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Measures of Central Tendency
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By- Prof. Saurabh Singh Saurabh.singh.21@gmail.com
Arithematic Mean
By- Prof. Saurabh Singh Saurabh.singh.21@gmail.com
1. Arithmetic Mean ( x ) Definition: “The arithmetic mean may be defined as the sum or aggregate of a series of items divided by their number” - W.I. King a. Simple Arithmetic Mean (Individual Series) i. Direct Method - x = ∑X N x = x 1 + x 2 + x 3 +…………+ xn N ii. Short-Cut Method – X = A + ∑dx N dx = x – A A = Assumed Mean N – No. of observations e.g- X = A + ∑dx N = 5 + 3 6 x dx (x-A) 3 - 2 4 - 1 5 0 6 1 7 2 8 3 ∑dx= 3
By- Prof. Saurabh Singh Saurabh.singh.21@gmail.com c. Simple Arithmetic Mean (Continuous Series) i. Direct Method - x = ∑fx ∑f ii. Short-Cut Method – X = A + ∑fdx ∑f dx = x – A A = Assumed Mean ∑f – sum of frequencies iii. Step Deviation Method – X = A + ∑fdx’ X i ∑f dx = x – A dx’ = dx/i or (x – A)/i A = Assumed Mean N – Sum of frequencies i = l 2 - l 1
By- Prof. Saurabh Singh Saurabh.singh.21@gmail.com e.g. (Direct Method) C.I. f x fx 0 - 10 2 5 10 10 - 20 7 15 105 20 - 30 13 25 325 30 - 40 5 35 175 40 - 50 3 45 135 ∑f = 30 ∑fx = 750 X = ∑fx ∑f X = 750 30 X = 25 e.g. (Short-Cut Method) C.I. f x dx (x-A) fdx 0 - 10 2 5 - 20 - 40 10 - 20 7 15 - 10 - 70 20 - 30 13 A = 25 0 0 30 - 40 5 35 10 50 40 - 50 3 45 20 60 50 - 60 10 55 30 300 ∑f = 40 ∑dx = 300 X = A + ∑fdx ∑f X = 25 + 300 40 X = 25 + 7.
By- Prof. Saurabh Singh Saurabh.singh.21@gmail.com
2. Median (M) - Median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. Formula - a) Individual Series- a. When the number of observations are odd M = (N+1) th term 2 b. When the number of observations are even M = Average of (N) th term + (N + 1) th term 2 2 b) Discrete Series- M = (N+1) th term or item 2 x f cf 5 3 3 6 4 7 8 6 13 9 4 17 10 3 20 15 5 25 ∑f = 25 M = (25 + 1) th term 2 = 13 th term = 8 (because value of 13 th term is 8) c) Continuous Series M = l 1 + (N/2) – pcf X i f
By- Prof. Saurabh Singh Saurabh.singh.21@gmail.com Where, l 1 - Lower limit of the Median Class pcf- Previous Cumulative Frequency of Median Class i = l 2 – l 1 f = Frequency of Median Class Median Class – where the (N/2) th term lies M = l 1 + (N/2) – pcf X i f N = 25 N/2 = 25/ = 12. It lies in the 20-30 class intervals; therefore the Median Class is 20- 30 M = 20 + {12.5– 7 } X 10 6 M = 20 + 0.917X 10 M = 20 + 9. M = 29.
3. Harmonic Mean – The mean of n numbers expressed as the reciprocal of the arithmetic mean of the reciprocals of the numbers. Harmonic mean is used to calculate the average of a set of numbers. Here the number of elements will be averaged and divided by the sum of the reciprocals of the elements. The Harmonic mean is always the lowest mean. x f cf 0 - 10 3 3 10 - 20 4 7 20 - 30 6 13 30 - 40 4 17 40 - 50 3 20 50 - 60 5 25 ∑f = 25
By- Prof. Saurabh Singh Saurabh.singh.21@gmail.com
4. Geometric Mean – Geometric mean is the n th root of the product of N items or values. Calculation of Geometric Mean (G): Individual Series Discrete & Continuous Series G = Antilog ∑ log X N G = Antilog ∑f log X N 5. Mode (Z) – Mode is the value that appears most frequently in a series i.e. it is the value of the item around which frequencies are most densely concentrated. Calculation of Mode: a) Individual Series: (i) By inspection – value repeated most. (ii) By converting individual series into discrete series. (iii) By empirical relationship between the averages - Z = 3M – 2X b) Discrete Series: (i) By inspection – value having highest frequency. (ii) By grouping. (iii) By empirical relationship. c) Continuous Series: (i) First calculate model class by inspection or by grouping. (ii) Then apply the following formula – Z = l 1 + f 1 - f 0 X i 2f 1 - f 0 - f 2
By- Prof. Saurabh Singh Saurabh.singh.21@gmail.com e.g. Z = 20 + 12 – 7 X 10 (2x12 – 7 – 5 ) Z = 20 + 5 X 10 12 Z = 20 + 4. Z = 24.
6. Weighted Arithmetic Average- Arithmetic mean computed by considering relative importance of each items is called weighted arithmetic mean. To give due importance to each item under consideration, we assign number called weight to each item in proportion to its relative importance. Weighted Arithmetic Mean is computed by using following formula: Direct Method : Xω = ∑xω ∑ω Where: Xω Stand for weighted arithmetic mean. x Stands for values of the items and ω Stands for weight of the item Example: A student obtained 40, 50, 60, 80, and 45 marks in the subjects of Math, Statistics, Physics, Chemistry and Biology respectively. Assuming C.I. f 0 - 10 3 10 - 20 7 f 0 20 - 30 12 f 1 30 - 40 5 f 2 40 - 50 3 ∑f = 30
By- Prof. Saurabh Singh Saurabh.singh.21@gmail.com Xω = 60 + (-75) 15 Xω = 60 – 5 Xω = 55
7. Quartiles- A statistical term describing a division of observations into four defined intervals based upon the values of the data and how they compare to the entire set of observations. Each quartile contains 25% of the total observations. Generally, the data is ordered from smallest to largest with those observations falling below 25% of all the data analyzed allocated within the 1st quartile Observations falling between 25.1% and 50% and allocated in the 2nd quartile, The observations falling between 51% and 75% allocated in the 3rd quartile, and finally the remaining observations allocated in the 4th quartile. Subjects Marks Obtained (x) dx Weight ω xω Math 40 - 20 5 - 100 Statistics 50 - 10 2 - 20 Physics 60 0 4 0 Chemistry 80 20 3 60 Biology 45 - 15 1 - 15 Total ∑ω = 15 ∑ xω = - 75
By- Prof. Saurabh Singh Saurabh.singh.21@gmail.com Computation of Quartiles in Individual Series Q 1 = N + 1 th value 4 Q 2 = 2 N + 1 th value or N + 1 th value 4 2 Q 3 = 3 N + 1 th value 4 Note: First arrange the observation in either ascending or descending order. Example: Calculate first and third quartile from the following data: 12, 15, 17, 18, 18, 19, 19, 20, 21, 22, 23, 24, 24, 24, 30, 35, 37, 38, 40 Solution: Number of observations, N = 20 Q 1 = N + 1 th value 4 = 20 + 1 th value = 5. th value 4 = 5 th value + 0.25 ( th value – 5 th value) = 18 + 0.25(19 – 18) = 18 + 0. = 18. Q 3 = 3 N + 1 th value 4 =3 20 + 1 th value = 15. th value 4 = 15 th value + 0.75 ( th value – 15 th value)
By- Prof. Saurabh Singh Saurabh.singh.21@gmail.com Q 3 = L 1 + 3N/4 – pcf x i or (L 2 – L 1 ) f
8. Deciles: It is a method of splitting up a set of ranked data into 10 equally large subsections. Formula, Individual Series and Discrete Series D 1 = N + 1 th value 10 D 2 = 2 N + 1 th value 10 … … … … D 9 = 9 N + 1 th value 10 Grouped Series D 1 = N th value 10 D 2 =2 N th value 10 D 3 =3 N th value 10 … … … … D 9 =9 N th value 10
By- Prof. Saurabh Singh Saurabh.singh.21@gmail.com Continuous Series Dk = L 1 + kN/10 – pcf x i or (L 2 – L 1 ) f k = 1, 2, 3………… 9 f = frequency of the k th decile class pcf = cumulative frequency of the class preceding the k th deciles class
9. Percentiles: A percentile (or centile) is the value of a variable below which a certain percent of observations fall. For example, the 20th percentile is the value (or score) below which 20 percent of the observations may be found. The term percentile and the related term percentile rank are often used in the reporting of scores from norm-referenced tests. Individual Series and Discrete Series P 1 = N + 1 th value 100 P 2 = 2 N + 1 th value 100 … … … … P 99 = 99 N + 1 th value 100 For Grouped Series P 1 = 1 N th value 100 P 2 = 2 N th value 100
