Measure of central tendency

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Measure of central tendency

  1. 1. Measures Of Central Tendency Quantitative Aptitude & Business Statistics
  2. 2. Quantitative aptitude & Business Statistics: Measures Of Central 2 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.
  3. 3. Quantitative aptitude & Business Statistics: Measures Of Central 3 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 .”
  4. 4. Quantitative aptitude & Business Statistics: Measures Of Central 4 Measures of Central Tendency
  5. 5. Quantitative aptitude & Business Statistics: Measures Of Central 5 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.
  6. 6. Quantitative aptitude & Business Statistics: Measures Of Central 6 Averages are central part of distribution and, therefore ,they are also called measures of central tendency.
  7. 7. Quantitative aptitude & Business Statistics: Measures Of Central 7 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)
  8. 8. Quantitative aptitude & Business Statistics: Measures Of Central 8 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
  9. 9. Quantitative aptitude & Business Statistics: Measures Of Central 9  4.It should be easily subjected to further mathematical calculations  5.It should be least affected by fluctuations of sampling
  10. 10. Quantitative aptitude & Business Statistics: Measures Of Central 10 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.
  11. 11. Quantitative aptitude & Business Statistics: Measures Of Central 11  The arithmetic mean is simply dividing the sum of variables by the total number of observations.
  12. 12. Quantitative aptitude & Business Statistics: Measures Of Central 12 Arithmetic Mean for raw data is given by n x n X n i i xxxx n ∑=++++ == 1......321
  13. 13. Quantitative aptitude & Business Statistics: Measures Of Central 13 Find mean for the data 17,16,21,18,13,16,12 and 11
  14. 14. Quantitative aptitude & Business Statistics: Measures Of Central 14 Arithmetic Mean for Discrete Series ∑ ∑ = =++++ = ++++ = n i i n i ii n xfxfxfxf f xf ffff X nn 1 1 321 ...... .... 332211
  15. 15. Quantitative aptitude & Business Statistics: Measures Of Central 15 Arithmetic Mean for Continuous Series C N fd AX ×+= ∑
  16. 16. Quantitative aptitude & Business Statistics: Measures Of Central 16 Calculation of Arithmetic mean in case of Continuous Series Marks 0- 10 10- 20 20- 30 30- 40 40- 50 50- 60 No. of Students 10 20 30 50 40 30 From the following data calculate Arithmetic mean
  17. 17. Quantitative aptitude & Business Statistics: Measures Of Central 17 Marks Mid values (X) No.of Students (f) d= X-45 10 f.d 0-10 5 10 -4 -40 10-20 15 20 -3 -60 20-30 25 30 -2 -60 30-40 35 50 -1 -50
  18. 18. Quantitative aptitude & Business Statistics: Measures Of Central 18 Marks Mid values (X) No.of Students (f) d= X-45 10 f.d 40-50 45 40 0 0 50-60 55 30 1 30 N=180 ∑fd=- 180
  19. 19. Quantitative aptitude & Business Statistics: Measures Of Central 19 Solution  Let us take assumed mean =45  Calculation from assumed mean  Mean = 35 180 10*180 45x = − +=×+= − ∑ C N fd A
  20. 20. Quantitative aptitude & Business Statistics: Measures Of Central 20 Calculation Of Arithmetic Mean in case of Less than series Marks less than /up to 10 20 30 40 50 60 No. of students 10 30 60 110 150 180
  21. 21. Quantitative aptitude & Business Statistics: Measures Of Central 21 Solution: Let us first convert Less than series into continuous series as follows Marks 0-10 10- 20 20- 30 30- 40 40- 50 50-60 No. of students 10 20 30 50 40 30 180- 150=30
  22. 22. Quantitative aptitude & Business Statistics: Measures Of Central 22 Calculation Of Arithmetic Mean in case of more than series Marks more than 0 10 20 30 40 50 60 No. of students 180 170 150 120 70 30 0
  23. 23. Quantitative aptitude & Business Statistics: Measures Of Central 23 Solution: Let us first convert More than series into continuous series as follows Marks 0-10 10- 20 20- 30 30- 40 40-50 50- 60 No. of students 10 20 30 50 40 30 180-170=10 170-150=20 70-30=40 30-0=30
  24. 24. Quantitative aptitude & Business Statistics: Measures Of Central 24 Calculation of Arithmetic Mean in case of Inclusive series  From the following data ,calculate Arithmetic Mean Marks 1-10 11-20 21- 30 31- 40 41- 50 51- 60 No. of Students 10 20 30 50 40 30
  25. 25. Quantitative aptitude & Business Statistics: Measures Of Central 25 Solution  Let us take assumed mean =45.5  Calculation from assumed mean  Mean = 35 180 10*180 45x = − +=×+= − ∑ C N fd A
  26. 26. Quantitative aptitude & Business Statistics: Measures Of Central 26 Marks Mid values No.of Students d=X-45.5 10 f.d 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
  27. 27. Quantitative aptitude & Business Statistics: Measures Of Central 27 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 Students 10 60 50 40 20
  28. 28. Quantitative aptitude & Business Statistics: Measures Of Central 28  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
  29. 29. Quantitative aptitude & Business Statistics: Measures Of Central 29  Calculation of Deviations from assumed mean  Mean= 778.32 180 10220 45x = − +=×+= − ∑ X C N fd A
  30. 30. Quantitative aptitude & Business Statistics: Measures Of Central 30 Marks Mid values No. of Students d= X-45.5 10 f.d 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
  31. 31. Quantitative aptitude & Business Statistics: Measures Of Central 31 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.
  32. 32. Quantitative aptitude & Business Statistics: Measures Of Central 32 Solution  Step 1;N1=10 N2=20  Step2:  =20 15;30 21 == XX 21 2211 12 NN XNXN X + + =
  33. 33. Quantitative aptitude & Business Statistics: Measures Of Central 33 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
  34. 34. Quantitative aptitude & Business Statistics: Measures Of Central 34  Weighted Arithmetic Mean is specially useful in problems relating to  1)Construction of Index numbers.  2)Standardised birth and death rates
  35. 35. Quantitative aptitude & Business Statistics: Measures Of Central 35  Weighted Arithmetic Mean is given by ∑ ∑ ∑ = W XW X w .
  36. 36. Quantitative aptitude & Business Statistics: Measures Of Central 36 Mathematical Properties of Arithmetic Mean  1.The Sum of the deviations of the items from arithmetic mean is always Zero. i.e.  2.The sum of squared deviations of the items from arithmetic mean is minimum or the least ( ) 0=−∑ XX ( ) 0 2 ≤−∑ XX
  37. 37. Quantitative aptitude & Business Statistics: Measures Of Central 37  3.The formula of Arithmetic mean can be extended to compute the combined average of two or more related series
  38. 38. Quantitative aptitude & Business Statistics: Measures Of Central 38  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 .
  39. 39. Quantitative aptitude & Business Statistics: Measures Of Central 39  Similarly When the value of the variable ‘X’ are multiplied by constant say k,arithmetic mean also multiplied the same quantity k .
  40. 40. Quantitative aptitude & Business Statistics: Measures Of Central 40  When the values of variable are divided by a constant say ‘d’ ,the arithmetic mean also divided by same quantity
  41. 41. Quantitative aptitude & Business Statistics: Measures Of Central 41 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.
  42. 42. Quantitative aptitude & Business Statistics: Measures Of Central 42  4.It is capable for further algebraic treatment so that its utility is enhanced
  43. 43. Quantitative aptitude & Business Statistics: Measures Of Central 43  6.The formula of arithmetic mean can be extended to compute the combined average of two or more related series.
  44. 44. Quantitative aptitude & Business Statistics: Measures Of Central 44  7.It has sampling stability .It is least affected by sampling fluctuations
  45. 45. Quantitative aptitude & Business Statistics: Measures Of Central 45 Limitations of Arithmetic Mean  1.Affected by extreme values i.e . Very small or very big values in the data unduly affect the value of mean because it is based on all the items of the series.
  46. 46. Quantitative aptitude & Business Statistics: Measures Of Central 46  2.Mean is not useful for studying the qualitative phenomenon.
  47. 47. Quantitative aptitude & Business Statistics: Measures Of Central 47 Median  The middle score of the distribution when all the scores have been ranked.  If there are an even number of scores, the median is the average of the two middle scores.
  48. 48. Quantitative aptitude & Business Statistics: Measures Of Central 48  In an ordered array, the median is the “middle” number If n or N is odd, the median is the middle number If n or N is even, the median is the average of the two middle numbers
  49. 49. Quantitative aptitude & Business Statistics: Measures Of Central 49 Potential Problem with Means Mean Mean Median Median
  50. 50. Quantitative aptitude & Business Statistics: Measures Of Central 50 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
  51. 51. Quantitative aptitude & Business Statistics: Measures Of Central 51 Median for raw data  When given observation are even  First arrange the items in ascending order then  Median (M)=Average of Item 2 1 2 + += NN
  52. 52. Quantitative aptitude & Business Statistics: Measures Of Central 52  Find the Median for the raw data  25,55,5,45,15 and 35  Solution ;Arrange the items  5,15,25,35,45,55,here N=6  Median =Average of 3rd and 4th item=30
  53. 53. Quantitative aptitude & Business Statistics: Measures Of Central 53 Median for raw data  When given observation are odd  First arrange the items in ascending order then  Median (M)=Size of Item 2 1+ = N
  54. 54. Quantitative aptitude & Business Statistics: Measures Of Central 54 Median for continuous series c f m N LM ×             − += 2 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 frequencies
  55. 55. Quantitative aptitude & Business Statistics: Measures Of Central 55 Quartiles  The values of variate that divides the series or the series or the distribution into four equal parts are known as Quartiles .
  56. 56. Quantitative aptitude & Business Statistics: Measures Of Central 56  The first Quartile (Q1),known as a lower Quartile is the value of variate below which 25% of the observations.
  57. 57. Quantitative aptitude & Business Statistics: Measures Of Central 57  The Second Quartile known as middle Quartile(Q2)known as middle Quartile or median ,the value of variates below which 50% of the observations
  58. 58. Quantitative aptitude & Business Statistics: Measures Of Central 58  The Third Quartile known as Upper Quartile(Q3)known as middle Quartile or median ,the value of variates below which 75 % of the observations.
  59. 59. Quantitative aptitude & Business Statistics: Measures Of Central 59  th N SizeQ 4 1 1 + = Item th N SizeQ 4 )1(3 3 + = Item
  60. 60. Quantitative aptitude & Business Statistics: Measures Of Central 60 Octiles  The values of variate that divides the series or the distribution into eight equal parts are known as Octiles .  Each octile contains 12.5% of the total number of observations .
  61. 61. Quantitative aptitude & Business Statistics: Measures Of Central 61  Since seven points are required to divide the data into 8 equal parts ,we have 7 octiles.
  62. 62. Quantitative aptitude & Business Statistics: Measures Of Central 62  th Nj SizeOj 8 )1( + = Item th N SizeO 8 )1(4 4 + = Item
  63. 63. Quantitative aptitude & Business Statistics: Measures Of Central 63 Deciles  The values of variate that divides the series or the distribution into Ten equal parts are known as Deciles .  Each Decile contains 10% of the total number of observations .
  64. 64. Quantitative aptitude & Business Statistics: Measures Of Central 64  Since 9 points are required to divide the data into 10 equal parts ,we have 9 deciles(D1 to D9)
  65. 65. Quantitative aptitude & Business Statistics: Measures Of Central 65  th Nj SizeDj 10 )1( + = Item th N SizeD 10 )1(5 5 + = Item
  66. 66. Quantitative aptitude & Business Statistics: Measures Of Central 66 Percentiles  The values of variate that divides the series or the distribution into hundred equal parts are known as Percentiles .  Each percentile contains 10% of the total number of observations .  Since 99 points are required to divide the data into 10 equal parts ,we have 99 deciles(p1 to p99)
  67. 67. Quantitative aptitude & Business Statistics: Measures Of Central 67  th Nj SizePj 100 )1( + = Item th N Sizep 100 )1(50 50 + = Item
  68. 68. Quantitative aptitude & Business Statistics: Measures Of Central 68 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
  69. 69. Quantitative aptitude & Business Statistics: Measures Of Central 69 Calculation of Median in case of Continuous Series Marks 0-10 10-20 20-30 30-40 40-50 50- 60 No. of Students 10 20 30 50 40 30 From the following data calculate Median
  70. 70. Quantitative aptitude & Business Statistics: Measures Of Central 70 Marks No. of Students (f) Cumulative Frequencies (c.f.) 0-10 10 10 10-20 20 30 20-30 30 60 30-40 50 110 40-50 40 150 50-60 30 180 N=180
  71. 71. Quantitative aptitude & Business Statistics: Measures Of Central 71  Calculate size of N/2 90 2 180 2 == N
  72. 72. Quantitative aptitude & Business Statistics: Measures Of Central 72 10 50 60 2 180 30 ×             − +=M 36630 =+=M
  73. 73. Quantitative aptitude & Business Statistics: Measures Of Central 73 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.
  74. 74. Quantitative aptitude & Business Statistics: Measures Of Central 74 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.
  75. 75. Quantitative aptitude & Business Statistics: Measures Of Central 75 0 X Y M Less than Cumulative curve More than Cumulative Curve Median By Graph Q3Q1 CI Frequency N/2 3N/4 N/4
  76. 76. Quantitative aptitude & Business Statistics: Measures Of Central 76 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 7 8 9 10 11 12 13 14 Mode = 9 0 1 2 3 4 5 6 No Mode
  77. 77. Quantitative aptitude & Business Statistics: Measures Of Central 77 Mode  The most frequent score in the distribution.  A distribution where a single score is most frequent has one mode and is called unimodal.
  78. 78. Quantitative aptitude & Business Statistics: Measures Of Central 78  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.
  79. 79. Quantitative aptitude & Business Statistics: Measures Of Central 79  Calculate the mode from the following data of marks obtained by 10 students.  20,30,31,32,25,25,30,31,30,32  Mode (Z)=30
  80. 80. Quantitative aptitude & Business Statistics: Measures Of Central 80 Mode for Continuous Series c fff ff LZ ×      −− − += 201 01 2 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
  81. 81. Quantitative aptitude & Business Statistics: Measures Of Central 81 Calculation of Mode :Continuous Series Marks 0- 10 10- 20 20- 30 30- 40 40- 50 50- 60 No. of Students 10 20 30 50 40 30 From the following data calculate Mode
  82. 82. Quantitative aptitude & Business Statistics: Measures Of Central 82 Marks No. of Students (f) 0-10 10 10-20 20 20-30 30 30-40 50 f1 40-50 40 50-60 30 N=180 f0 f2
  83. 83. Quantitative aptitude & Business Statistics: Measures Of Central 83 667.36667.630 10 4030502 6050 30 2 201 01 =+= ×      −−× − += ×      −− − += Z c fff ff LZ
  84. 84. Quantitative aptitude & Business Statistics: Measures Of Central 84 x0 Y Z 10 20 30 40 50 60 10 20 30 40 50 Calculation Mode Graphically
  85. 85. Quantitative aptitude & Business Statistics: Measures Of Central 85 Relationship between Mean, Median and Mode  The distance between Mean and Median is about one third of distance between the mean and the mode.
  86. 86. Quantitative aptitude & Business Statistics: Measures Of Central 86 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
  87. 87. Quantitative aptitude & Business Statistics: Measures Of Central 87 Example  For a moderately skewed distribution of marks in statistics for a group of 200 students ,the mean mark and median mark were found to be 55.60 and 52.40.what is the modal mark?
  88. 88. Quantitative aptitude & Business Statistics: Measures Of Central 88 Solution  Since in this case mean=55.60and median =52.40 applying ,we get  Mode=3median -2Mean  =3(52.40)-2(55.60)  Mode =46
  89. 89. Quantitative aptitude & Business Statistics: Measures Of Central 89 Example  If Y=2+1.50X and mode of X is 15 ,What is mode of Y  Solution  Y m=2+1.50*15=24.50
  90. 90. Quantitative aptitude & Business Statistics: Measures Of Central 90 Merits of Mode  1.Mode is the only suitable average e.g. ,modal size of garments, shoes.,etc  2.It is not affected by extreme values.  3.Its value can be determined graphically.
  91. 91. Quantitative aptitude & Business Statistics: Measures Of Central 91 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.
  92. 92. Quantitative aptitude & Business Statistics: Measures Of Central 92  3.It is not based on all the items of the series  4.Its value is significantly affected by the size of the class intervals
  93. 93. Quantitative aptitude & Business Statistics: Measures Of Central 93 Geometric mean nn i i n niG x xxxxx /1 1 21       = = ∏= 
  94. 94. Quantitative aptitude & Business Statistics: Measures Of Central 94  Take the logarithms of each item of variable and obtain their total i.e ∑ log X  Calculate G M as follows         = ∑ n X AntiMG log log.
  95. 95. Quantitative aptitude & Business Statistics: Measures Of Central 95 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         = ∑ N Xf AntiMG log. log.
  96. 96. Quantitative aptitude & Business Statistics: Measures Of Central 96 Merits of Geometric Mean  1.It is based on all items of the series .  2 It is rigidly defined  3.It is capable for algebraic treatment.
  97. 97. Quantitative aptitude & Business Statistics: Measures Of Central 97  4.It is useful for averaging ratios and percentages rates are increase or decrease
  98. 98. Quantitative aptitude & Business Statistics: Measures Of Central 98 Limitations of Geometric Mean  1.Its difficult to understand and calculate.  2.It cannot be computed when there are both negative and positive values in a series
  99. 99. Quantitative aptitude & Business Statistics: Measures Of Central 99  3.It is biased for small values as it gives more weight to small values .
  100. 100. Quantitative aptitude & Business Statistics: Measures Of Central 100 Calculation of G.M :Individual Series  From the following data calculate Geometric Mean Roll No 1 2 3 4 5 6 Marks 5 15 25 35 45 55
  101. 101. Quantitative aptitude & Business Statistics: Measures Of Central 101 Computation of G.M :Individual Series X log X 5 0.6990 15 1.1761 25 1.3979 35 1.5441 45 1.6532 55 1.7404 ∑log X=8.2107
  102. 102. Quantitative aptitude & Business Statistics: Measures Of Central 102 36.23 )3685.1log( 6 2107.8 log log. = =       =         = ∑ Anti Al n X AntiMG
  103. 103. Quantitative aptitude & Business Statistics: Measures Of Central 103  Find the average rate of increase population which in the first decade has increased by 10% ,in the second decade by 20% and third by 30%
  104. 104. Quantitative aptitude & Business Statistics: Measures Of Central 104 Decade % rise Population at the end of the decade logx 1 2 3 10 20 30 110 120 130 2.0414 2.0792 2.1139 ∑log X=6.2345
  105. 105. Quantitative aptitude & Business Statistics: Measures Of Central 105 8.119 )0782.2log( 2345.6 log log. = =       =         = ∑ Anti Al n X AntiMG Average Rate of increase in Population is 19.8%
  106. 106. Quantitative aptitude & Business Statistics: Measures Of Central 106 Weighted Geometric Mean         = ∑ ∑ w Xw AntiMG log. log.
  107. 107. Quantitative aptitude & Business Statistics: Measures Of Central 107 Harmonic Mean (H.M)  Harmonic Mean of various items of a series is the reciprocal of the arithmetic mean of their reciprocal .Symbolically, nXXXX N MH 1 ....... 111 . 321 ++++ =
  108. 108. Quantitative aptitude & Business Statistics: Measures Of Central 108  Where X1,X2,X3…….X n refer to the value of various series.  N= total no. of series
  109. 109. Quantitative aptitude & Business Statistics: Measures Of Central 109 Merits of Harmonic Mean  1.It is based on all items of the series .  2 It is rigidly defined  3.It is capable for algebraic treatment.
  110. 110. Quantitative aptitude & Business Statistics: Measures Of Central 110  4.It is useful for averaging measuring the time ,Speed etc
  111. 111. Quantitative aptitude & Business Statistics: Measures Of Central 111 Limitations of Harmonic Mean  1.Its difficult to understand and calculate.  2.It cannot be computed when one or more items are zero
  112. 112. Quantitative aptitude & Business Statistics: Measures Of Central 112  3.It gives more weight to smallest values . Hence it is not suitable for analyzing economic data .
  113. 113. Quantitative aptitude & Business Statistics: Measures Of Central 113 Calculation of H.M :Individual Series  From the following data calculate Harmonic Mean Roll No 1 2 3 4 5 6 Mark s 5 15 25 35 45 55
  114. 114. Quantitative aptitude & Business Statistics: Measures Of Central 114 Computation of H.M :Individual Series X l/x 5 0.2000 15 0.0666 25 0.0400 35 0.0286 45 0.0222 55 0.0182 ∑(1/x)=0.3756
  115. 115. Quantitative aptitude & Business Statistics: Measures Of Central 115 9744.15 3576.0 6 1 1 = = = ∑ = n i i H x n x
  116. 116. Quantitative aptitude & Business Statistics: Measures Of Central 116  Compute AM ,GM and HM for the numbers 6,8,12,36  AM=(6+81+12++36)/4=15.50  GM=(6.8.12.36)1/4=12  H.M=9.93 36 1 12 1 8 1 6 1 4 . +++ =MH
  117. 117. Quantitative aptitude & Business Statistics: Measures Of Central 117 Weighted Harmonic Mean ∑ ∑= )( i i i X w w HM
  118. 118. Quantitative aptitude & Business Statistics: Measures Of Central 118  Find the weighted AM and HM of first n natural numbers ,the weights being equal to the squares of the Corresponding numbers. X 1 2 3 …n W 12 22 32 ..n2
  119. 119. Quantitative aptitude & Business Statistics: Measures Of Central 119  Weighted ∑ ∑= Wi XiWi AM . )12(2 )1(3 + + = n nn       ++       + = ++++ ++++ 6 )12)(1( 4 )1( .....321 .....321 22 2222 3333 nnn nn n n
  120. 120. Quantitative aptitude & Business Statistics: Measures Of Central 120 ∑ ∑= )( i i i X w w HM 3 12 2 )1( 6 )12)(1( .....321 .....321 23222 + =     +       ++ = ++++ ++++ n nn nnn n n
  121. 121. Quantitative aptitude & Business Statistics: Measures Of Central 121  The AM and GM of two observations are 5 and 4 respectively ,Find the two observations.  Solution : Let the Two numbers are a and b given  ( a+b)/2=10 ;a + b=10  GM=4 ab=16  (a-b)2=(a+b)2-4ab=100-64=36  a-b=6 a=8 and b=2
  122. 122. Quantitative aptitude & Business Statistics: Measures Of Central 122  The relationship between AM ,GM and HM  G2=A.H
  123. 123. Quantitative aptitude & Business Statistics: Measures Of Central 123  1.The empirical relationship among mean, median and mode is ______  (a) mode=2median–3mean  (b) mode=3median-2mean  (c) mode=3mean-2median  (d) mode=2mean-3median
  124. 124. Quantitative aptitude & Business Statistics: Measures Of Central 124  1. The empirical relationship among mean, median and mode is ______  (a) mode=2median–3mean  (b) mode=3median-2mean  (c) mode=3mean-2median  (d) mode=2mean-3median
  125. 125. Quantitative aptitude & Business Statistics: Measures Of Central 125 2. In a asymmetrical distribution ____  (a) AM = GM = HM  (b) AM<GM<AM  (c) AM<GM>HM  (d) HMGMAM ≠≠
  126. 126. Quantitative aptitude & Business Statistics: Measures Of Central 126 2. In a asymmetrical distribution ____ (a) AM = GM = HM (b) AM<GM<AM (c) AM<GM>HM (d) HMGMAM ≠≠
  127. 127. Quantitative aptitude & Business Statistics: Measures Of Central 127  3. The points of intersection of the “less than and more than” ogive corresponds to ___  (a) mean  (b) mode  (c) median  (d) all of above
  128. 128. Quantitative aptitude & Business Statistics: Measures Of Central 128  .3.The points of intersection of the “less than and more than” ogive corresponds to ___  (a) mean  (b) mode  (c) median  (d) all of above
  129. 129. Quantitative aptitude & Business Statistics: Measures Of Central 129 •4. Pooled mean is also called  (a) mean  (b) geometric mean  (c) grouped mean  (d) none of these
  130. 130. Quantitative aptitude & Business Statistics: Measures Of Central 130  4. Pooled mean is also called  (a) mean  (b) geometric mean  (c) grouped mean  (d) none of these
  131. 131. Quantitative aptitude & Business Statistics: Measures Of Central 131  5. Relation between mean, median and mode is  (a)mean–mode=2(mean-median)  (b)mean–median=3(mean–mode)  (c) mean–median=2(mean– mode  (d)mean–mode=3(mean–median)
  132. 132. Quantitative aptitude & Business Statistics: Measures Of Central 132  5. Relation between mean, median and mode is  (a)mean–mode=2(mean-median)  (b)mean–median=3(mean–mode)  (c) mean–median=2(mean– mode  (d)mean–mode=3(mean–median)
  133. 133. Quantitative aptitude & Business Statistics: Measures Of Central 133  6. The geometric mean of 9, 81, 729 is _____  (a) 9  (b) 27  (c) 81  (d) none of these
  134. 134. Quantitative aptitude & Business Statistics: Measures Of Central 134  6. The geometric mean of 9, 81, 729 is _____  (a) 9  (b) 27  (c) 81  (d) none of these
  135. 135. Quantitative aptitude & Business Statistics: Measures Of Central 135  7. The mean of the data set of 1000 items is 5. From each item 3 is subtracted and then each number is multiplied by 2. The new mean will be _____  (a) 4  (b) 5  (c) 6  (d) 7
  136. 136. Quantitative aptitude & Business Statistics: Measures Of Central 136  7. The mean of the data set of 1000 items is 5. From each item 3 is subtracted and then each number is multiplied by 2. The new mean will be  (a) 4  (b) 5  (c) 6  (d) 7
  137. 137. Quantitative aptitude & Business Statistics: Measures Of Central 137  8. If each item is reduced by 15, AM is ____  (a) reduced by 15  (b) increased by 15  (c) reduced by 10  (d) none of these
  138. 138. Quantitative aptitude & Business Statistics: Measures Of Central 138  8. If each item is reduced by 15, AM is ____  (a) reduced by 15  (b) increased by 15  (c) reduced by 10  (d) none of these
  139. 139. Quantitative aptitude & Business Statistics: Measures Of Central 139  9. In a series of values if one value is 0 ____  (a)both GM and HM are zero  (b)both GM and HM are intermediate  (c) GM is intermediate and HM is zero (d)GM is zero and HM is intermediate
  140. 140. Quantitative aptitude & Business Statistics: Measures Of Central 140  9. In a series of values if one value is 0 ____  (a) both GM and HM are zero  (b)both GM and HM are intermediate  (c) GM is intermediate and HM is zero  (d)GM is zero and HM is intermediate
  141. 141. Quantitative aptitude & Business Statistics: Measures Of Central 141  10.Histogram is useful to determine graphically the value of  (a) Mean  (b) Mode  (c) Median  (d) all of above
  142. 142. Quantitative aptitude & Business Statistics: Measures Of Central 142  10.Histogram is useful to determine graphically the value of  (a) Mean  (b) Mode  (c) Median  (d) all of above
  143. 143. Quantitative aptitude & Business Statistics: Measures Of Central 143  11.The positional measure of central Tendency  (a) Arithmetic Mean  (b) Geometric Mean  (c) Harmonic Mean  (d) Median
  144. 144. Quantitative aptitude & Business Statistics: Measures Of Central 144  11.The positional measure of central Tendency  (a) Arithmetic Mean  (b) Geometric Mean  (c) Harmonic Mean  (d) Median
  145. 145. Quantitative aptitude & Business Statistics: Measures Of Central 145  12.The average has relevance for  (a) Homogeneous population  (b) Heterogeneous population  (c) Both  (d) none 
  146. 146. Quantitative aptitude & Business Statistics: Measures Of Central 146  12.The average has relevance for  (a) Homogeneous population  (b) Heterogeneous population  (c) Both  (d) none 
  147. 147. Quantitative aptitude & Business Statistics: Measures Of Central 147  13.The sum of individual observations is Zero When taken from  (a) Mean  (b) Mode  (C) Median  (d) All the above 
  148. 148. Quantitative aptitude & Business Statistics: Measures Of Central 148  13.The sum of individual observations is Zero When taken from  (a) Mean  (b) Mode  (C) Median  (d) All the above
  149. 149. Quantitative aptitude & Business Statistics: Measures Of Central 149  14.The sum of absolute deviations from median is  (a) Minimum  (b) Zero  (C) Maximum  (d) A negative figure
  150. 150. Quantitative aptitude & Business Statistics: Measures Of Central 150  14.The sum of absolute deviations from median is  (a) Minimum  (b) Zero  (C) Maximum  (d) A negative figure
  151. 151. Quantitative aptitude & Business Statistics: Measures Of Central 151  15.The mean of first natural numbers (a)n/2 (b)n-1/2 (c)(n+1)/2 (d) none
  152. 152. Quantitative aptitude & Business Statistics: Measures Of Central 152  15.The mean of first natural numbers (a)n/2 (b)n-1/2 (c)(n+1)/2 (d) none
  153. 153. Quantitative aptitude & Business Statistics: Measures Of Central 153  16.The calculation of Speed and velocity (a) G.M (b) A.M (c) H.M (d) none is used
  154. 154. Quantitative aptitude & Business Statistics: Measures Of Central 154  16.The calculation of Speed and velocity (a)G.M (b)A.M (c)H.M (d)none is used
  155. 155. Quantitative aptitude & Business Statistics: Measures Of Central 155  17. The class having maximum frequency is called  A) Modal class  B) Median class  C) Mean Class  D) None of these
  156. 156. Quantitative aptitude & Business Statistics: Measures Of Central 156  17. The class having maximum frequency is called  A) Modal class  B) Median class  C) Mean Class  D) None of these
  157. 157. Quantitative aptitude & Business Statistics: Measures Of Central 157  18. The mode of the numbers 7, 7, 9, 7, 10, 15, 15, 15, 10 is  A) 7  B) 10  C) 15  D) 7 and 15
  158. 158. Quantitative aptitude & Business Statistics: Measures Of Central 158  18. The mode of the numbers 7, 7, 9, 7, 10, 15, 15, 15, 10 is  A) 7  B) 10  C) 15  D) 7 and 15
  159. 159. Quantitative aptitude & Business Statistics: Measures Of Central 159  19. Which of the following measures of central tendency is based on only 50% of the central values?  A) Mean  B) Mode  C) Median  D) Both (a) and (b)
  160. 160. Quantitative aptitude & Business Statistics: Measures Of Central 160  19. Which of the following measures of central tendency is based on only 50% of the central values?  A) Mean  B) Mode  C) Median  D) Both (a) and (b)
  161. 161. Quantitative aptitude & Business Statistics: Measures Of Central 161  20. What is the value of the first quartile for observations 15, 18, 10, 20, 23, 28, 12, 16?  A) 17  B) 16  C) 15.75  D) 12
  162. 162. Quantitative aptitude & Business Statistics: Measures Of Central 162  20. What is the value of the first quartile for observations 15, 18, 10, 20, 23, 28, 12, 16?  A) 17  B) 16  C) 15.75  D) 12
  163. 163. Quantitative aptitude & Business Statistics: Measures Of Central 163  21. The third decile for the numbers 15, 10, 20, 25, 18, 11, 9, 12 is  A) 13  B) 10.70  C) 11  D) 11.50
  164. 164. Quantitative aptitude & Business Statistics: Measures Of Central 164  21. The third decile for the numbers 15, 10, 20, 25, 18, 11, 9, 12 is  A) 13  B) 10.70  C) 11  D) 11.50
  165. 165. Quantitative aptitude & Business Statistics: Measures Of Central 165  22. In case of an even number of observations which of the following is median?  A) Any of the two middle-most value..  B) The simple average of these two middle values  C) The weighted average of these two middle values.  D) Any of these
  166. 166. Quantitative aptitude & Business Statistics: Measures Of Central 166  22. In case of an even number of observations which of the following is median?  A) Any of the two middle-most value..  B) The simple average of these two middle values  C) The weighted average of these two middle values.  D) Any of these
  167. 167. Quantitative aptitude & Business Statistics: Measures Of Central 167  23. A variable is known to be _______ if it can assume any value from a given interval.  A) Discrete  B) Continuous  C) Attribute  D) Characteristic
  168. 168. Quantitative aptitude & Business Statistics: Measures Of Central 168  23. A variable is known to be _______ if it can assume any value from a given interval.  A) Discrete  B) Continuous  C) Attribute  D) Characteristic
  169. 169. Quantitative aptitude & Business Statistics: Measures Of Central 169  24. Ogive is used to obtain.  A) Mean  B) Mode  C) Quartiles  D) All of these
  170. 170. Quantitative aptitude & Business Statistics: Measures Of Central 170  24. Ogive is used to obtain.  A) Mean  B) Mode  C) Quartiles  D) All of these
  171. 171. Quantitative aptitude & Business Statistics: Measures Of Central 171  25. The presence of extreme observations does not affect  A) A.M.  B) Median  C) Mode  D) Any of these
  172. 172. Quantitative aptitude & Business Statistics: Measures Of Central 172  25. The presence of extreme observations does not affect  A) A.M.  B) Median  C) Mode  D) Any of these
  173. 173. THE END Measures Of Central Tendency

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