Following are the important methods of studying dispersion -
1) Numerical Methods :
a) Methods of limits
ii) Inter-quartile range
iii) Percentile Range
b) Methods of average deviation:
i) Quartile Deviation
ii) Mean Deviation
iii) Standard Deviation
(2) Graphic Method - Lorenz Curve :
i) Range : The difference between the value of the smallest item and the value of the largest item of the series is called range.
Range = Largest item – Smallest item
Co-efficient of Range = L - S/ L + S
Merits of Range : -
- Simple and easy to be computed.
- It takes minimum time to calculate.
- Helpful in quality control of products.
Demerits of Range : -
Not based on all the items.
- Subject to fluctuation/uncertain measure.
- Cannot be computed in case of open-end distributions.
- As it is not based on all the values it is not considered as a good or appropriate measure.
ii) Inter-Quartile Range : Inter-quartile range represents the difference between the third-quartile and the first quartile. It is also known as the range of middle 50% values.
Inter-quartile range = Q3 – Q1
- It is easy to calculate.
- Can be measured in open end distributions.
- It is least affected by the uncertainty of the extreme values.
- It does not represent all the values.
- It is an uncertain measure.
iii) Percentile Range : It is the difference between the values of the 90th and 10 th percentile. It is based on the middle 80% items of the series.
Percentile Range = P90 – P10
Merits and Demerits : Its use is limited. Percentile range has almost the same merits and demerits as those of interquartile range.
iv) Quartile Deviation / Semi – Interquartile Range :
Quartile Deviation gives the average amount by which the two quartiles differ from the median. Quartile deviation is an absolute measure of dispersion.
It is calculated by using the formula
- It is easy to calculate and understand.
- It has a special utility in measuring variation in open end distributions.
- QD is not affected by the presence of extreme values
- It is very much affected by sampling fluctuations.
- It does not give an idea of the formulation of the series.
- It is not capable of further algebraic treatment.
v) Mean Deviation ( δ) : Mean Deviation is also known as average deviation or first measure of dispersion. It is the average difference between the items in a distribution and the median mean or mode of that series.
- It is simple to understand and easy to compute.
- Least affected by the extreme values.
- It can be computed from any average, mean, median or mode.
- Signs are ignored therefore mathematically it is incorrect and not a significant measure.
- Does not give accurate results.
vi) Standard Deviation (σ) :
Standard Deviation was introduced by Karl Pearson in 1823. It is the most important and widely used measure of studying dispersion, as it is free from those defects from which the earlier methods suffer and satisfies most of the properties of a good measure of dispersion.
Standard Deviation is the square root of the average of the square deviations from the arithmetic mean of a distribution.
Co-efficient of Standard Deviation :
Standard deviation is an absolute measure. When comparison of variability in two or more series is required, relative measure of standard deviation is computed which is called coefficient of standard deviation
Computation of Standard Deviation :
Coefficient of S.D. = σ / X
Variance = σ2
Coefficient of variation : Coefficient of variation is used for the comparative study of stability or homogeneity in more than two or more series.
Merits : -
- Based on all the items.
- Well-defined and definite measure of dispersion.
- Much importance is given to the extreme values.
vii) Graphical Method - Lorenz Curve :
This curve was devised by Dr. Max O‘Lorenz. He used this technique to show inequality of wealth and income of a group of people. It is simple, attractive and effective measure of dispersion, yet it is not scientific since it does not provide a figure to measure dispersion.
Merits : -
- Easy to understand from the graph.
- Comparison can be done in two or more series.
- Concentration or density of frequency can be known from the curve.
Demerits : -
It is not a numerical measure; therefore it is not definite as a mathematical measure.
- Drawing of curve requires more time and labour