Central tendency

mean,median and mode

1. MEASURES OF CENTRAL TENDENCY BY: Mrs.Keerthi Samuel Asst.Professor ,

2. INTRODUCTION AND DEFINITION Measures of central tendency is a typical value of the entire data. It describes the characteristics of the entire data. It is the value around which other values are distributed. It gives us the mental picture of the central value which is generally called average or central tendency. This value always lies between maximum and minimum values and generally , it is located in the center or middle of the distribution. Mean, median and mode are commonly used averages.

3. CHARACTERISTICS It should be easy to understand It should be rigidly defined It should be easy to calculate and simple to follow It should be based on all the observations of the series. It should not be affected by fluctuations of sampling

4. CLASSIFICATION

5. MERITS Easy to understand Easy to calculate It takes into account each and every observation Sampling fluctuations do not affect the value of mean It is easy for further calculation

6. DEMERITS It is influenced by very big or very small numbers. example: the mean of 10,80,90,1,4 is 37 . This mean 37 neither represents big number like 80,90 nor represent small numbers like 10,1,4 properly. Sometimes it looks ridiculous when we calculate average number of children born to the women. We may get a mean value of 2.5 children. This id evidently absurd. It is not a positional average like median and mode. Sometimes it may give false conclusions.

7. ARTHEMATIC MEAN Mean is the simplest measurement of central tendency and is widely used measure. It is relatively a stable measure of central tendency. Mean is defined as sum of the values divided by the number of values in the case of ungrouped data. Ungrouped data: ∑= symbol of summation n= number of observations This formula is used in case of individual observations where frequencies are not given.  

8. ARTHEMATIC MEAN The duration of survival of eight patients after the diagnosis of blood cancer is as follows . Find the mean duration of survival 5,3,7,4,30,2,3,2 = 5+3+7+4+30+2+3+2 =7 8  

9. GROUPED DATA-WITHOUT CLASS INTERVAL If the number of observations are many it is a laborious process to calculate mean without grouping them into frequency tables. In this method the values of the variables are multiplied by their respective frequencies and the products so obtained are totaled The total is divided by the total number of frequencies STEPS: Multiply each variable by its frequency ( fx ) Add all the ( fx ) = (∑ fx ) Divide (∑ fx ) by the total frequency (∑f) The formula is  

10. GROUPED DATA-WITHOUT CLASS INTERVAL = sum of fx = total frequency This method corresponds to the direct method of calculation of mean for grouped data. This also can be called as grouped data without class intervals  

11. The following data is the number of illness for students in a hostel. Find the mean number of illness per student NUMBER OF ILLNESS(X) NUMBER OF STUDENTS(F) 1 2 3 4 5 6 7 24 76 114 115 86 51 26 18

12. =1530/510=3 Mean number of illness per student = 3   NUMBER OF ILLNESS NUMBER OF STUDENTS fx 1 2 3 4 5 6 7 24 76 114 115 86 51 26 18 76 228 345 344 255 156 126

13. The following data gives the distribution of students according to their age in college. find the mean age of students of the class NUMBER OF ILLNESS NUMBER OF STUDENTS 18 40 19 30 20 20 21 10 22 10

14. =2120/110=19.2 Mean age of the students of the class is= 19.2   NUMBER OF ILLNESS NUMBER OF STUDENTS fx 18 40 720 19 30 570 20 20 400 21 10 210 22 10 220 =110 =2120 NUMBER OF ILLNESS NUMBER OF STUDENTS fx 18 40 720 19 30 570 20 20 400 21 10 210 22 10 220

15. GROUPED DATA WITH CLASS INTERVAL When the range of Values of the variable is large, the values of the variables are grouped into appropriate class intervals and the corresponding frequencies will be grouped into frequency distribution. Mean  

16. Protein intake of 400 females is given in the following table. Find the mean. PROTEIN INTAKE NO. OF FEMALES 15-25 30 25-35 40 35-45 100 45-55 110 55-65 80 65-75 30 75-85 10

17. =19000/400=47.50   PROTEIN INTAKE NO. OF FEMALES Mid point of CI(m) F(m) 15-25 30 20 600 25-35 40 30 1200 35-45 100 40 4000 45-55 110 50 5500 55-65 80 60 4800 65-75 30 70 2100 75-85 10 80 800 PROTEIN INTAKE NO. OF FEMALES Mid point of CI(m) F(m) 15-25 30 20 600 25-35 40 30 1200 35-45 100 40 4000 45-55 110 50 5500 55-65 80 60 4800 65-75 30 70 2100 75-85 10 80 800

18. Calculate the weighted mean of the grouped data weights of 50 students given , calculate mean Weight in kgs students 65-69 6 60-64 5 55-59 10 50-54 9 45-49 9 40-44 8 35-39 7

19. =2768/54=51.25   Weight in kgs students Mid point of CI F(x) 65-69 6 67 402 60-64 5 62 310 55-59 10 57 570 50-54 9 52 468 45-49 9 47 423 40-44 8 42 336 35-39 7 37 259 Weight in kgs students Mid point of CI F(x) 65-69 6 67 402 60-64 5 62 310 55-59 10 57 570 50-54 9 52 468 45-49 9 47 423 40-44 8 42 336 35-39 7 37 259

20. MEDIAN

21. MEDIAN A value that divides a distribution into two equal halves or central or middle value of a series of observations, when the observed values are arranged in ascending or descending order of magnitude. Median is just the 50 th perceive value. Median is denoted by’Md ’ Like mean, median can also be calculated. Ungrouped data Discrete series Continous series

22. MEDIAN-UNGROUPED DATA STEP-I: Arrange the data in ascending or descending order of magnitude STEP-2: If the number of items are odd, n+1/2 th item gives the serial number of the median. If the number of items are even, the average of n/2th and n/2+1 th item gives the serial number of the median.

23. MEDIAN-PROBLEM 1. The diastolic blood pressure of 10 individuals is as follows. find the median 83,75,81,79,71,95,75,77,84,90 Arrange in ascending order n/2= 10/2 = 5 th item in the serial =10/2+1=6 th item in serial Ascending order: 71,75,75,77,79,81,83,84,90,95 5 th item is 79 and 6 th item is 81 Average of 5 th and 6 th values is : 79+81/2=160/2=80 Median = 80  

24. MEDIAN-PROBLEM 2. In a hospital ward the following are the number of days of stay of patients. Find the median days of stay in the hospital 13,42,8,9,7,3,6,52,82,11,11,10,2 Ascending order: 2,3,6,7,8,8,9,10,11,11,13,42,52 Patients are in odd number th value =13+1/2=14/2= 7 th value in the series Median=9  

25. MEDIAN-PROBLEM(GROUPED) Find the median from the following data Step-I : find the cumulative frequencies Step-II : value Step-III : check the obtained value in relation to cumulative frequencie and the corresponding x value will be the median Size of shoe 4 5 6 7 8 9 frequency 10 15 22 16 12 5  

26. = 80+1/2=81/2=40.5 th value Here 40.5 th value lies between 25 and 47 of cumulative frequency and falls on 47.So, we take the corresponding x value of cf 47. The corresponding ‘x’ value of cf is 6. Median is 6 .   SIZE OF SHOE F cf 4 10 10 5 15 25 6 22 47 7 16 63 8 12 75 9 5 80 80

27. MEDIAN-PROBLEM(GROUPED) Find the median from the following data Step-I : find the cumulative frequencies Step-II : value Step-III : check the obtained value in relation to cumulative frequencies and the corresponding x value will be the median Size of shoe 4 5 6 7 8 9 frequency 10 15 22 16 12 5  

28. MEDIAN-PROBLEM(GROUPED WITH CI) l= lower limit of the median class N= total frequency m= the cumulative frequency value above the median class F= the frequency corresponding to the median class C=class Interval  

29. MEDIAN-PROBLEM(GROUPED WITH CI)   Protein intake Number of families 15-25 30 25-35 40 35-45 100 45-55 110 55-65 80 65-75 30 75-85 10

30. MEDIAN-PROBLEM(GROUPED WITH CI)   Protein intake Number of families cf 15-25 30 30 25-35 40 70 35-45 100 170-m or cf 45 (l) -55 110 (f) 280-Median class 55-65 80 360 65-75 30 390 75-85 10 400 N=400

31. MEDIAN-PROBLEM(GROUPED WITH CI) N=Total frequency; N/2= 400/2=200 The value 200 falls on 280 in the cumulative frequency. Median class value is 280. the cumulative frequency value above the median class m=170. The frequency corresponding to the median class is f=110, the lower limit corresponding to the median class is l=45;c=10. = 45+ x10 =45+ =47.73grams  

32. MEDIAN ADVANTAGES: It is simple to calculate and easy to understand It can be located by inspection The value of median is unaffected by extreme values Its value generally lies in distribution. DISADVANTAGES: Median is not familiar average like mean. If the data is large , it is tedious to arrange the data in ascending order. Since, it is a positional average, the value is not influenced by each and every observation. The median value is affected by sampling fluctuations.

33. MODE Mode is the most frequently occurring value in a series of observations. Mode is also a positional average. This is the another measure of central tendency which is least influenced by the size of the individual observations. Mode is more useful in certain type od observations , where it is required to know the value of observation, which has high influence in the series. Ex: when studying the age of attack , or onset of a disease it is advisable to know the maximum number of persons are affected rather than the mean age of onset or median age of onset. Mode also can be calculated from the following relationship, if median and mean are known. Mode=3(median)-2(mean)

34. MODE-TYPES The most frequently occurring value in the series of observations. Ex: 4,5,8,6,7,5,9,5 Mode is 5 BIMODEL: If two sets of observations are same or distribution with two modes is bimodal Ex: 4,8,9,10,4,6,12,10 Bimodel:4 and 10 MULTIMODEL: If more than two sets of observations are same or distribution with more than two modes is multimodel . Ex: 5,8,9,10,5,9,16,8 Multimodel:5,8,9

35. MODE-UNGROUPED DATA The diastolic BP of 20 individuals is 85,75,81,79,71,95,75,77,75,90,71,75,79,95,75,77,84,75,81,75 The mode is 75 If all sets of values are same there is no mode Ex:3,7,3,4,7,4,9,9 If different values are there is no mode Ex:31,33,34,36,37,39,40

36. MODE-GROUPED DATA( discreete ) Wages employees 145 3 170 16 180 8 190 20 200 6 210 2 Since the highest frequency is 20 the mode will be 190

37. MODE-GROUPED DATA( Continous ) f = modal class frequency l= lower limit of the modal class = frequency preceding the model class = frequency succeeding the model class h= height of the class (or) class interval of model class  

38. MODE-GROUPED DATA( Continous ) Protein No.of families 15-25 30 25-35 40 35-45 100-f1 45(l)-55 110 -f 55-65 80-f2 65-75 30 75-85 10  

39. = =45+4.4 =49.4 Mode = 49.4  

40. MERITS It is easy to understand and not affected by the extreme values It is a positional average and can be located easily by inspection

41. DEMERITS It is an average, which is ill defined and indeterminate In the case of bimodal class, the calculation is difficult as it involves grouping and analysis It is not based on all observations The exact location is uncertain Mode is not used in biological and medical sciences