measures of central tendency

1. MEASURES OF CENTRAL TENDENCY Biostatistics -Bhumi

2. Central tendency -It is a statistical measure that identifies a single value as representative of the entire set of distribution. -Central tendency means central value of a statistical series

3. average of numbers • Sum of all values divided by total number of values • Most frequent value • Middle number in ordered data set

4. Aim -To provide an accurate description of the entire data

5. Statistical average • Mathematical average • Positional average Median Mode Geometric Mean (G) Arithmetic Mean (͞x) Harmonic Mean (H)

6. Important characteristics of measures of central tendency • It should be based on all the observations of the series. • It should be easy to calculate & simple to understand. • Is should not be affected by extreme values. • It should be rigidly defined. • It should be capable of further mathematical treatment. • It should be least affected by the fluctuation of sampling.

7. Mean • nth root of the product of all observations. • Reciprocal of the arithmetic mean of the reciprocals of the observations • Sum of all observations divided by the total number of observations.

8. Arithmetic mean • Arithmetic mean of a set of observations is defined as the sum of all observations divided by the total number of observations. • Denoted by ͞x • Called as “average value”.

10. For Individual series or Ungrouped data For e.g., if x1. x2, x3, x4,……xn be the values of n observations, then the arithmetic mean is, By direct method By shortcut method Where, n= total no. of observations Where, a= assumed mean (deviation)

11. Example 1: Calculate the arithmetic mean of marks obtained in pharmaceutics by 10 students. • by direct method, Here, total no. of students(n)= 10 ⸫ Arithmetic mean

12. by Shortcut method, Here, consider assumed mean a=14 Then calculate the deviations from assumed mean, ⸫ d = x- a ⸫

13. For discrete series or ungrouped frequency distribution For e.g., distribution with x1, x2,…xn & frequencies f1, f2,…fnthen, = =

14. By direct method By shortcut method

15. Example 2: Calculate the arithmetic mean from the following table that shows marks secured in Pharmaceutical engineering by students. • Multiply frequency (f) with (x),

16. So, • Ʃ • a= 58 (assumed mean) • By direct method • By shortcut method

17. For Continuous series or grouped frequency distribution In this, we take midpoints of each class. If m1, m2,..mn are the midpoints of the class with frequencies f1, f2,…fn, then arithmetic mean is calculated as, By direct method By shortcut method Where, d = m-a a = assumed mean ͞x=

18. Example 3: Calculate the arithmetic mean of the following distribution of patient of hypertension as per their weights. Solution:

19. By direct method By shortcut method Mean weight of patients Mean weight of patients

20. Weighted arithmetic mean () • While calculating a simple arithmetic mean, it was assumed that all items of the data were of equal importance. When these are not of equal importance, weights are assigned to them in proportion to their relative importance. • If w1, w2,…wn denote the weights (importance) given to the variables x1, x2,…xn respectively, then the weighted arithmetic mean is given by,

21. How to calculate combined mean? Example : Calculate the combined arithmetic mean of medical & paramedical staff working in Govt. hospital of Valsad district in last 3 years. Solution: First calculate total staff in each category & average staff

23. Merits • Easy to understand and easy to calculate • It is based upon all the observations • Capable of further mathematical treatment • It is affected by sampling fluctuations. Hence it is more stable. • It is correctly or rigidly defined. • Used for further calculations like standard deviation.

24. Demerits • Affected by extreme values(either low or high) • It cannot be obtained even if a single value is missing. • It cannot be determined by inspection • It cannot be used if we are dealing with qualitative characteristics, which cannot be measured quantitatively like caste, religion, sex, etc.

25. Geometric Mean Geometric mean of a set of observations is defined as the nth root of product. • For e.g., if x1. x2, x3, x4,……xn be the values of n observations, then the geometric mean is, • Actual calculations can be made with following formula,

26. For e.g., distribution with x1, x2,…xn & frequencies f1, f2,…fnthen, Where, • If G1 and G2 are the geometric mean of two groups of sizes n1 and n2 then the geometric mean of combination is

27. If G1 and G2 are the geometric mean of two groups of sizes n each, then combination geometric mean is

28. Example: The daily income of helpers (in ₹) in one of ayurvedic formulation industries are given as follow: 700,900,800,850,750 Calculate the geometric mean. Solution:

29. ⸫ Geometric mean of daily income is ₹ 796.74

30. Harmonic mean It is defined as the reciprocal of the arithmetic mean of the reciprocals of the given items. or

31. If the distribution is discrete or continuous, then Where,

32. Example: The weights of 5 capsules (in gm) are given below. 1.20,1.18,1.23,1.17,1.19 Calculate Harmonic mean. Solution: ⸫Harmonic mean of weight for 5 capsules is 1,19 gm.

33. Median It is defined as the size of the item which lies at the centre when all items are arranged in either descending or ascending order of their magnitude, Where, N = Total number of items True for discrete distribution

34. If the distribution is continuous then median is, Where, L= lower limit of median class c.f.= cumulative frequency of class f = frequency of the median class I = interval of median class

35. Example 1: Calculate the median of following marks obtained in English paper. 10,11,12,8,9 Solution: Arranging all values in ascending order, 8,9,10,11,12 N = Number of items or values

36. Example 1: Calculate the median of following weight of tablets. 140,138,130,150,135,145 Solution: Arranging all values in ascending order, 130,135,138,140,145,150 = average of 3rd and 4th value

37. Example 3: The daily expenditure of 60 students of Pharmacy is as follows. Find out median. Solution: The series is continuous or grouped data with intervals. Median (class interval) = 80-90 So, L = lower limit of class = 80 I = width of lass interval = 10

38. c.f. = cumulative frequency of class interval in which median lies = 18 , f = frequency of class interval in which median lies = 12 Median

39. Merits • It is rigidly defined. • It is easy to understand and easy to calculate. • It is not affected by extreme values. • It can be calculated for distribution with open-end classes. • It is the only average to be used while dealing with qualitative data. • It can be determined by graphically.

40. Demerits • In case of even number of observations median cannot be determined exactly. • It is not based on all the observations. • It is not capable of further mathematical treatment.

41. Mode • Mode is that value in a series which occurs with the highest frequency. • It represents the value occurring most often in a data. For Individual series, Example 1: Find the mode of following values of weights of 8 tablets in mg. 30,30,25,28,25,30,26,21 Solution: Arranging the given values in the form of frequency table.

42. So here, maximum frequency is 3. Hence, the Mode = 30 For Discrete series, Example 2: Fine paid by students of college given in the following distribution. Calculate the mode. Solution: The maximum frequency is 19. Hence, the Mode = 136

43. For Continuous series, • In this, class of maximum frequency is called modal class and value of mode belongs to this class. • For this, mode is determined by following formula. Where, = frequency of modal class = frequency of previous class = frequency of the next class = class interval of modal class

44. Example 3: Compute the mode from following data. Solution: The highest frequency is 52. Hence the modal class is 18-22 and modal frequency is 52. = 52, = 47, = 36, = 4

45. Merits • It is readily comprehensible and easy to calculate. • It is not affected by extreme values. • It can be conveniently located even if the frequency distribution has class intervals of unequal magnitude. • Open-end classes also do not pose any problem in the location of mode. • It is the average to be used to find the ideal size.

46. Demerits • It is ill defined. • It is not based upon all the observations. • It is not capable of further mathematical treatment. • As compared with mean, mode is affected to a great extent by fluctuations of sampling.

47. RELATION BETWEEN MEAN, MEDIAN AND MODE Mode = 3Median- 2Mean

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