Session 8 MEASURES Of DISPERSION

Learning Objectives  Why do we study dispersion?  Measures of Dispersion Range Standard Deviation Variance

DISPERSION Definition : Dispersion is the degree of the scatter of observation about a central values. It is the degree of variation of the variable about central tendency.

CENTRAL TENDENCY  Means:  Family A:( )/6 = 60/6 = 10  Family B:( )/6 = 60/6 = 10  Family C:( )/6 = 60/6 = 10  Medians:  For each, average the middle two & get 10 But the distributions clearly differ!

Measures of Dispersion Measures of Dispersion Absolute Measures Relative Measures

Measures of Dispersion Range Inter Quartile Range Quartile Deviation Variance Standard Deviation Mean Deviation

RANGE  It is the difference between the value of smallest observation and the value of the largest observation present in the distribution.  R = L-S  Coefficient of Range= (L-S)/(L+S)

MERITS OF RANGE  Simple to understand  easy to calculate  Quickest way to get a measure of dispersion

LIMITATIONS OF RANGE  Not based on all observation  Influenced by extreme value  It Cannot computed for open – end data  It Fail to explain the scatter around average

USES OF RANGE  Used for making quality control chart  For study the fluctuation in financial and share market

Variance Definition Variance is average squared deviation from arithmetic mean Ungrouped data σ²= Σ(x-x) ² n Grouped data σ²= Σf(x-x) ² n

Standard Deviation Standard Deviation is the square root of the variance Standard deviation is also known as root of mean squared deviation. For ungrouped data S.D. (σ)= Σ(x-x ) ² n For Grouped data S.D.(σ )= Σ(x-x ) ² n

Properties of Standard Deviation  Standard Deviation is independent of change of origin.  Standard Deviation is dependent on the change of scale.  Standard Deviation is the minimum root-mean squared deviation.

Coefficient of variation  Coefficient of variation is relative measure of dispersion based on S.D.  Coefficient of Variation= S.D./Mean  Coefficient of variation is used in problem situation where we want to compare the variability, homogeneity, stability, uniformity & consistency of two or more data set.

Skewness and Kurtosis  Skewness is refer to the study of the frequency distribution curve.  Kurtosis is concerned with the flatness or peakness of the frequency distribution curve.

(a) Positive skewed, (b) Negative skewed

Difference between Dispersion & Skewness  Dispersion is concern with the amount of variation rather than with its direction.  Skewness tell us about the direction of the variation.