Dispersion

Introduction of Dispersion, measures, ranges, quartile deviation, mean deviation, standard deviation,

1. PRESTIGE INSTITUTE OF MANAGEMENT GWALIOR Presented By:- Deepanshu Goyal Devendra Singh Rathore Md. Ibrahim Azaz Meghna Sharma Sanjay Singh Rajawat Presented to:- Prof. Rajesh Gupta Topic – Measures of Dispersion

2. INTRODUCTION OF DISPERSION An average is a single value which represents a set of values in a distribution. It is the central value which typically represents the entire distribution. Dispersion, on the other hand, indicates the extend to which the individual value fall away from the average or a the central value. This measure brings out how to distribution with the same average value may have wide differences in the spread of individual values around the central value.

3. DEFINITION “Dispersion is the measure of variations of the items”. - A.E. Bowley “Dispersion or spread is the degree of the scatter or variation of the variable about a central value”. - Brooks & Dick

4. MEASURES OF DISPERSION Absolute Measure Range Quartile Deviation Mean Deviation Standard Deviation Relative Measure Co-Efficient of Range Co-Efficient of Quartile Deviation Co-Efficient of Mean Deviation Co- Efficient of Variance

5. RANGE It is defined as the difference between the smallest and the largest observations in a given set of data. Formula is R = L – S Ex. Find out the range of the given distribution: 1, 3, 5, 9, 11 The range is 11 – 1 = 10 .

6. QUARTILE DEVIATION It is the second measure of dispersion, no doubt improved version over the range. It is based on the quartiles so while calculating this may require upper quartile (Q3) and lower quartile (Q1) and then is divided by 2. Hence it is half of the difference between two quartiles it is also a semi inter quartile range. The formula of Quartile Deviation is (Q D) = Q3 - Q1 2

7. MEAN DEVIATION Mean Deviation is also known as average deviation. In this case, deviation taken from any average especially Mean, Median or Mode. While taking deviation we have to ignore negative items and consider all of them as positive. The formula is given below

8. MEAN DEVIATION The formula of MD is given below MD =  d N (deviation taken from mean) MD =  m N (deviation taken from median) MD =  z N (deviation taken from mode)

9. STANDARD DEVIATION It is defined as “The mean of the squares of deviations of all the observation from their mean.” It’s square root is called “standard deviation”. Usually it is denoted by =

10. STANDARD DEVIATION The standard deviation is represented by the Greek letter (sigma). It is always calculated from the arithmetic mean, median and mode is not considered. While looking at the earlier measures of dispersion all of them suffer from one or the other demerit i.e. Range – It suffer from a serious drawback considers only 2 values and neglects all the other values of the series.

11. STANDARD DEVIATION Quartile deviation considers only 50% of the item and ignores the other 50% of items in the series. Mean deviation no doubt an improved measure but ignores negative signs without any basis. Karl Pearson after observing all these things has given us a more scientific formula for calculating or measuring dispersion. While calculating SD we take deviations of individual observations from their AM and then each squares. The sum of the squares is divided by the number of observations. The square root of this sum is knows as standard deviation.

12. NO OF YOUNG ADULTS VISIT TO THE LIBRARY IN 10 DAYS (X) d=X - A.M d 2 40 -26 676 44 -22 484 54 -12 144 60 -6 36 62 -4 16 64 -2 4 70 4 16 80 14 196 90 24 596 96 30 900 N=10  X=660  d 2 = 3048

13. STANDARD DEVIATION AM =  X N = 660 = 66 AM 10   SD = √ ∑ d 2 N SD =√ 3048 = 17.46 10