Absolute Measure Of Dispersion 1r5c2j

Statistics MCQs on Basic Statistics

Absolute Measure of Dispersion

1. Range

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Absolute measures gives an idea about the amount of dispersion/ spread in a set of observations. These quantities measures the dispersion in the same units as the units of original data. Absolute measures cannot be used to compare the variation of two or more series/ data set. A measure of absolute dispersion does not in itself, tell whether the variation is large or small. Commonly used absolute measures are 2. Quartile Deviation 3. Mean Deviation

4. Variance or Standard Deviation

The details about these absolute measures are: Range

Range is the dierence between the largest value and the smallest value in the data set. For ungrouped data, let X0 is the smallest value and Xn is the largest value in a data set then the range (R) is dened as R = Xn − X0 . For grouped data Range can be calculated in three dierent ways

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R=Mid point of highest class - Mid point of lowest class

R=Upper class limit of highest class-Lower class limit of lower class R=Upper class boundary of highest class - Lower class boundary of lowest class Quartile Deviation (Semi-Interquantile Range)

Quartile deviation dened as the dierence between the third and rst quartiles, and half of this range is called the semiinterquartile range (SIQD) or simply quartile deviation (QD). 1 . QD = Q3 −Q 2 The Quartile Deviation is superior to range as it is not aected 1

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Statistics MCQs on Basic Statistics

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by extremely large or small observations, any how it does not give any information about the position of observation lying outside the two quantities. It is not amenable to mathematical treatment and is greatly aected by sampling variability. Although Quartile Deviation is not widely used as measure of dispersion, but it is used in situations in which extreme observations are thought to be unrepresentative/ misleading. Quartile Deviation is not based on all observation therefore it is aected by extreme observations. The range "Median ± QD" contains approximately 50% of the data. Note:

Mean Deviation (Average Deviation)

The Mean Deviation is dened as the arithmetic mean of the deviations measured either from mean or from the median. All these deviations are counted as positive to avoid the diculty arising from the property that the sum of deviations of observations from their mean is zero. MD = MD =

|X−X| n

P

fP |X−X| f

P

˜ |X−X| n

for ungrouped data for mean for grouped data for mean

for ungrouped data for median

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MD =

P

MD =

P

˜ fP |X−X| f

for grouped data for median

Mean Deviation can be calculated about other central tendencies but it is least when deviations are taken as median. The Mean Deviation gives more information than range or the Quartile Deviation as it is based on all the observed values. The Mean Deviation does not give undue weight to occasional large deviations, so it should likely to be used in situation where such deviation are likely to occur. Variance and Standard Deviation

It is dened as the mean of the squares of deviations of all the observations from their mean. Traditionally for population 2

Muhammad Imdadullah

Statistics MCQs on Basic Statistics variance is denoted by σ 2 (sigma square) and for sample data denoted by S 2 or s2 . Symbolically S2 = σ2 = σ2 =

P (Xi −µ)2 N P (Xi −X)2 n P 2 f (X P i −µ) f P 2 f (X Pi −X) f

Population Variance for ungrouped data sample Variance for ungrouped data

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σ2 =

Population Variance for grouped data Sample Variance for grouped data

The variance is denoted by V ar(X) for random variable X . The term variance was introduced by R. A. Fisher (1890-1982) in 1918. The variance is in square of units and the variance is a large number compared to observation themselves. Note that there are alternative formulas to compute Variance or Standard Deviations. The positive square root of the variance is called Standard Deviation (SD) to express the deviation in the same units as the original observation themselves.It is a measure of the average spread about the mean and symbolically dened as σ2 = 2

σ2 = 2

qP

(Xi −µ)2 N

Population SD for ungrouped data

(Xi −X)2 n

Sample SD for ungrouded data

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§ =

qP

σ =

qP

2 f (X P i −µ) f

qP

2 f (X Pi −X) f

Population SD for grouped data Sample SD for grouped data

Standard Deviation is most useful measure of dispersion is credited with the name Standard Deviation by Karl Pearson (1857-1936). In Psome text Sample Standard Deviation is 2 i −X) dened as S 2 = (Xn−1 on the basis of the argument that knowledge of any n − 1 deviations determines the remaining deviations as the sum of n deviations must be zero. In fact this is an unbiased estimator of the population variance 3

Muhammad Imdadullah

Statistics MCQs on Basic Statistics The Standard Deviation has a denite mathematical measure, it utilizes all the observed values and is amenable to mathematical treatment but aected by extreme values.

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σ2.

4

Muhammad Imdadullah