Measures of central tendency

Measures of central tendency

1. Measures of Central Tendency Mod 5

2. Average A single significant figure, which sums up the characteristics of a group of figures. It conveys the general idea of the whole group It is generally located at the center or middle of the distribution

3. Various Measures of Central tendency are; Arithmetic Mean Median Mode Geometric Mean Harmonic Mean

4. 1. Arithmetic Mean It is a Mathematical average It represents the whole data using a single figure It is the simplest and most widely used measure

5. Mean in an Individual Series If the individual values in a set of variables are, x1, x2, x3…….. xn , then the Arithmetic mean will be;

6. Mean of a discrete frequency distribution Let the discrete variables be x1, x2, x3……. xn Let the corresponding frequencies be f1, f2, f3…… fn Then the arithmetic mean = Σ fx N (N= Σ f)

7. Mean in Continuous Frequency distribution Here, you have to consider the mid values of each class as ‘x’ Then the same formula of discrete frequency distribution can be applied. Class Frequency (f) Mid point of class (x) fx 0-10 14 5 14×5 = 70 10-20 18 15 18 ×15 = 270 Σ f = N Σ fx

8. Find the arithmetic mean. Temperature Number of days -40 to -30 10 -30 to -20 28 -20 to -10 30 -10 to 0 42 0 to 10 65 10 to 20 180 20 to 30 10

9. Arithmetic Mean = 1565/365 = 4.28 degree Celsius Temperature Number of days (f) Mid Point (x) fx -40 to -30 10 -35 -350 -30 to -20 28 -25 -700 -20 to -10 30 -15 -450 -10 to 0 42 -5 -210 0 to 10 65 5 325 10 to 20 180 15 2700 20 to 30 10 25 250 N = 365 Σ fx = 1565

10. Find the AM Marks Number of Students 20-29 10 30-39 8 40-49 6 50-59 4 60-69 2

11. Arithmetic Mean = 1135/30 = 37.83 Marks Number of Students (f) Midpoint (x) fx 20-29 10 24.5 245 30-39 8 34.5 276 40-49 6 44.5 267 50-59 4 54.5 218 60-69 2 64.5 129 N = 30 Σ fx = 1135

12. Find the AM Marks Number of Students Less than 10 5 Less than 20 17 Less than 30 31 Less than 40 41 Less than 50 49

13. Am = 1265/49 = 25.81 Marks Number of Students Class New frequency (f) Midpoint (x) fx Less than 10 5 0-10 5 5 25 Less than 20 17 10-20 12 15 180 Less than 30 31 20-30 14 25 350 Less than 40 41 30-40 10 35 350 Less than 50 49 40-50 8 45 360 N = 49 Σ fx = 1265

14. 2. Median Median is the value of that item which occupies the central position, when the items are arranged in ascending or descending order of their magnitude. Therefore, Median is the value of that item, which has equal number of items above and below. Hence Median is a Positional average.

15. Median of Individual Series Arrange the items in ascending or descending order Then take the (n+1)/2th item. When ‘n’ is even, there will be two middle terms, then the median will be the average of them.

16. Median of Discrete frequency distribution Here, median will be the size of (N+1)/2th item. Here N = Σ f Mode = N+1/2th item = 56/2 = 28 th Item. Age Frequency Cumulative Frequency 2 14 14 6 22 36 10 19 55 N ( Σ f) = 55

17. Find the Median Marks Frequency CF 5 3 8 12 10 8 15 7 20 5 25 4

18. Median of a Continuous Frequency Distribution The first step is to find the Median class Median class corresponds to the cumulative frequency which includes N/2 Here also, N = Σ f Once the Median class is found, the Median has to be calculated using the interpolation formula.

19. Class Frequency C.F 10-30 8 8 30-50 6 14 50-70 12 26 N=26 Median = 26/2th item, that is 13 th item. 13 is included in the CF 14, which means Median class is 30-50 Median =

20. L1 = Lower class limit of the median class Cf = Cumulative frequency of the class just proceeding the median class f = Frequency of the median class C = Interval of the Median class

21. Calculate the Median Class Frequency CF 0-10 8 10-20 12 20-30 20 30-40 23 40-50 18 50-60 7 60-70 2

22. Calculate the Median Class Frequency 10-20 4 20-40 10 40-70 26 70-120 8 120-140 2

23. 3. Mode Mode is the value of the item of a series, which occurs most frequently. When no item appears to have repeating more than others, the mode is considered as ‘ ill defined’ In such cases, mode is found using the formula, Mode = 3 Median - 2 Mean

24. Mode in Individual Series Consider the most repeated item in the given series.

25. Mode in Discrete frequency distribution Mode will be that particular item with the maximum frequency.

26. Mode in Continuous Frequency Distribution First step is to find the class with the maximum frequency. The class with the maximum frequency is then termed as Model Class From the Model class, you have to calculate the Mode using interpolation formula.

27. L1 = Lower limit of the Model Class F1 = Frequency of the Model class Fo = Frequency of the class just preceding the Model class F2 = Frequency of the class just succeeding the Model class

28. Calculate Mode Wages No: of Workers 0-10 15 10-20 20 20-30 25 30-40 24 40-50 12 50-60 31 60-70 71 70-80 52

29. Calculate Mean, Median & Mode Class Frequency 10-15 4 15-20 8 20-25 18 25-30 30 30-35 20 35-40 10 40-45 5 45-50 2

30. Class Frequency Mid Point (X) fx CF 10-15 4 12.5 50 4 15-20 8 17.5 140 12 20-25 18 22.5 405 30 25-30 30 27.5 825 60 30-35 20 27.5 650 80 35-40 10 32.5 375 90 40-45 5 37.5 212.5 95 45-50 2 42.5 95 97 N = 97 Σ fx = 2752.5

31. Answers AM = 28.37 Median Class = 25-30 Median = 28.08 Model Class = 25-30 Mode = 27.72

32. Find The mean marks obtained by all 50 boys in a class is 40 and mean marks of all 30 girls of the same class is 46. Find the mean marks of the class.

33. Find Average monthly production of a certain factory for the first 9 months is 2584 units and for the remaining 3 months is 2416 units. Calculate the average monthly production for the year.

34. Find The mean weight of 150 students in a certain class is 60 Kgs . The mean weight of boys and girls in the class is 70 and 55 kgs respectively. Find the number of boys and girls in the class.

35. Finding Log Values Step 1 Consider log (12.69) Finding the number of digits on the left hand side of the decimal point. Eg : 12.69 (Number of digits to the left of the decimal point is 2) So here, n = 2

36. Step 2 Find n-1 Here, in case of 12.69, n=2 and n-1 = 1 This means that the value of log(12.69) would start with 1

37. 00

38. Step 3 Consider the first two digits of the number (12.69) and plot the same in the vertical bar of the Log chart Here, the first two numbers is 12

39. Step 4 Consider the 3 rd digit of the number and plot the corresponding value in the horizontal bar of the chart Here the number is 12.69, so 3 rd number is 6 Find the value in the chart which corresponds to the intersection of both the vertical and horizontal figures. That is the number corresponds to 12 and 6

40. The intersection value = 1004

41. Step 5 Consider the fourth digit of the number and find its corresponding value in the mean difference column of the chart. (Number = 12.69, 4 th digit is 9) Find the intersection point of the vertical column and the mean difference value. Add the corresponding mean difference value to the previous intersection figure (Here 1004). Write the resulting number after the decimal point

42. So the value now becomes 1004 + 31 = 1035 Therefore log (12.69) = 1.1035

43. Find the log values for the following; 57.5 87.75 53.5 73.5 81.75

44. Calculating the Antilog Values. Consider Antilog (2.8675) Step one is to calculate the Antilog value corresponds to .8675 using Antilog table (same procedures apply as in the case of Log) The resultant value is 7362 + 8 = 7370

45. Step 2 Step two is to find where to put the decimal point Consider the number to the left of decimal point, which is ‘2’ in this case (2.8675). Add 1 to that number. (2+1 = 3) This mean that you have to put the decimal point after the first 3 digits. So AL (2.8675) = 737.0

46. Geometric Mean If there are ‘n’ values in a series, then their GM is defined as the n’th root of the product of those n values It is a mathematical average

47. Geometric Mean in Individual Series This can be expressed in another form as;

48. Calculate the geometric mean for the following figures 57.5, 87.75, 53.5, 73.5, 81.75

49. Geometric Mean in Discrete/Continuous Series

50. Find GM Size Frequency 5 2 8 3 10 4 12 1

51. Find GM Weight Number 100-104 24 105-109 30 110-114 45 115-119 65 120-124 72 125-129 84 130-134 124 135-139 58

52. Find GM Marks Number of Students 0-10 5 10-20 7 20-30 15 30-40 25 40-50 8

53. Marks f Mid Point (X) Log (x) F × log (x) 0-10 5 5 10-20 7 15 20-30 15 25 30-40 25 35 40-50 8 45 60

54. Marks f Mid Point (X) Log (x) F × log (x) 0-10 5 5 0.6990 3.495 10-20 7 15 1.1761 8.2327 20-30 15 25 1.3979 20.9685 30-40 25 35 1.5441 38.6025 40-50 8 45 1.6532 13.2256 60 84.5243

55. Merits and Limitations of GM Helps in averaging ratios and percentages Not affected by extreme values Difficult to understand Not applicable on negative values

56. Harmonic Mean Harmonic Mean of a set of n Values is defined as the reciprocal of the mean of the reciprocals of those values Harmonic Mean is used when it is desired to give the greatest weight to the smallest items. It is predominantly used in averaging rates .

57. Harmonic Mean – Individual Series

58. Harmonic Mean in Individual Series

59. Find Harmonic Mean for the following 2,3,4,5

60. Find HM X 1/X 2 0.50 3 0.33 4 0.25 5 0.20 Σ 1.28

61. Find Harmonic Mean for the following 8,14,16,10,24,18

62. X 1/X 8 0.125 14 0.071 16 0.062 10 0.1 24 0.042 18 0.055 0.455 n/0.455 = 6/0.455 = 13.18

63. HM in a Frequency Distribution

64. Size Frequency 6 20 10 40 14 30 18 10 Find HM

65. Size (X) Frequency (f) 1/X f (1/x) 6 20 0.1667 3.334 10 40 0.1000 4.000 14 30 0.0714 2.142 18 10 0.0556 0.556 N = 100 Σ = 10.032 HM = 100/10.032 = 9.97

66. Class Frequency 10-20 4 20-30 6 30-40 10 40-50 7 50-60 3 Find HM

67. Class Frequency Mid Point (x) 1 x f× 1 x 10-20 4 15 0.067 0.268 20-30 6 25 0.040 0.240 30-40 10 35 0.029 0.290 40-50 7 45 0.022 0.154 50-60 3 55 0.018 0.054 30 1.006 HM = 30/1.003 = 29.82

68. Find GM & HM Weight Frequency 10-20 11 20-30 19 30-40 9 40-50 10 50-60 17 60-70 16 70-80 15 80-90 23

69. Weight Frequency Mid Point (X) Log(x) F × log (x) 10-20 11 15 1.1761 12.94 20-30 19 25 1.3971 26.54 30-40 9 35 1.5441 13.90 40-50 10 45 1.6532 16.53 50-60 17 55 1.7404 29.59 60-70 16 65 1.8129 29.00 70-80 15 75 1.8751 28.13 80-90 23 85 1.9294 44.38 120 201.01 GM = AL (201.01/120) = AL (1.6751) = 47.33

70. Measures of Dispersion (Measures of Variability)

71. Dispersion Dispersion refers to the variability in the size of items. It represents the spread or scatter of the value in a series. Measures of dispersion measure the variability in a series It tell us the extent to which the values of a series differ from their average or among themselves.

72. Absolute Measures of Dispersion Range Quartile Deviation Mean Deviation Standard Deviation

73. 1. Range Range is the simplest possible measure of dispersion It is the difference between the highest and lowest values in a series Range of an individual series ; = Highest Value- Lowest Value (H-L) Range measures the maximum variation in the values of a series.

74. Coefficient of Range Coefficient of Range is the relative measure based on range. C oefficient of Range  

75. Uses of Range Used to study Data, for which variations are low. Application in Diagnosing Applications in Quality Control Applications in Weather forecasting etc

76. 2. Quartile Deviation (Semi Interquartile range) Defined as half the distance between the third and first Quartiles Quartile Deviation Coefficient of Quartile Deviation  

77. QD in Individual Series First, arrange the variables in the ascending order Q1 th item Q3 ×3 th item Quartile Deviation  

78. Problem 1 Find the QD & Coefficient of QD A) 23, 25, 8, 10, 9, 29, 45, 85, 10, 16, 24

79. QD in discrete frequency distribution Q1 th item (N= Σ f) Q3 ×3 th item Quartile Deviation  

80. P1: Find QD Size Frequency 5 3 8 10 10 15 12 20 19 8 20 7 32 6

81. P2: Find QD Age Number 57 15 59 20 60 32 61 35 62 33 63 22 64 20 66 10 68 8

82. QD in Continuous frequency Distribution Q1 th item (N= Σ f) Q3 ×3 th item Quartile Deviation  

83. Interpolation formula for calculating QD Q1 Q3  

84. P1: Find QD Class Frequency 0-10 15 10-20 30 20-30 53 30-40 75 40-50 100 50-60 110 60-70 115 70-80 125

85. P2 Find QD & Coefficient of QD Class Frequency 10-19 5 20-29 8 30-39 17 40-49 29 50-59 30 60-69 20 70-79 10 80-89 1

86. P2: Find QD & Coefficient of QD Wages Workers Below 5 4 Below 10 10 Below 15 13 Below 20 21 Below 25 33 Below 30 40

87. 3. Mean Deviation Mean deviation is defined as the Arithmetic mean of the deviations of all the values in a series from their average . For the purpose of the calculation of MD, all the deviations are considered positive irrespective of their sign. = deviations from the average, without sign  

88. Coefficient of Mean Deviation =  

89. Mean Deviation in an Individual Series P1: Find the MD from Mean for the following values; 25, 63 85, 75, 62, 70, 83, 28, 30, 12

90. Mean Deviation in an Individual Series P2: Find the MD from the Median for the following values; 5, 28, 33, 44, 83, 87, 96, 99, 25, 35, 82

91. Mean Deviation in Discrete Frequency Distribution

92. P1: Find MD from Mean & Coefficient of MD No. of Children (x) 1 2 3 4 5 6 No. of families (f) 171 82 50 25 13 7 2

93. P2: Find MD from mean & Coefficient of MD X 10 11 12 13 14 Frequency 3 12 18 12 3

94. MD in Continuous Frequency Distribution

95. Find MD about Arithmetic Mean Marks (x) 0-10 10-20 20-30 30-40 40-50 50-60 60-70 Freq. (f) 4 6 10 20 10 6 4

96. X F 0-10 18 10-20 16 20-30 15 30-40 12 40-50 10 50-60 5 60-70 2 70-80 2 Calculate the MD about Median

97. 4. Standard Deviation Standard Deviation is the square root of the mean of the squares of the deviations of all the values of a series from their Arithmetic Mean It is calculated as the square root of variance by determining the variation between each data point relative to the mean . If the data points are further from the mean, there is a higher deviation within the data set; thus, the more spread out the data, the higher the standard deviation.

98. SD in Individual Series SD = Coefficient of Variance =  

99. Find the Variance and Standard Deviation 5,8,7,11,9,10,8,2,4,6

100. SD in Discrete/Continuous Frequency Distribution SD =

101. Marks Number of Students 2 8 4 10 6 16 8 9 10 7 Find the Variance and Standard Deviation

102. Class Frequency 0-2 2 2-4 4 4-6 6 6-8 4 8-10 2 10-12 6 Find the Variance and Standard Deviation

103. From the above table, showing the runs scored by two batsmen in their last 8 innings’, find out who is more consistent and who is more efficient. Batsman A 10 12 80 70 60 100 4 Batsman B 8 9 7 10 5 9 10 8

104. Measures of Skewness

105. Symmetry

106. Symmetric Distributions A frequency distribution is said to be Symmetric, if the frequencies are distributed symmetrically or evenly on either side of the average. In other words, the number of items above the mean and below the mean would be the same. For such distributions, Q3and Q1 would be equidistant from median

107. Skewness Skewness means lack of Symmetry. The word skewness literally denotes assymetry . If a frequency distribution is Skewed, there will be more items on one side of the average than the other side. Such distributions will have a long tail on one side and a shorter one on the other side. Most of the Economic data have skewed distributions.

108. Positive Skewness Skewness is said to be positive, when Mean is greater than Median and Median is greater than Mode. ( Mean ˃ Median ˃ Mode ) Here, the curve is skewed to the right. More than half the area falls to the right side of the highest ordinate.

109. Negative Skewness Skewness is said to be Negative, when Mean is less than Median and Median is less than Mode. ( Mean ˂ Median ˂ Mode ) Here, the curve is skewed to the left. More than half the area falls to the left side of the highest ordinate.

110. What is Skewness?

111. X F FX 1 2 2 2 4 8 3 6 18 4 8 32 5 6 30 6 4 24 7 2 14 Mean Median Mode Normal Distribution

112. X F FX 1 10 10 2 12 24 3 10 30 4 8 32 5 4 20 6 2 12 7 2 14 Mode Mean Positive Skewness

113. X F FX 1 1 2 2 2 4 3 4 12 4 6 24 5 10 50 6 12 72 7 10 70 Mode Mean N egative Skewness

114. Measures of Kurtosis

115. Kurtosis Kurtosis indicates whether a distribution in flat topped or peaked. Thus, Kurtosis is a measure of peakedness . When a frequency curve is more peaked than the normal curve, it is called Lepto Kurtic When it is more flat topped than the normal curve, it is called Platy Kurtic . When the curve is neither peaked nor flat topped, it is called Meso Kurtic

116. Index Numbers

117. Meaning An Index number is a statistical device for measuring changes in the magnitude of a group of related variables during a specific period in comparison to their level in some other period .

118. BSE SENSEX Published since 1 January 1986, the S&P BSE SENSEX is regarded as the pulse of the domestic stock markets in India. The base value of the SENSEX was taken as 100 on 1 April 1979 and its base year as 1978–79.

119. Consumer Price Index A Consumer Price Index measures changes in the price level of market basket of consumer goods and services purchased by households . In I ndia the base year is 1982.

120. Methods for Index Number Construction Simple Aggregative Method Weighted aggregative Method Simple average relative method Weighted average of price relative

121. 1. Simple Aggregative Method Simple Index Number =  

122. 2. Weighted Aggregative Method Laspeyer’s Index Number Paasche’s Index Number Fisher’s Index Number

123. Laspeyer’s Index number Laspeyer’s Index number =   P0, P1 = Price in Base year and Current year Q0 = Quantity in the base year

124. Paasche’s Index number Paasche’s Index number =   P0, P1 = Price in Base year and Current year Q1 = Quantity in the Current year

125. Find out Index number. Commodity Unit of consumption in base year Price in Base year Price in current year A 200 1.00 1.20 B 50 3.00 3.50 C 50 4.00 5.00 D 20 20.00 30.00 E 40 2.00 5.00 F 50 10.00 15.00 G 60 2.00 2.50 H 40 15.00 18.00

126. Find out Index number. Commodity Quantity Consumed (2009) Price 2005 2009 A 50 32 40 B 35 30 42 C 55 16 24 D 45 40 52 E 15 45 42

127. Fisher’s Index number Fisher’s Index number =   P0, P1 = Price in Base year and Current year Q0,Q1 = Quantity in the Base year &Current year

128. Find the weighted aggregative index number Commodity Price Quantity 2009 2010 2009 2010 A 4 7 10 8 B 5 9 8 6 C 6 8 15 12 D 2 2 5 6

129. 3. Simple Average relative method Under this method, the price relative of each item is individually calculated and the average of all such values would be the Index number. Here, Price Relative; I = Price Index =  

130. Calculate the Simple index number using average relative method. Item Base year price Current year price 1 5 7 2 10 12 3 15 25 4 20 18 5 8 9

131. 3. Weighted Average of Price relative method Here, we will be assigning some arbitrary numbers as weight. The price relative would be found and then multiplied by the concerned weight. Here, Price Relative; I = Price Index =  

132. Calculate the Price Index for the following data. Commodities V (Weight) Price (2008) Price (2009) A 40 16 20 B 25 40 60 C 5 2 2 D 20 5 6 E 10 2 1

133. Consumer Price Index A Consumer Price Index measures changes in the price level of market basket of consumer goods and services purchased by households . In I ndia the base year is 1982.

134. Steps in the Construction of CPI number. Decisions about class of people & Scope Decisions about the items to be selected Family budget enquiry Obtaining price quotations Selection of base period and weightages Selection of suitable indexing number

135. Food (35%) Rent (15%) Clothing (20%) Fuel (10%) Misc (20%) Price (2006) 150 30 75 25 40 Price (2008) 145 30 65 23 35