Dr. Ansari Khurshid Ahmed -- Characteristics & Limitations of Median and Mode-Illustrations.pptx

Median and Mode are Measures of Central Tendency having their own characteristics and limitations. Median is a counting average whereas Mode is inspectional average. Both Median and Mode are used under some special circumstances. Median is useful when a data is skewed or contains scores in the form of ranks or grades. Mode is useful to understand most trending item.

1. Measures of Central Tendency (Characteristics and Limitations -Median & Mode) (Few Illustrations) For B.Ed. Students

2. Dr. Ansari Khurshid Ahmed Associate Professor Marathwada College of Education Chh . Sambhajinagar (Aurangabad)

3. Characteristic of Median Median is a reliable measure of central tendency when a series of scores contains few very high or very low scores at its extremes. Illustration: X 1 : 10 11 12 13 14 15 16 17 18 M = 126/9 = 14 Median = 14 X 2 : 10 11 12 13 14 15 16 17 90 M = 198/9 = 22 Median = 14 Discussion : Mean of First Series of scores (which is not skewed) is 14 which is acceptable representative of all the scores. But the second series of scores is skewed as 8 out of total 9 scores are at the lower end of the series whereas only one score 90 is at the upper end of the series. There is a huge gap between 17 and 90. The mean of second series is 22 which does not seem to be a representative of majority of scores whose values are less than 22. But Median is 14 which can be considered as representative of the second series of scores, as 90 is an exceptional case.

4. Characteristic of Median Median is a reliable measure of central tendency in case of open end table. Illustration: (Open End Table) A table that does not have intervals of specified length at upper end or lower end (or both the ends) of the table. Median = l + (N/2 – F)/f * i = 59.5 + (20 – 16)/10 * 10 = 59.5 + 4 =63.5 Discussion : In order to compute Mean, mid-point of each class interval is required. The mid-point of the class interval ‘Less than 50’ can not be determined accurately. Hence in this case, Mean can not be determined accurately. But Median can be determined accurately because Mid points of all the class intervals are not required to compute Median. Class Interval (X) Frequency (f) 80 - 89 6 70 – 79 8 60 – 69 10 50 – 59 5 Less than 50 11

5. Characteristic of Median When scores of a series are expressed in the form of Rank or Grade then Median can be determined. Illustration: X : A B D C A C E D A B D (Scores in the form of Rank or grades) X : A A A B B C C D D D E Median = C Discussion : In order to compute Median, scores of the series are to be arranged in either ascending order or descending order. The grade at the centre of the given series, when arranged in descending order, is C. Hence C is Median of the series.

6. Characteristic of Median Median can be easily determined without converting units of measurement when units are unequal. Illustration: X : 2m 10cm 30cm 15cm 0.5m 55cm 1.2m 62cm 12cm 45cm 0.23m X : 10cm 12cm 15cm 0.23m 30cm 45cm 0.5m 55cm 62cm 1.2m 2m (arranged in ascending order) Median = 45cm Discussion : In order to compute Median, scores of the series are to be arranged in either ascending order or descending order for which there is no need to convert the units of measurement. But for computing Mean, all the scores have to be in the same unit in order to add them.

7. Limitation of Median For computing Median, data have to be arranged in ascending or descending order which goes difficult for a long series of scores Illustration: X : 17 15 19 20 25 78 44 21 12 62 51 37 28 92 48 52 64 11 33 32 81 54 52 53 61 7 41 20 23 24 27 90 89 55 24 42 13 63 36 42 40 10 20 15 16 51 17 81 9 42 32 21 51 62 47 45 23 29 36 34 57 59 54 56 Discussion : In order to compute Median, scores of the series are to be arranged in either ascending order or descending order. In the above example, the series contains 64 scores. It will be difficult to arrange all 64 scores manually in ascending or descending order. But scores of a small series can be arranged easily without any error.

8. Limitation of Median Median can not be used to compute important statistical measures like Standard Deviation and Coefficient of Correlation. Illustration: Standard Deviation = σ = √(Ʃx 2 /N) where x = X – M Coefficient of Correlation = r = [ Ʃ(x i – M x )(y i – M y )]/[√(x i – M x ) 2 (y i – M y ) 2 ] Where M x = Mean of the values of x variable M y = Mean of the values of y variable Discussion : The formula of Standard Deviation and Coefficient of Correlation clearly reveal that unless we compute Mean, we can not compute them. If we know the value of Median or Mode as measure of central tendency, even then for computing the above two important statistical measures, value of Mean is required.

9. Limitation of Median Median can not be used when due weightage is to be given to all the scores. Illustration: X : 12 15 19 25 26 42 43 44 45 Median = 26 Mean = (12+15+19+25+26+42+43+44+45)/9 = 279/9 = 30.11 Discussion : In order to compute Mean, all the scores of the series have to be added. Which indicates that while computing Mean, weightage of each and every score of the series is taken into account. But while computing Median, only the score which is at the middle of the series is considered. Values of the scores above and below the middle score are not considered for computing Median.

10. Limitation of Median Median can not be used when due weightage is to be given to all the scores. Illustration: X : 12 15 19 25 26 42 43 44 45 Median = 26 Mean = (12+15+19+25+26+42+43+44+45)/9 = 279/9 = 30.11 Discussion : In order to compute Mean, all the scores of the series have to be added. Which indicates that while computing Mean, weightage of each and every score of the series is taken into account. But while computing Median, only the score which is at the middle of the series is considered. Values of the scores above and below the middle score are not considered for computing Median.

11. Characteristic of Mode Mode can be easily determined. Illustration: X : 10 11 12 13 15 15 15 19 25 42 43 44 45 45 Mode = 15 Discussion : In order to compute Mode, we have to simply find out the score which occurs most repeatedly, just by inspecting the given series of scores. But for computing Median or Mean, we have to go through a lengthy procedure and use formula.

12. Characteristic of Mode Mode is an average which is most frequently used in the day today life. Illustration: Suppose we want to know which car is most popular, we observe frequency (number) of the cars running on the road or frequency (number) of the cars sold from various showrooms. The car whose frequency is highest out of the cars running on the road or sold from various showrooms, is considered as most popular car or car of the season. Here we are making use of the concept of mode to know most trending car. Similarly, fashion or trend of any item, like shoe or dress, is understood by observing their frequency i.e. number of people using them. The shoe or dress used by highest number of people ( highest frequency) are considered as fashion or trend.

13. Limitation of Mode Mode changes due to changes in grouping scheme. Illustration: Table A Table B Crude Mode = 77 Crude Mode = 74.5 Class Interval (X) Frequency (f) 80 – 89 8 70 -- 79 11 60 -- 69 3 N 22 Class Interval (X) Frequency (f) 85--89 3 80--84 5 75--79 7 70--74 4 65--69 2 60--64 1 N 22

14. Limitation of Mode Discussion: The size of class intervals in Table A is 5. The frequency of the class interval 75 – 79 is 7 which is highest. Hence crude mode of table A will be the mid-point of the class interval 75 – 79 which is 77 . When the same scores are grouped by taking class size 10 as shown in the Table B, the frequency of the class interval 70 – 79 becomes highest i.e. 11. Hence the Crude Mode of the Table B will be mid point of the class interval 70 – 79 which is 74.5 . So when grouping of scores changes from 5 to 10, the mode changes from 77 to 74.5.

15. Limitation of Mode Mode is not reliable measure of central tendency with small groups of scores. Illustration: X 1 : 11 12 12 15 16 19 20 25 26 30 Mode = 12 X 2 : 11 12 13 15 16 19 20 25 25 30 Mode = 25 Discussion: Mode is the score which occurs most repeatedly in the series. Due to slight change in the scores ( 12 to 13 and 25 to 26), Mode changes from 12 to 25. This generally happens when a series contains few scores as shown in the above example.