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Measures of Dispersion/ Variability Dr Faris Al Lami MB ChB PhD Measures of dispersion & variability • They measure the variability in the values of observations in the set. • They also called measures of variation, spread and scatter. Measures of dispersion & variability • If all values are the same the dispersion is zero. • If the values are homogenous and close to each other the dispersion is small. • If the value are so different the dispersion is large. Measures of dispersion • • • • • Range: Is the difference between the largest and smallest value R=XL- XS R=Range XL= largest value, XS= smallest value Properties of the range: ØSimple to calculate ØEasy to understand ØIt neglect all values in the center and depend on the extreme value, extreme value are dependent on sample size Properties of the range: ØIt is not based on all observations ØIt is not amenable for further mathematic treatment Øshould be used in conjunction with other measures of variability Variance: The mean sum of squares of the deviation from the mean. e.g. if the data is: 1,2,3,4,5. The mean for these data=3 the difference of each value in the set from the mean: 1-3= -2 2-3= -1 3-3= 0 4-3= 1 5-3= 2 • The summation of the differences =zero • Summation of square of the differences is not zero Variance: • Population Variance (sigma squared) 2 2 ∑(X- μ) • α =---------------N 2 α= sigma squared(pop.var) X=observation value μ= population mean N=population size 2 ∑x =summa on of squared 2 (∑ X)=squared of summa on 2 2 2 • α =[ N ∑x – (∑ X) ] / N.N Variance: • Sample Variance _2 2 ∑ (X- X ) • S=---------------OR n-1 2 2 [ n∑X – (∑X) ] s= ---------------------n(n-1) 2 2 • S= sample variance • n= sample size Variance: • Variance can never be a negative value • All observations are considered • The problem with the variance is the squared unit Standard deviation (SD): • It is the square root of the variance • SD=√sigma square= ± sigma(α)---- for population 2 • Sd= √S = ± S----for sample Standard deviation (SD): • The standard deviation measured the variability between observations in the sample or the population from the mean of that sample or that population. • The unit is not squared • SD is the most widely used measure of dispersion Standard Error of the mean(SE) • • • It measures the variability or dispersion of the sample mean from population mean It is used to estimate the population mean, and to estimate differences between populations means SE=SD/√ n Coefficient of variation (CV): • • • • It expresses the SD as a percentage of the mean CV= S /mean X100 (mean of the sample) It has no unit It is used to compare dispersion in two sets of data especially when the units are different Coefficient of variation (CV): • It measures relative rather than absolute variation • It takes in consideration all values in the set EXERCISE • For the same 15 patients in the previous example , calculate measures of dispersion. Pat. no Distance (mile)(X) 2 Pat. no Distance (mile)(X) 2 X 1 2 3 4 5 9 11 3 X 25 81 121 9 5 12 144 13 12 144 6 7 13 12 169 144 14 15 15 5 225 25 8 6 36 Total 141 1575 9 10 11 12 13 7 3 15 169 49 9 225 Range R=XL- XS =15-3 =12 mile Variance & sd 2 2 2 n∑X – (∑X) s= ---------------------n(n-1) 2 =(15)(1575) – (141) / 15 x 14 2 =17.8 mile sd= √17.8 = ± 4.2 mile Standard Error • SE=SD/√ n • =4.2/√15 = 4.2/3.87 = 1.085 mile Coefficient of Variation • CV= S /mean X100 • = 4.2 mile/ 9.4 mile X 100% • =44.7% EXERCISE The following are the hemoglobin values (gm/dl) of 10 children receiving treatment for hemolytic anemia: 9.1,10.0, 11.4, 12.4, 9.8,8.3, 9.9, 9.1, 7.5, 6.7 Compute the sample mean, median, variance, and standard deviation EXERCISE • A sample of 11 patients admitted to a psychiatric ward experienced the following lengths of stay, calculate measures of central tendency and dispersion. No. length No. length 1 29 7 28 2 14 8 14 3 11 9 18 4 24 10 22 5 14 11 14 6 14 total