Measure of central tendency

1. Measure of central tendency • In a representative sample, the values of a series of data have a tendency to cluster around a certain point usually at the center of the series. This tendency of cluster the values around the center of the series is usually called central tendency and its numerical measures are called measures of central tendency.

2. 1)Summation Notation ( Σ )The symbol ΣX is used to denote the sum of all the X’s. by definitionΣX =X1 + X2 + X3 + . . . +XnSimilarly ΣY =Y1 + Y2 + Y3 + . . . +Yn ΣX =X1 + X2 + X3 + . . . +Xn 2 2 2 2 2

3. Types of averages The most used averages are 1)The arithmetic mean 2)The geometric mean 3)The median and 4)The mode

4. The Arithmetic Mean • The arithmetic mean is the sum of a set of observations, divided by the number of such observations. If x is a variable then arithmetic mean is denoted by or. • For ungrouped data: Arithmetic mean, X (read as X bar) or by “µ” is given by X = ΣX n When the data are grouped into a frequency distribution, we calculate arithmetic mean by using the following formula X = Σ fX Σf

5. Example1: Calculate Arithmetic Mean. 2, 4, 9 Solution: X = X = ΣX n 2+4+9 =5 3

6. Example 2: Calculate Arithmetic Mean x f fx 1 5 5 2 13 26 3 4 12 22 43 =Σf =Σfx Σfx 43 X = = =1.95 Σf 22

7. Example: Calculate Arithmetic Mean 392.5 =Σfx 49 =Σf Σfx 392.5 8.01 = X = = 49 Σf

8. Combined Mean If k subgroups of data consisting of n1, n2 ,n3, . . . nk (Σni =n) observations have respective means X1, X2 , X3 , . . . Xn , then X the combined mean for all the data is given by X = n1x1 + n2x2 + n3x3 + . . . nkxn n1 + n2+ n3+… nk

9. Example: Average Profit was $2.00 per order on 200 small orders and $4.60 per order on 50 large orders. Find average profit per order on all 250 orders. Solution: n1 = 200 n2 = 50 x1 = 2.00 x2 = 4.60 X = X = n1X1 + n2X2 n1 + n2 200(2.00) + 50 (4.60) 630 = 200 +50 250 = $2.52

10. Total Value property of Arithmetic Mean ΣX = n X (To find ΣX multiply the mean by the number of data.) Example: Compute the total sales to 400 customers if the average sale per customer was $26.25. Solution: n=400 X=26.25 ΣX = n X = 400x 26.25= $10500

11. Merits (AM) • Merits of Arithmetic Mean • It is rigidly defined. • It is easy to understand and easy to calculate. • It is based upon all the observations • It is amenable to algebraic treatment. • It is less affected by sampling fluctuation.

12. Demerits (AM) • Demerits of Arithmetic Mean • If a single observation is missing or lost arithmetic mean cannot be calculated. • It is affected much by extreme values. • It cannot be calculated if the extreme class is open. • It cannot be determined by inspection nor it can be located graphically. • It cannot be used for qualitative data.

13. Use of Arithmetic Mean • It is widely used almost in all the areas in economics, business, social science etc. • It is extensively used for business forecasting and time series analysis. • It is easily used for comparing two or more sets of data.

14. Geometric Mean • The geometric mean of a set of non-zero positive observations is the n-th root of their product. It is usually denoted by GM or G and expressed as • Geometric mean, • = (Product of the items)

15. For ungroup data • Let be n non-zero positive observations in a series of data. Then the geometric mean is given below, • Or,

16. continue • Log GM

17. Example:Computation of GM Class Interval Frequencies Mid values 0-10 1 5 5 0.6910 10.4835 10-20 20 15 1.1761 23.5220 20-30 30 25 1.3979 41.9370 30-40 15 35 1.5441 23.1615 40-50 10 45 1.6532 16.5320 115.6360

18. Continue • We know,

19. Merits of Geometric Mean • It is rigidly defined. • It is based upon all the observations. • It is suitable for further mathematical treatment. • It is less affected by sampling fluctuation. • Demerits of Geometric Mean: • It is not easy to understand. • If any observation is zero or negative GM cannot be calculate.

20. Uses of Geometric Mean • The GM is used to calculate the averages of ratios and percentages. • It is used also for computing average rate of increase or decrease. • It is useful for the construction of index numbers.

21. Harmonic Mean • The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the observations. It is usually denoted by H.M. and expressed as For ungrouped data ,

22. For grouped data • Harmonic Mean,

23. Merits of Harmonic Mean • It is rigidly defined. • It is based upon all the observations. • It is amenable to algebraic treatment. • It is less affected by sampling fluctuation.

24. Demerits of Harmonic Mean • It is not easy to understand. • If any observation is zero then HM cannot be defined. • The value of the HM may not occur in the series. • Uses of Harmonic Mean • The H.M is used when the observations are expressed of rates, speeds, prices etc.

25. The Median The median of a set of values arranged in ascending order of magnitude is defined as the middle value if the number of values is odd and the mean of the two middle values if the number of values is even. When the data are grouped into a frequency distribution, we calculate median by using the following formula

26. Formula Represents the median class of the distribution. Lower limit of the median class. Total number of observations. Cumulative frequency of the just pre median class Length of the median class

27. Example: Find the median of 18, 30, 44, 60, 31, 22, 68 Solution: Arrange the data in ascending order 18, 22, 30, 31, 44, 60, 68 N is odd therefore The median is position (N+1)/2 = Position (7+1)/2 Position (8/2) = Position 4 Therefore Median= 31

28. Merits & Demerits of Median • Merits of Median • It is rigidly defined. • It is easy to understand and easy to calculate. • It is based upon all the observations. • It is not affected by extreme values. • Demerits of Median • It is not based upon all the observations. • It is not easy to algebraic treatment. • It is affected much by sampling fluctuation.

29. Uses of Median • It is widely used when the observations are not quantitative. • It verily helps us when there exists open interval. • It is useful for comparing two or more sets of qualitative data.

30. Example: Find the median of 25, 13, 16, 30, 20 and 14 Solution: Arrange the data in ascending order 13, 14, 16, 20, 25, 30 Here N = 6 which is even Therefore Median = average of two middle values = (16+20)/2 =36/2 Median = 18

31. Mode • Mode is the value of a distribution for which the frequency is maximum. In other words, mode is the value of a variable, which occurs with highest frequency. Where, L=Lower limit of modal class (modal class is that class for which the frequency) Difference between the frequency of the modal class and pre modal class.

32. Continue • Difference between the frequency of the modal class and post modal class. • Length of the modal class. • Computation of Mode

33. Continue Class IntervalFrequencies( ) 10-20 30 20-30 25 30-40 15 40-50 20 50-60 18 60-70 10 We know, Here, In the class 10-20 the frequency is maximum. So in the class 10-20 the mode contain.

34. Continue • Hence,

35. Merits of Mode • It is rigidly defined. • It is easy to understand and easy to calculate. • It is not affected by extreme values. • It can be calculated when there exists open interval.

36. Demerits of Mode • It indefinite • It is not clearly defined in case of bi-modal or multimodal distribution. • It is not suitable for further algebraic treatment.

37. Example1: Determine the mode. 26, 28, 28, 28, 28, 28, 28,30,30,32,34,34 Solution: As the most repeated value is 28 Therefore Mode= 28 Example2: Calculate mode: x 12 13 14 15 16 17 18 19 f 20 31 38 45 61 34 21 9 Solution: The maximum frequency is 61. therefore Mode = 16