variables about central value” OR “The degree to which numerical data tend to spread about an average value is called the Variation or dispersion .

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A.G.Tapashetti, A.S.P.College of Commerce, Bijapur Measures of Dispersion serves the following objects: To determine the reliability of an average. To compare the variability of different Distribution. To control the variability. To facilitate the use of other Statistical Techniques

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A.G.Tapashetti, A.S.P.College of Commerce, Bijapur It should be rigidly defined. It should be simple to understand & easy to calculate. It should be based upon all values of given data. It should be capable of further mathematical treatment. It should have sampling stability. It should be not be unduly affected by extreme values. Requisites of a Good Measures of Dispersion:

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A.G.Tapashetti, A.S.P.College of Commerce, Bijapur Absolute & Relative Measures Of Dispersion. Absolute Measures of Dispersion: The measures of dispersion which are expressed in terms of original units of a data are termed as Absolute Measures. Relative Measures of Dispersion: Relative measures of dispersion, are also known as coefficients of dispersion, are obtained as ratios or percentages. These are pure numbers independent of the units of measurement.

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A.G.Tapashetti, A.S.P.College of Commerce, Bijapur Range 2. Quartile Deviation or Semi-inter quartile Range. 3. Mean Deviation. 4. Standard Deviation.

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A.G.Tapashetti, A.S.P.College of Commerce, Bijapur Definition: For Unclassified data: Range is defined as the difference between the largest and the smallest values of the data, Symbolically, R = L – S Where L = Largest value, S = Smallest value, R = Range The relative measure of range is defined as,

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A.G.Tapashetti, A.S.P.College of Commerce, Bijapur Definition: For Classified data: Range is defined as the difference between the upper boundary of last class interval and the lower boundary of first class boundary of the distribution. Symbolically, R = ULI – LFI Where ULI = upper boundary of last class interval, LFI = lower boundary of first class interval, R = Range The relative measure of range is defined as,

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A.G.Tapashetti, A.S.P.College of Commerce, Bijapur 1. Range is rigidly defined. Range is simple to understand and easy to calculate. Range is not based upon all observation of given data. Range is not capable for further mathematical treatment. Range is much affected by extreme values. Range is much affected by sampling variation. Range can not be calculated for open end classes without any assumptions.

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A.G.Tapashetti, A.S.P.College of Commerce, Bijapur What is the range of the following data:4 8 1 6 6 2 9 3 6 9 Soln: The largest score (L) is 9; The smallest score (S) is 1; Range= R =L - S = 9 - 1 = 8. Coefficient of Range = R = = = = 0.8

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A.G.Tapashetti, A.S.P.College of Commerce, Bijapur Definition: Quartile Deviation(Q.D.) is defined as Q.D. = Where Q3 = Upper (Third) quartile, Q1 = Lower (First) quartile The relative measure of quartile deviation is defined as, Coefficient of Q.D. =

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A.G.Tapashetti, A.S.P.College of Commerce, Bijapur MERITS OF QUARTILE DEVIATION : Quartile deviation is rigidly defined. Quartile deviation is simple to understand and easy to calculate. Quartile deviation is not affected by extreme values. Quartile deviation can be calculated for open end classes without assumptions DEMERITS OF QUARTILE DEVIATION : Quartile deviation is not based upon all observations of data. Quartile deviation is not capable of further mathematical treatment. Quartile deviation is much affected by sampling fluctuations

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A.G.Tapashetti, A.S.P.College of Commerce, Bijapur MEAN DEVIATION : Range and Quartile deviation are not based upon all observations. They are positional measures of dispersion. They do not show any scatter of the observations from an average. The mean deviation based upon all the observations.

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A.G.Tapashetti, A.S.P.College of Commerce, Bijapur MEAN DEVIATION : Definition: For Unclassified data: Let x1, x2,…., xn are n observations of given data. If n values x1, x2,…., xn have an Arithmetic mean then are the deviations of values from mean. Mean deviation about mean is defined as follow,

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A.G.Tapashetti, A.S.P.College of Commerce, Bijapur Similarly, If Me is median of given data, Then Men deviation about median is given by, If Mo is the mode of given data. Than Mean Deviation about mode is,

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A.G.Tapashetti, A.S.P.College of Commerce, Bijapur For Classified data: Let the variable X has values x1, x2,…., xn with frequencies f1, f2,…., fn If n values x1, x2,…., xn have an Arithmetic mean  then Mean deviation about mean is defined as follow,

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A.G.Tapashetti, A.S.P.College of Commerce, Bijapur Similarly, If Me is median of given data, Then Men deviation about median is given by, If Mo is the mode of given data. Than Mean Deviation about mode is,

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A.G.Tapashetti, A.S.P.College of Commerce, Bijapur Merits of Mean Deviation 1. Mean deviation is rigidly defined. 2. Mean deviation is simple to understand and easy to calculate. 3. Mean deviation is based upon all observations, Demerits of Mean Deviation The greatest drawback of Mean deviation is that algebraic signs are ignored while taking deviations from items. 2. Mean deviation is not capable of further mathematical treatment. 3. Mean deviation is much affected by sampling variation. 4. Mean deviation is much affected by extreme values. 5. There no hard & fast rule in the selection of particular average, with respect to which the deviation are computed. 6. Mean deviation can not be calculated for open end classes without any assumptions.

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A.G.Tapashetti, A.S.P.College of Commerce, Bijapur Definition: For Classified data: Let variables X has values x1, x2,…., xn with frequencies f1, f2,…., fn . If n values x1, x2,…., xn have an Arithmetic mean Than Standard deviation is given by Where N = Total frequency

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A.G.Tapashetti, A.S.P.College of Commerce, Bijapur STANDERD DEVIATION Definition: For Unclassified data: Let x1, x2,…., xn are n observations of given data. If n values x1, x2,…., xn have an Arithmetic mean Than Standard deviation is given by

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A.G.Tapashetti, A.S.P.College of Commerce, Bijapur Coefficient of variation (C.V.) When this is expressed as percentage, that is multiplied by 100, it is called Coefficient of variation. The coefficient of variation is the ratio of standard deviation to the arithmetic mean expressed as percentage.

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A.G.Tapashetti, A.S.P.College of Commerce, Bijapur Merits of Standard deviation 1. Standard deviation is rigidly defined. 2. Standard deviation is based upon all observations. 3. Standard deviation is capable of further mathematical treatment. 4. Standard deviation is less affected by sampling variations Demerits of Standard deviation: Standard deviation is not simple to understand and not easy to calculate. Standard deviation is much affected by extreme values. Standard deviation can not be calculated for open end classes without any assumptions.

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