VARIABILITY TEST

DISPERSION AND VARIABILITY

The measure of central tendency tell us only how typical is the average value of a set of measurements but fails to describe how spread out are these measurement about there average value. To completely describe a given data set, the study should also describe the different measures of its variability. The measure of variability is a quality that describes the spread of variability of the observations or measurements in the data set. Some common measures of variability are the rage, the variance and standard deviation and the coefficient of variation.

The Range

 Range- The range us the difference between the highest value and the lowest value in the ungrouped data set. In grouped data set, the range is the different between the upper limit of the highest class interval and the lower limit of the lowest class interval.

            R  = HV- LV ( ungrouped data)

            R_G = UL of the highest class – LL of the lowest class ( grouped data )

Eg.

            Given the ungrouped data 2,3,4,7,6,4,2,7,9,2,3,4,10

              R- 10- 2= 8

            For grouped data given below the range is

R-    75-1  = 74

Classes	Frequency (fi)	Class Mark (xi)
1-15	11	8.0
16-30	10	23.0
31-45	6	38
46-60	2	53
61-75	1	68

Properties of the range:

1.     It is quick but rough measure of dispersion

2.     The larger the value of the range, the more dispersed are the observations.

3.     It considers on the lowest and the highest values in population

The Variance

Variance- The variance, denoted by ó ²  is the mean of the squared deviation of the observations from their arithmetic mean. The formula is given by

ó ²   = Σ(Xj-ù)² / N for ungrouped data

ó ²   =  Σ f ( Xj- ùg)/ Σ f   for grouped data

Eg. From the original sample

   ó ²   = [(2-4.92)² +(3-4.92)² +(4-4.92)² +….+(10-4.92)² ] /59  = 1.47

For the grouped data

               ó ²   =[11(8-24)² +10(23-24)² +….+1(68-24)² /30  = 254

Properties of  Variance

1.     The variance is always non negative

2.     The variance is easy to manipulate for further mathematical treatment .

3.     The variance make use of all observations.

4.     The variance comes in a unit of measure that is the square of the unit of measure of the given set of values.

The Standard Deviation

            Standard Deviation: The standard Deviation, denote by ó, is the positive root of the variance. The formula is given by

Ó  =   √   ó ²        =     √Σ(Xj-ù)² / N for ungrouped data

Ó = √        ó ²        =    √Σ f ( Xj- ùg)/ Σ f   for grouped data

Eg.

Ó  =   √   1.47 =  1.21              ungrouped data

Ó = √     254  = 15.94               for grouped data

Properties of standard deviation

1.     The SD is always non negative

2.     The SD is easy to manipulate for further mathematical treatment

3.     The SD makes use of all observations.

4.     The SD has a unit of measure which is the same as the unit of measure of the given set of values.

The Coefficient of Variation

Coefficient of Variation- the CV is the ratio of the SD and the mean which is in percent.  The formula is given by

CV =( Ó/ ù) x 100

Properties of Coefficient of Variation

1.     It is unit less quantity

2.     It can used to compare the dispersion of two or more population  measured in the same or different units.