UNIT I - CENTRAL TENDENCY - Unit I Measures of Central Tendency Dispersion Content Measures of Central Tendency Arithmetic Mean Median Mode Comparison

# UNIT I - CENTRAL TENDENCY - Unit I Measures of Central...

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Unit I: Measures of Central Tendency & Dispersion Content: Measures of Central Tendency: Arithmetic Mean, Median, Mode, Comparison of Mean, Median and Mode. Measures Dispersion: Range, Quartile Deviation, Mean Deviation, Standard Deviation, Relative Dispersion: Coefficient of Variance. Average A 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 Average 1. It should be rigidly defined 2. It should be based on all the observations of the series 3. It should be capable of further algebraic treatment 4. It should be easy to calculate and simple to follow 5. It should not be affected by fluctuations of sampling Types of Averages Mathematical Averages 1. Arithmetic Average or Mean 2. Geometric Mean 3. Harmonic Mean Average of Position 1. Median 2. Mode Arithmetic 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 observations Direct Method
´ X = 1 N ( X 1 + X 2 + X 3 + … X n ) X = 1 N ∑ X X = ∑ X N Where, X = Arithmetic Average, X = Values of the variable, ∑ = Summation or Total, N = number of items. Example 1 Calculate the Simple Arithmetic Average of the following items Size of the item 20 50 72 28 53 74 34 54 75 39 59 78 42 64 79 Solution: Calculation of Arithmetic Mean by Direct Method Size of Items (X) 20 28 34 39 42 50 53 54 59 64 72 74 75 78 79
∑X = 821 Arithmetic Average, X = ∑ X N X = 821 15 = 54.73 Calculation of Arithmetic Mean by Short Cut Method X = A + ∑dX N X = Arithmetic Average, A= Assumed Arithmetic Average, X = Values of the variable, ∑ = Summation or Total, N = number of items. Size of Items Deviation from an Assumed Mean X dX = X-50 20 -30 28 -22 34 -16 39 -11 42 -8 50 0 53 3 54 4 59 9 64 14 72 22 74 24 75 25 78 28 79 29 N=15 ∑dX=+71 X = A + ∑dX N X = 50 + 71 15 X = 50+ 4.73 = 54.73
Calculation of Arithmetic Average in a discrete Series by Direct Method If f1, f2, f3 etc. stand respectively for the frequencies of the values X 1 , X 2 , X 3 etc., ´ X = 1 N ( f 1 X 1 + f 2 X 2 + f 3 X 3 + … f n X n ) OR X = ∑ fX N = ∑ fX ∑ f Example 2 The following table gives the number of children born per family in 735 families. Calculate the average number of children born per family. Number of Children Born per Family Number of Families Number of Children Born per Family Number of Families 0 96 7 20 1 108 8 11 2 154 9 6 3 126 10 5 4 95 11 5 5 62 12 1 6 45 13 1 Solution Number of Children Born per Family Number of Families (X) (f) (fX) 0 96 0 1 108 108 2 154 308 3 126 378 4 95 380 5 62 310 6 45 270 7 20 140 8 11 88
9 6 54 10 5 50 11 5 55 12 1 12 13 1 13 ∑f = 735 ∑fX = 2166 Arithmetic Average or X = ∑ fX N = ∑ fX ∑ f X = 2166 735 = 2.9 = 3 Children approximately The average number of children born per family is 3 approximately. Calculation of Arithmetic Average in a discrete Series by Short Cut Method If f1, f2, f3 etc. stand respectively for the frequencies of the values X 1 , X 2 , X 3 etc., X = A + ∑fdX N Where, ∑fdX = the total of the products of the deviations from the assumed average and the respective frequencies of the items.