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Revision of All Topics Revision Of All Topics Quantitative Aptitude & Business Statistics Def:Measures of Central Tendency • A single expression representing the whole group,is selected which may convey a fairly adequate idea about the whole group. • This single expression is known as average. Quantitative Aptituide & Business 2 Statistics: Revision Of All Toics Averages are central part of distribution and, therefore ,they are also called measures of central tendency. Quantitative Aptituide & Business 3 Statistics: Revision Of All Toics Types of Measures central tendency: There are five types ,namely 1.Arithmetic Mean (A.M) 2.Median 3.Mode 4.Geometric Mean (G.M) 5.Harmonic Mean (H.M) Quantitative Aptituide & Business 4 Statistics: Revision Of All Toics Arithmetic Mean (A.M) The most commonly used measure of central tendency. When people ask about the “average" of a group of scores, they usually are referring to the mean. Quantitative Aptituide & Business 5 Statistics: Revision Of All Toics • The arithmetic mean is simply dividing the sum of variables by the total number of observations. Quantitative Aptituide & Business 6 Statistics: Revision Of All Toics Arithmetic Mean for Ungrouped data is given by n ∑ xi + + + + X = x1 x2 x3 ...... xn = i=1 n n Quantitative Aptituide & Business 7 Statistics: Revision Of All Toics Arithmetic Mean for Discrete Series n ∑ fi xi f1x1 + f 2 x2 + f 3 x3 +......+ f n xn i=1 X = = n f1 + f2 + f3 + .... + fn ∑ fi i=1 Quantitative Aptituide & Business 8 Statistics: Revision Of All Toics Arithmetic Mean for Continuous Series fd X = A + ∑ ×C N Quantitative Aptituide & Business 9 Statistics: Revision Of All Toics Weighted Arithmetic Mean • The term ‘ weight’ stands for the relative importance of the different items of the series. Weighted Arithmetic Mean refers to the Arithmetic Mean calculated after assigning weights to different values of variable. It is suitable where the relative importance of different items of variable is not same Quantitative Aptituide & Business 10 Statistics: Revision Of All Toics • Weighted Arithmetic Mean is specially useful in problems relating to • 1)Construction of Index numbers. • 2)Standardised birth and death rates Quantitative Aptituide & Business 11 Statistics: Revision Of All Toics • Weighted Arithmetic Mean is given by W.X = ∑ ∑W Quantitative Aptituide & Business 12 Statistics: Revision Of All Toics Mathematical Properties of Arithmetic Mean • 1.The Sum of the deviations of the items from arithmetic mean is always Zero. i.e. ∑(X − X ) = 0 • 2.The sum of squared deviations of the items from arithmetic mean is minimum or the least 2 ∑(X − X ) ≤ 0 Quantitative Aptituide & Business 13 Statistics: Revision Of All Toics • 3.The formula of Arithmetic mean can be extended to compute the combined average of two or more related series Quantitative Aptituide & Business 14 Statistics: Revision Of All Toics • 4.If each of the values of a variable ‘X’ is increased or decreased by some constant C, the arithmetic mean also increased or decreased by C . Quantitative Aptituide & Business 15 Statistics: Revision Of All Toics • Similarly When the value of the variable ‘X’ are multiplied by constant say k,arithmetic mean also multiplied the same quantity k . Quantitative Aptituide & Business 16 Statistics: Revision Of All Toics • When the values of variable are divided by a constant say ‘d’ ,the arithmetic mean also divided by same quantity Quantitative Aptituide & Business 17 Statistics: Revision Of All Toics Median 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14 Median = 5 Median = 5 Quantitative Aptituide & Business 18 Statistics: Revision Of All Toics Median for raw data • When given observation are even • First arrange the items in ascending order then N N + 1 • Median (M)=Average of + Item 2 2 Quantitative Aptituide & Business 19 Statistics: Revision Of All Toics Median for raw data • When given observation are odd • First arrange the items in ascending order then • Median (M)=Size of N + 1 Item = 2 Quantitative Aptituide & Business 20 Statistics: Revision Of All Toics Median for continuous series N − m M = L + 2 × c f Where M= Median; L=Lower limit of the Median Class,m=Cumulative frequency above median class f=Frequency of the median class N=Sum of frequenciesQuantitative Aptituide & Business 21 Statistics: Revision Of All Toics Merits of Median • 1.Median is not affected by extreme values . • 2.It is more suitable average for dealing with qualitative data ie.where ranks are given. • 3.It can be determined by graphically. Quantitative Aptituide & Business 22 Statistics: Revision Of All Toics Limitations of Median 1.It is not based all the items of the series . 2.It is not capable of algebraic treatment .Its formula can not be extended to calculate combined median of two or more related groups. Quantitative Aptituide & Business 23 Statistics: Revision Of All Toics Mode • A measure of central tendency • Value that occurs most often • Not affected by extreme values • Used for either numerical or categorical data • There may be no mode or several modes 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Mode = 9 No Mode Quantitative Aptituide & Business 24 Statistics: Revision Of All Toics • A distribution that consists of only one of each score has n modes. • When there are ties for the most frequent score, the distribution is bimodal if two scores tie or multimodal if more than two scores tie. Quantitative Aptituide & Business 25 Statistics: Revision Of All Toics Mode for Continuous Series f1 − f0 Z = L + ×c 2 f1 − f0 − f2 Where Z= Mode ;L=Lower limit of the Mode Class f0 =frequency of the pre modal class f1=frequency of the modal class f2=frequency of the post modal class C=Class interval of Modal Class Quantitative Aptituide & Business 26 Statistics: Revision Of All Toics Relationship between Mean, Median and Mode • The distance between Mean and Median is about one third of distance between the mean and the mode. Quantitative Aptituide & Business 27 Statistics: Revision Of All Toics Karl Pearson has expressed the relationship as follows. Mean –Mode=(Mean-Median)/3 Mean-Median=3(Mean-Mode) Mode =3Median-2Mean Mean=(3Median-Mode)/2 Quantitative Aptituide & Business 28 Statistics: Revision Of All Toics Limitations of Mode • 1.In case of bimodal /multi modal series ,mode cannot be determined. • 2.It is not capable for further algebraic treatment, combined mode of two or more series cannot be determined. Quantitative Aptituide & Business 29 Statistics: Revision Of All Toics Geometric mean n xG = x1x2 xi xn 1/ n n = ∏ xi i=1 Quantitative Aptituide & Business 30 Statistics: Revision Of All Toics Computation of G.M - Discrete Series • Take the logarithms of each item of variable and multiply with the respective frequencies obtain their total i.e ∑ f .log X • Calculate G M as follows f .log X ∑ G.M = Anti log N Quantitative Aptituide & Business 31 Statistics: Revision Of All Toics Weighted Geometric Mean w.log X = ∑ G.M Anti log ∑ w Quantitative Aptituide & Business 32 Statistics: Revision Of All Toics • It is useful for averaging ratios and percentages rates are increase or decrease Quantitative Aptituide & Business 33 Statistics: Revision Of All Toics Harmonic Mean (H.M) • Harmonic Mean of various items of a series is the reciprocal of the arithmetic mean of their reciprocal .Symbolically, N H.M = 1 1 1 1 + + + ....... + X 1 X 2 X 3 X n Quantitative Aptituide & Business 34 Statistics: Revision Of All Toics • 4.It is useful for averaging measuring the time ,Speed etc Quantitative Aptituide & Business 35 Statistics: Revision Of All Toics Quartiles • N + 1 Q = Size th Item 1 4 3(N +1) Q = Size th Item 3 4 Quantitative Aptituide & Business 36 Statistics: Revision Of All Toics Octiles • j(N +1) O j = Size thItem 8 4(N +1) O = Size th Item 4 8 Quantitative Aptituide & Business 37 Statistics: Revision Of All Toics Deciles • j(N +1) D = Size th j 10 Item 5(N +1) D = Size th 5 10 Item Quantitative Aptituide & Business 38 Statistics: Revision Of All Toics Relation Ship Between Partition Values 1.Q1=O2=P25 value of variate which exactly 25% of the total number of observations 2.Q2=D5=P50,value of variate which exactly 50% of the total number of observations. 3. Q3=O6=P75,value of variate which exactly 75% of the total number of observations Quantitative Aptituide & Business 39 Statistics: Revision Of All Toics Measures of Dispersion Why Study Dispersion? • An average, such as the mean or the median only locates the centre of the data. • An average does not tell us anything about the spread of the data. Quantitative Aptituide & Business 41 Statistics: Revision Of All Toics Properties of Good Measure of Dispersion • Simple to understand and easy to calculate • Rigidly defined • Based on all items • A meanable to algebraic treatment • Sampling stability • Not unduly affected by Extreme items. Quantitative Aptituide & Business 42 Statistics: Revision Of All Toics Absolute Measure of Dispersion Based on selected Based on all items items 1.Range 1.Mean Deviation 2.Inter Quartile Range 2.Standard Deviation Quantitative Aptituide & Business 43 Statistics: Revision Of All Toics Relative measures of Dispersion Based on Based on Selected all items items 1.Coefficient of 1.Coefficient of MD Range 2.Coefficient of SD & Coefficient of Variation 2.CoefficientQuantitative of Aptituide & Business 44 QD Statistics: Revision Of All Toics The Range • The simplest measure of dispersion is the range. • For ungrouped data, the range is the difference between the highest and lowest values in a set of data. Quantitative Aptituide & Business 45 Statistics: Revision Of All Toics • RANGE = Highest Value - Lowest Value Quantitative Aptituide & Business 46 Statistics: Revision Of All Toics Coefficient of Range • Coefficient of Range = L − S L + S Quantitative Aptituide & Business 47 Statistics: Revision Of All Toics Interquartile Range • The interquartile range is used to overcome the problem of outlying observations.
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