Unit I: Measures of Central Tendency & DispersionContent: Measures of Central Tendency:Arithmetic Mean, Median, Mode, Comparison of Mean, Median andMode. Measures Dispersion:Range, Quartile Deviation, Mean Deviation, Standard Deviation, RelativeDispersion: Coefficient of Variance.AverageA measure of central tendency is a typical value around which other figures congregate. This single value is a point around which the individual items cluster. Characteristics of a Good Average1.It should be rigidly defined2.It should be based on all the observations of the series3.It should be capable of further algebraic treatment4.It should be easy to calculate and simple to follow5.It should not be affected by fluctuations of samplingTypes of AveragesMathematical Averages1.Arithmetic Average or Mean 2.Geometric Mean3.Harmonic Mean Average of Position1.Median 2.ModeArithmetic Average (Mean)Arithmetic Average or Mean of a series is the figure obtained by dividing the total values of the various items by their number. Calculation of Arithmetic Average in a series of individual observationsDirect Method
´X=1N(X1+X2+X3+… Xn)X=1N∑ X∨X=∑ XNWhere, X= Arithmetic Average, X = Values of the variable, ∑ = Summation or Total, N = number of items. Example 1Calculate the Simple Arithmetic Average of the following itemsSize of the item205072285374345475395978426479Solution: Calculation of Arithmetic Mean by Direct MethodSize of Items (X)202834394250535459647274757879
∑X = 821Arithmetic Average, X=∑ XNX=82115=54.73Calculation of Arithmetic Mean by Short Cut Method X=A+∑dXNX= Arithmetic Average, A= Assumed Arithmetic Average, X = Values of the variable, ∑ = Summation or Total, N = number of items. Size of ItemsDeviation from an Assumed MeanXdX = X-5020-3028-2234-1639-1142-8500533544599641472227424752578287929N=15∑dX=+71 X=A+∑dXNX=50+7115X= 50+ 4.73 = 54.73
Calculation of Arithmetic Average in a discrete Series by Direct MethodIf f1, f2, f3 etc. stand respectively for the frequencies of the values X1, X2, X3etc.,´X=1N(f1X1+f2X2+f3X3+… fnXn)OR X=∑ fXN=∑ fX∑ fExample 2The following table gives the number of children born per family in 735 families. Calculate the average number of children born per family. Number of ChildrenBorn per FamilyNumber ofFamiliesNumber of ChildrenBorn per FamilyNumber ofFamilies09672011088112154963126105495115562121645131SolutionNumber ofChildrenBorn perFamilyNumber ofFamilies(X)(f)(fX)096011081082154308312637849538056231064527072014081188
965410550115551211213113∑f=735∑fX=2166Arithmetic Average or X=∑ fXN=∑ fX∑ fX=2166735=2.9=3Children approximately The average number of children born per family is 3 approximately. Calculation of Arithmetic Average in a discrete Series by Short Cut MethodIf f1, f2, f3 etc. stand respectively for the frequencies of the values X1, X2, X3etc.,X=A+∑fdXNWhere, ∑fdX= the total of the products of the deviations from the assumed average and the respective frequencies of the items.