ABSOLUTE MEASURES OF DISPERSION

MEANING OF DISPERSION

In Statistics,This term is used commonly to mean scatter,Deviation, Fluctuation, Spread or variability of

data. The degree to which the individual values of the

variate scatter away from the average or the central value, is called a dispersion.

Why study dispersion??

Dispersion is used to denote a lack of uniformity in item values of a given variable.

• Important tool of statistics for biologist because biological phenomena are more variable than that of physical and chemical sciences.

• Absolute Measures of Dispersion: The measures of dispersion which are expressed in terms of original units of a data are termed as Absolute Measures.

• Relative Measures of Dispersion: Relative measures of dispersion, are also known as coefficients of dispersion, are obtained as ratios or percentages. These are pure numbers independent of the units of measurement and used to compare two or more sets of data values.

Absolute Measures• Range• Quartile Deviation• Mean Deviation• Standard DeviationRelative Measure• Co-efficient of Range• Co-efficient of Quartile Deviation• Co-efficient of mean Deviation• co-efficient of Variation.

RANGE

Difference b/w the maximum and the minimum oservations in the data set is the range for that data set.

R = L-Swhere,R= rangeL=largest value of variableS=Smallest value of variable

For instance, 18 plots of 1 cubic m were selected . No. Of earthworm in each

plot were 435, 420, 416, 436, 439, 506, 415, 469, 500, 496, 450, 465, 441, 475, 491, 481, 431 and 471.

For calculating range,Arrange data in array : 415, 416, 420, 431, 435,436, 439, 441,

445, 450, 465, 471, 475, 481, 491, 496, 500, 506.R=L-S

Therefore, R= 506-416 = 90

For grouped data,Range= upper limit of last class- lower limit of first classFor instance,

Here, L= 80 , S= 11 ; Range = 80-11 = 69

No. of clusters 11-20 21-30 31-40 41-50 51-60 61-70 71-80

No. of plants 6 10 12 15 11 7 4

MERITS OF RANGE :-• Easiest o calculate and simplest to understand.• Gives a quick answer.

DEMERITS OF RANGE :-• It gives a rough answer.• It is not based on all observations.• It changes from one sample to the next in a

population.• It can’t be calculated in open-end distributions.• It is affected by sampling fluctuations.

MEAN DEVIATIONIt is the average of the absolute values of thedeviation from the mean (or median or mode).Mean deviation or MD or

where, MD= mean deviation; x = deviation from actual mean = not considering sign (+ve or -ve )

Deviation,

Nx

XXx

MERITS AND DEMERITS OF MEAN DEVIATION

Mean deviation is easy to calculate but since mean deviation has less mathematical value , it is rarely applied for biological statistical analysis.

It is also not meaningful because negative sign of deviations is ignored.

QUARTILE DEVIATIONThe half distance between 75th percentile i.e., 3rd quartile (Q3 ) and 25th percentile i.e., 1st quartile (Q1 ) is called Quartile deviation or semi-interquartile range.

Formula : (Q3- Q2) + (Q2- Q1) 2 = Q3- Q1

2For grouped data, Q1= ; Q3=

Here, L= Lower limit of class interval where Q1 and Q3 falls

F= Cumm. frequency just above the Q1 and Q3 classes.fq= frequency of Q1 and Q3 classes. i= length of class interval.

ifqFN

)4/( i

fqFNL

)4/3(

Cummulative frequency table-

Make out the solution for N=48

Length of Earthworm(cm)

15-20 20-25 25-30 30-35 35-40 40-45 45-50 50-55 55-60

frequency 4 3 8 9 14 3 3 2 2

Class interval

15-20 20-25 25-30 30-35 35-40 40-45 45-50 50-55 55-60

f 4 3 8 9 14 3 3 2 2

c.f. 4 7 15 24 38 41 44 46 48

MERITS OF QUARTILE DEVIATION• It is a better measure of dispersion as it is not based

on two extreme values like range but rather on middle 50 % observations.

• It is the only measure of dispersion which can be used for open end distributions.

STANDARD DEVIATIONIt may be defined as “the square root of the

airthmetic mean of the squares of deviations from the airthmetic mean.”

MERITS AND DEMERITS OF STD. DEVIATION• Std. Dev. summarizes the deviation of a large

distribution from mean in one figure used as a unit of variation.

• It indicates whether the variation of difference of a individual from the mean is real or by chance.

• Std. Dev. helps in finding the suitable size of sample for valid conclusions.

• It helps in calculating the Standard error.DEMERITS- It gives weightage to only extreme values.

The process of squaring deviations and then taking square root involves lengthy calculations.

RELATIONSHIP OF STANDARD DEVIATION TO OTHER MEASURES OF DISPERSION:

There is a very interesting relatinship between the standard deviation, Mean deviation and Quartile deviation :

QD: MD: SD = 2/3 : 4/3 : 1 = 10 : 12 : 15