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Measures of Central Tendency Measures Of Central Tendency Quantitative Aptitude & Business Statistics Statistics in Plural Sense as Statistical data. Statistics in Plural Sense refers to numerical data of any phenomena placed in relation to each other. For example ,numerical data relating to population ,production, price level, national income, crimes, literacy ,unemployment ,houses etc., Statistical in Singular Scene as Statistical method. Quantitative aptitude & Business 2 Statistics: Measures Of Central According to Prof.Horace Secrist: “By Statistics we mean aggregate of facts affected to marked extend by multiplicity of causes numerically expressed, enumerated or estimated according to reasonable standard of accuracy ,collected in a systematic manner for a pre determined purpose and placed in relation to each other .” Quantitative aptitude & Business 3 Statistics: Measures Of Central Measures of Central Tendency Quantitative aptitude & Business 4 Statistics: Measures Of Central 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 aptitude & Business 5 Statistics: Measures Of Central Averages are central part of distribution and, therefore ,they are also called measures of central tendency. Quantitative aptitude & Business 6 Statistics: Measures Of Central 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 aptitude & Business 7 Statistics: Measures Of Central Features of a good average 1.It should be rigidly defined 2.It should be easy to understand and easy to calculate 3.It should be based on all the observations of the data Quantitative aptitude & Business 8 Statistics: Measures Of Central 4.It should be easily subjected to further mathematical calculations 5.It should be least affected by fluctuations of sampling Quantitative aptitude & Business 9 Statistics: Measures Of Central 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 aptitude & Business 10 Statistics: Measures Of Central The arithmetic mean is simply dividing the sum of variables by the total number of observations. Quantitative aptitude & Business 11 Statistics: Measures Of Central Arithmetic Mean for raw data is given by n ∑ xi + + + + X = x1 x2 x3 ...... xn = i=1 n n Quantitative aptitude & Business 12 Statistics: Measures Of Central Find mean for the data 17,16,21,18,13,16,12 and 11 Quantitative aptitude & Business 13 Statistics: Measures Of Central Arithmetic Mean for Discrete Series n ∑ fi xi f1x1 + f 2 x2 + f3 x3 +......+ f n xn i=1 X = = n f1 + f2 + f3 + .... + fn ∑ fi i=1 Quantitative aptitude & Business 14 Statistics: Measures Of Central Arithmetic Mean for Continuous Series fd X = A + ∑ ×C N Quantitative aptitude & Business 15 Statistics: Measures Of Central Calculation of Arithmetic mean in case of Continuous Series From the following data calculate Arithmetic mean Marks 0- 10- 20- 30- 40- 50- 10 20 30 40 50 60 No. of 10 20 30 50 40 30 Students Quantitative aptitude & Business 16 Statistics: Measures Of Central Marks Mid No.of d= X-45 f.d values Students 10 (X) (f) 0-10 5 10 -4 -40 10-20 15 20 -3 -60 20-30 25 30 -2 -60 30-40 35 50 -1 -50 Quantitative aptitude & Business 17 Statistics: Measures Of Central Marks Mid No.of d= X-45 f.d values Students 10 (X) (f) 40-50 45 40 0 0 50-60 55 30 1 30 N=180 ∑fd=- 180 Quantitative aptitude & Business 18 Statistics: Measures Of Central Solution Let us take assumed mean =45 Calculation from assumed mean − fd −180*10 x = A + ∑ ×C = 45 + N 180 Mean = = 35 Quantitative aptitude & Business 19 Statistics: Measures Of Central Calculation Of Arithmetic Mean in case of Less than series Marks 10 20 30 40 50 60 less than /up to No. of 10 30 60 110 150 180 students Quantitative aptitude & Business 20 Statistics: Measures Of Central Solution: Let us first convert Less than series into continuous series as follows Marks 0-10 10- 20- 30- 40- 50-60 20 30 40 50 No. of 10 20 30 50 40 30 students 180- 150=30 Quantitative aptitude & Business 21 Statistics: Measures Of Central Calculation Of Arithmetic Mean in case of more than series Marks 0 10 20 30 40 50 60 more than No. of 180 170 150 120 70 30 0 students Quantitative aptitude & Business 22 Statistics: Measures Of Central Solution: Let us first convert More than series into continuous series as follows Marks 0-10 10- 20- 30- 40-50 50- 20 30 40 60 No. of 10 20 30 50 40 30 students 180-170=10 170-150=20 30-0=30 70-30=40 Quantitative aptitude & Business 23 Statistics: Measures Of Central Calculation of Arithmetic Mean in case of Inclusive series From the following data ,calculate Arithmetic Mean Marks 1-10 11-20 21- 31- 41- 51- 30 40 50 60 No. of 10 20 30 50 40 30 Students Quantitative aptitude & Business 24 Statistics: Measures Of Central Solution Let us take assumed mean =45.5 Calculation from assumed mean − fd −180*10 x = A + ∑ ×C = 45 + Mean = N 180 = 35 Quantitative aptitude & Business 25 Statistics: Measures Of Central Marks Mid No.of d=X-45.5 f.d values Students 10 0.5-10.5 5.5 10 -4 -40 10.5-20.5 15.5 20 -3 -60 20.5-30.5 25.5 30 -2 -60 30.5-40.5 35.5 50 -1 -50 40.5-50.5 45.5 40 0 0 50.5-60.5 55.5 30 1 30 N=180 ∑fd= -180 Quantitative aptitude & Business 26 Statistics: Measures Of Central Calculation of Arithmetic Mean in case of continuous exclusive series when class intervals are unequal From the following data ,calculate Arithmetic Mean Marks 0-10 10-30 30-40 40-50 50-60 No. of 10 60 50 40 20 Students Quantitative aptitude & Business 27 Statistics: Measures Of Central Since class intervals are unequal, frequencies have been adjusted to make the class intervals equal on the assumption that they are equally distributed throughout the class Let us take assumed mean =45 Quantitative aptitude & Business 28 Statistics: Measures Of Central Calculation of Deviations from assumed mean − fd − 220X10 Mean= x = A + ∑ ×C = 45 + N 180 = 32.778 Quantitative aptitude & Business 29 Statistics: Measures Of Central Marks Mid No. of d= X-45.5 f.d values Students 10 0-10 5 10 -4 -40 10-20 15 30 -3 -90 20-30 25 30 -2 -60 30-40 35 50 -1 -50 40-50 45 40 0 0 50-60 55 20 1 30 N=180 ∑fd=-220 Quantitative aptitude & Business 30 Statistics: Measures Of Central Combined Arithmetic Mean (A.M) An average daily wages of 10 workers in a factory ‘A’ is Rs.30 and an average daily wages of 20 workers in a factory B’ is Rs.15.Find the average daily wages of all the workers of both the factories. Quantitative aptitude & Business 31 Statistics: Measures Of Central Solution Step 1;N1=10 N2=20 X = 30; X = 15 1 2 Step2: N X + N X X = 1 1 2 2 12 N1 + N 2 =20 Quantitative aptitude & Business 32 Statistics: Measures Of Central 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 aptitude & Business 33 Statistics: Measures Of Central Weighted Arithmetic Mean is specially useful in problems relating to 1)Construction of Index numbers. 2)Standardised birth and death rates Quantitative aptitude & Business 34 Statistics: Measures Of Central Weighted Arithmetic Mean is given by ∑W.X ∑ X w = ∑W Quantitative aptitude & Business 35 Statistics: Measures Of Central 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 2 the least ∑(X − X ) ≤ 0 Quantitative aptitude & Business 36 Statistics: Measures Of Central 3.The formula of Arithmetic mean can be extended to compute the combined average of two or more related series Quantitative aptitude & Business 37 Statistics: Measures Of Central 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 aptitude & Business 38 Statistics: Measures Of Central Similarly When the value of the variable ‘X’ are multiplied by constant say k,arithmetic mean also multiplied the same quantity k . Quantitative aptitude & Business 39 Statistics: Measures Of Central When the values of variable are divided by a constant say ‘d’ ,the arithmetic mean also divided by same quantity Quantitative aptitude & Business 40 Statistics: Measures Of Central Merits Of Arithmetic Mean 1.Its easy to understand and easy to calculate. 2.It is based on all the items of the samples. 3.It is rigidly defined by a mathematical formula so that the same answer is derived by every one who computes it. Quantitative aptitude & Business 41 Statistics: Measures Of Central 4.It is capable for further algebraic treatment so that its utility is enhanced Quantitative aptitude & Business 42 Statistics: Measures Of Central 6.The formula of arithmetic mean can be extended to compute the combined average of two or more related series.
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