mean median mode

students will be able to understand various measures of central tendency and also will be able to calculate mean median and mode for individual discrete and continuous series.

1. Dr. Mitali Gupta Dr. Ambedkar Institute of Management Studies & Research Measures of Central Tendency

2. A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. As such, measures of central tendency are sometimes called measures of central location. Definition

3. Measures of Central Tendency Mean- mean is the total of all the values, divided by the number of values . Median- median is the middle number in a list of numbers ordered from smallest to largest . Mode- mode is the value that appears most often in a set of data .

4. Frequency A frequency is the number of times that a particular data value occurs in a data set. A series of data that is formed along with the frequencies of their occurrences is called a frequency series. A frequency series is again, of three types viz. Individual series     Discrete series, and   Continuous series. Frequency Series

5. Mean The mean (or average) is the most popular and well known measure of central tendency. It can be used with both discrete and continuous data, although its use is most often with continuous. The mean is equal to the sum of all the values in the data set divided by the number of values in the data set. So, if we have values in a data set and they have values x1,x2,x3… xn , the sample mean, usually denoted by (pronounced "x bar"), is:

6. Merits of Arithmetic Mean: 1 . It is a mathematical mean. 2. It is easy to calculate. 3. It is always definite. 4. It establishes a simple relation among the values of data series.

7. Demerits of Arithmetic Mean: 1 . It cannot be determined by simple visual observation of data. 2. It cannot be presented by chart or graph. 3. It is not suitable for qualitative studies. 4. Sometimes it does not show representation of series.

8. Ari t h m e t i c Mean for Individual Series ( Mar k s) X 65 55 42 58 94 86 ∑ X= 4

9. Ari t h m e t i c Mean for Discrete Series (Marks) Freq f x X F X.F 20 8 160 30 12 360 40 20 800 50 10 500 60 6 360 70 4 N=60 280 ∑fx=2460 Discrete Series

10. Arithmetic Mean (Continuous Series) Marks (X) F MV F.MV 0-10 5 5 25 10-20 10 15 150 20-30 25 25 625 30-40 30 35 1050 40-50 20 45 900 50-60 10 N=100 55 550 ∑ fmv =3300

11. Median The median is that value of the variable which divides the group into two equal parts, one part comprising all values greater and the other values less than the median .” Median is the middle value of the series when items are arranged either in ascending or descending order.

12. Merits of Median It is easy to calculate and simple to understand. In many situations median can be located simply by inspection. It is not affected by the extreme values i.e. the largest and smallest values. Because it is a positional average and not dependent on magnitude . Median is the best measure of central tendency when we deal with qualitative data, where ranking is preferred instead of measurement or counting.

13. Demerits It is not based on all the observations of the series . It is not capable of further algebraic treatment like mean, geometric mean and harmonic mean.

14. Calculation of Median (Individual Series) Wages (in ascending order): 108, 110, 112, 115, 116, 120, 140 F r o m t h e f o ll o w i n g d a t a o f t h e w a g e s o f 7 w o r ke rs co m pu t e the median wage

15. Calculation of Median (Discrete Series) From the following data find the value of median: Income (Rs) No. person (f) Cumulative freq (cf) 8 1 6 1 6 1 2 4 4 1 5 2 6 6 6 1 8 3 9 6 2 2 11 6 2 5 6 12 2 N=122

16. Calculation of Median (Continuous Series) Calculate the median for the following frequency di Marks No. students (f) Cumulative freq (cf) 5-10 7 7 10-15 1 5 2 2 15-20 2 4 4 6 20-25 3 1 7 7 25-30 4 2 11 9 30-35 3 14 9 35-40 2 6 17 5 40-45 1 5 19 45-50 1 20 N=200

17. Mode The mode is the number that appears most frequently in a set. A set of numbers may have one mode, more than one mode, or no mode at all.

18. Merits of Mode It is easy to understand and simple to calculate. It is not affected by extremely large or small values . It can be located just by inspection in ungrouped data It can be located graphically.

19. Demerits of Mode It is not well defined. It is not based on all the values. It is stable for large values so it will not be well defined if the data consists of a small number of values. It is not capable of further mathematical treatment.

20. Calculation of Mode (Individual Series) From the following data calculate the value of mode: 3 X No. of times occurred 5 3 2 8 4 2 5 5 3 Max 4 8 1 5 9 1 9 3 Mo=5 4

21. Calculation of Mode (Discrete Series) Calculate the mode from the following data: size of garments No. people 28 1 29 2 30 4 31 6 5 Max Mo=31 32 5 33 1 5

22. Calculation of Mode for Continues Series

23. THANK YOU