Measures of Central tendency

This slideshow explains the important measures of central tendency in statistics. It deals with Mean, mode and median; its characteristics, its computation, merits and demerits. This slideshow will be useful to students, teachers and managers.

1. Measures of Central tendency Sir Parashurambhau College Pune, India [email protected] 1

2. Measures of central tendency In statistics, collection of data is the first step, but this is not sufficient to arrive at any conclusion. It is essential to represent data by a single value that shows conclusion of data at that particular value. This is possible by the measures of central tendency. Generally, it is seen that the values of variables have a tendency to cluster around a particular value. This tendency is called as Central tendency. 2

3. Measures of central tendency Agricultural scientist, genetics, plant breeder, manager, teacher, production engineer, etc. often talk about averages in the context of average weight gain, average milk production per day, average income of worker, average marks of students etc. These averages simply summarize a set of data in a single value. These are representative of data and around which large portion of observations gathered. The averages can be presented as one of the three common measures of central tendency viz. mean, mode and median. 3

4. P roperties of a good measure of central tendency It should be rigidly defined. It should be unique i.e. only one value. It should be based on all observations (values). It should have sampling stability i.e. it should not be affected by sampling fluctuations. It should be capable for the further mathematical calculations It should not be affected by extreme values. 4

5. Arithmetic Mean(AM) Mean: It is a widely used measure of central tendency. There are three types of means, i ) Arithmetic Mean (AM), Geometric Mean (GM) and Harmonic Mean (HM). The selection of mean depends on a particular type of data. Arithmetic Mean (A.M.): It is popularly known as average. It is the most commonly used measure. It is calculated by dividing sum of all the individual observations by the total number of observations. A.M. = Sum of observations/total number of observations 5

6. Arithmetic Mean for individual observations (i.e. ungrouped data) It is computed by summing up of the observations and dividing the sum by total number of observations. A.M. =  X =  X/n A.M. for a discrete series (grouped data): For computation of arithmetic mean in a discrete series the values of variables are multiplied by their respective frequencies. However total number of observations is equal to the total number of frequencies. X =  ( fx )/n f= frequency of individuals in each class. x= class value of measurements n = number of observations. 6

7. Merits of Arithmetic mean It is simple to understand and easy to calculate. It is calculated by the perfect formulae so every one gets same results. It is not necessary to arrange the data. It provides good measure for comparison between two or more groups. It takes into account all the observations in the series. Subsequent calculations and algebraic treatments can be done in better way than median and mode. It is unique i.e. for a given set of data, there can be only one mean. It does not change much even when repeated samples are taken for the same population. 7

8. Demerits of Arithmetic Mean It is largely affected by the extreme values of data. It does not consider increasing and decreasing trend. It is a good measure of central tendency only when the distribution is normal (Bell shaped) and it is not useful in case of ‘U’ shaped distribution. In some cases it may lead to meaning less results. It cannot be calculated in open-end classes without making assumptions about the size of the class interval. 8

9. Important Applications of AM It is mostly used in practical statistics. It is used to calculate many other estimates in statistics. It is the most used method by any measurement used by common people to get the average of any data. It is essential to calculate Mean deviation, variance, standard deviation, etc. 9

10. Geometric mean (G.M) It usually a more suitable as a measure of central tendency when the values changes exponentially. It is extensively used in microbiological and serological research. If there are two observations G.M. is the square root of product of two observations. If there are three observations it is a cube root of the product of three observations, if there are ‘n’ observations it is the n th root of the product of ‘n’ observations. 10

11. Merits and Demerits of GM Merits of GM: It is based on all observations It is rigidly defined. It is not much affected by fluctuations of sampling. It is suitable for averaging rates, ratios and percentages. It gives larger weightage to the small items. Demerits of GM: It is difficult to understand and calculate. It cannot be calculated when there are both negative and positive values. If one or more values are “ 0”, then it would also be “0”. The geometric mean cannot be used if any of the original observations are negative, since negative number has no logarithm. 11

12. Calculations of GM in a series of individual observations Steps involved: Find logs of the given values (x 1 ,x 2 ,x 3 etc ) Take the sum of the given values. Sum of logs is divided by number of observations. Find out antilog of the values 12

13. Harmonic Mean (H.M.) It is suitable measure of central tendency when the data pertained to speed, rates and time. H.M. (for ungrouped data) = n /  (1/n) H.M.(for grouped data) =  f /  (f/x) Relation ship between A.M., GM and H.M. It has been observed that A.M is greater than G.M. while G.M. is greater than H.M. Therefore A.M.  G.M.  H.M. 13

14. Merits and Demerits of GM Merits: It is rigidly defined. It is based on all observations of a series, It gives greater weightage to the smaller items. It is useful to study the average price, speed, time, distance etc. It is not much affected by sampling fluctuations. Demerits: It is not easy to calculate and understand. It cannot be calculated if one of the value is ‘0’ 14

15. Median It is a positional average. It is the "middle" value in the list of numbers when observations are arranged in ascending or descending order. It is a value separating the higher half from the lower half of a data sample. If a series of observations is 23,25,27,29,31,33, then median will be 27. 23 25 27 Median 29 31 33 15

16. Merits of median It is easy to define and understand. It is unique i.e. it has single value. It is simple to calculate. It is not affected by extreme values of data. It indicates exactly the middle item and hence it can be easily explained. It can be determined by mere inspection in some cases. It is useful when the classes are open and when the distribution is unequal (skewed distributions). 16

17. Demerits of median For calculating the median, it is necessary to arrange the data in ascending or descending order, however it is not for calculation of mean and mode. If the number of observations is large, then arranging the data is tedious and time consuming. In this relative importance cannot be judged properly. It is a proportional average and its value is not determined by each and every observation. It largely affected by sampling fluctuations. It can not be used for further mathematical processing. 17

18. Median for individual observations This involves the following steps: 1. Arrange the data in the ascending order. 2. Median is located by finding the size of n+1 / 2 th item. Median = Size of n+1 / 2 th item Where n = number of observations. 18

19. Median for continuous series It involves following steps. 1. The first step is to determine median class by using n/2 as the rank [not (n+1)/ 2 ]. 2. To determine cumulative frequencies. This is done by adding the number of frequencies that are in each interval to the frequencies in the next class. 3. The cumulative frequency tells us where n/2 falls. Median = L + [ ( n/2 – cf ) / f ] x i Where L = lower limit of median class Cf = cumulative frequency of the class preceding the median clas f = frequency of the median class, i = class interval. 19

20. Application of median Median is used when distribution is markedly skewed and when small number of measurements is given. Usually, a median is computed when we are interested, whether cases fall with in the upper or lower half of distribution and not particularly in how far they are from the central point. 20

21. Mode It is the value or values that occur most often in distribution. It is nothing but the most frequently occurring value. It is symbolized by ‘Mo’. E.g. Following observations has been recorded 10,15, 12, 13, 14, 15, 12,11,13,15 Value Frequency 1 11 1 12 2 13 2 14 1 15 3 15 occurs is having highest frequency so 15 is mode 21

22. Merits of Mode It is simple and precise. It is easy to calculate and can be determined by a mere observation of data. It is not affected by extreme values of variables. It is applicable to quantitative as well as qualitative data. It can be calculated from a grouped frequency distribution having open-end classes and with class intervals of unequal lengths. The value of mode can be ascertained graphically. 22

23. Demerits of Mode It is largely affected by sampling fluctuations. It is not always possible to find out well-defined mode. It can not be used for further mathematical calculations. It is not based on all observations. The value of mode can not be determined in bimodal distribution. 23

24. Mode of individual observations The value which is occurring maximum number of times is the modal value. Mode in discrete series: It can be determined by looking to that value of variable around which the items are most heavily concentrated. In frequency distribution, modal class is that class interval which contains the highest frequency of observations and the mode is the mid- point of this class interval. 24

25. Mode of continuous series Mode = Lm + d 1 /(d 1 +d 2 ) x c Lm = lower limit of modal class, d 1 = frequency of modal class – frequency of the preceding class, (ignore signs) d 2 = frequency of modal class – frequency of the succeeding class ( ignore signs). c = class interval of modal class. 25

26. Significance of central tendency A measure of central tendency tells only about the general level of magnitudes of the distribution, but it fails to give any idea of the variability of the observations, as to the individual values scatter around to what extent or spread about the average. To know the extent of the spread about these variations the measure called Dispersion is used. In general, greater the spread from the mean, greater is the variability. The different measures of variability (dispersion) are range, mean deviation, variance, standard deviation, etc.   26