Measures of Dispersion - Revision Notes

 CBSE Class 11 Economics

Revision Notes
Chapter - 6
MEASURES OF DISPERSION

Points to remember

Dispersion is a measure of the variation of the items from central value.
The measures of dispersion is important to compare uniformity, consistency and reliability amongst variables / series.

Absolute measures of dispersion are expressed in terms of original unit of series.

Relative measures are expressed in ratios or percentage of average, also known as coefficients of dispersion.

Measures of Dispersion:
(i) Range
(ii) Inter quartile range
(iii) Quartile deviation or Semi-Inter-quartile range
(iv) Mean deviation
(v) Standard Deviation
(vi) Lorenz curve

Range: Range is defined as the difference between two extreme observations i.e. the largest and the smallest value.

Symbolically, R = L-S
Where R = Range
L = Largest Value
S = Smallest value
Coefficient of range = 

Inter Quartile Range: 
Inter quartile range is the difference between upper quartile and lower quartile.
Inter-quartile range = 
Where  = Third quartile or upper quartile.
 = First quartile or lower quartile

Quartile Deviation:
Quartile deviation is known as half of difference of third quartile () and first quartile (). It is also known as semi inter quartile range.

Where Q.D. = Quartile deviation
 = Third quartile or upper quartile.
 = First quartile of lower quartile.
Coefficient of quartile deviation= Q3−Q1Q3−Q1

Mean Deviation:
Mean deviation/average deviation is the arithmetic mean of the deviations of various items from their average (mean, median or mode) generally from the median.

Calculation of mean deviation
Individual Series 
Discrete Series 
Continuous Series 

Where,
MD = Mean deviation
| D | = Deviations from mean or median ignoring + Signs
N = Number of item (Individual Series)
N = Total number of Frequencies (Discrete and continuous series)
F = Number of frequencies.
Coefficient of mean deviation

Merit of Mean deviation:

  1. As in case of X, every term is taken in account hence, it is certainly a better measure than other measures of dispersion i.e. Range, Percentile Range or Quartile Range.
  2. Mean deviation is extensively used in other fields such as Economics, Business, Commerce or any other field of such type.
  3. It has least sampling fluctuations as compared to Range, Percentile Range and Quartile Deviation.
  4. When comparison is needed this is perhaps the best measure between two or more series.
  5. This calculation has its base upon measurement than an estimate.
  6. Mean Deviation is rigidly defined; one of the main focus point of any measure used for statistical Analysis.
  7. It we calculate it from median it is less affected by extreme terms.
  8. As it is based on the deviations about an average, it gives us better measure for comparison.

Demerits of Mean Deviation:

  1. If average is in fractions, it is difficult to compile M.D.
  2. Main property is absent, It is not capable of further Algebraic Treatment.
  3. Not so easy to calculate to calculate X, M or Z first and then to go for other measures.
  4. If it is calculated from Z it is not much reliable as Mode (Z) is not the true representative of the series.
  5. M.D. and its co-efficient taken from X, M and Z often differ.
  6. As +,- signs are ignored which is not possible mathematically. Algebraically we have to proceed for Standard Deviation; or another measure of dispersion.
  7. As for mean, open and series cannot be taken for the true result.
  8. If Range increases in case the sample increases, Average deviation also increases but not in the same ratio.
  9. For Sociological studies, it is almost not used.

Standard Deviation:

Standard deviation is the best and widely used measure of dispersion. Standard deviation is the square root of the arithmetic mean of the squares of deviation of its items from their arithmetic mean. Calculation of standard deviation in individual series.
Actual mean method.

Where = Standard Deviation
 = Sum total of square of Deviation taken from Mean
N = Number of items

Shortcut Method or assumed mean method:

Where  = Square of deviation taken from assumed mean.

Calculation of standard deviation in discrete series:
Actual mean method or direct method

Where  = S.D.
 = Sum total of the squared deviations multiplied by frequency
N = Number of pair of observations.

Shortcut method or assumed method:

 = S.D.
 = Sum total of the squared deviations Multiplied by frequency
 = Sum total of deviations multiplied by frequency.
N = Number of pair of observations.

Step deviation method:
X C
 = Standard Deviation
 = Sum total of the squared step deviations multiplied by frequency.
 = Sum total of step deviations multiplied by frequency
C = Common factor
N = Number of pair of observation

Individual Series:
1. Actual Mean Method

2. Assumed Mean Method

Discrete/Continuous Series:
1. Actual Mean Method

2. Assumed Mean Method

3. Step Deviation Method
X C

Merits of standard donation:
1. Based on all values
2. Rigidly defined
3. Less effect of fluctuations
4. Capable of algebraic treatment

Demerits of standard donation:
1. Difficult to compute
2. More stress on extreme items
3. Dependent on unit of measurement.

Coefficient of variation:
When two or more groups of similar data are to be compared with respect to stability (or uniformly or consistency or homogeneity). Coefficient of variation is the most appropriate measures. It is the ratio of the standard deviation to the mean.

Where C.V. = Coefficient of variation
= Standard deviation
X = Arithmetic mean

LORENZ CURVE:
The Lorenz curve devised by Dr. Max O. Lorenz is a graphic method of studying dispersion.
The Lorenz curve always lies-below the line of equal distribution, unless the distribution is uniform.
The Area between the line of equal distribution and the plotted curve gives the extent of inequality in the items. The larger the area, more is the inequality.

Application Lorenz Curve:
(i) Distribution of income
(ii) Distribution of wealth
(iii) Distribution of wages
(iv) Distribution of production
(v) Distribution of population

Construction of Lorenz Curve:
1. Series is converted into a cumulative frequency series. The cumulative sum of items is assumed to be 100 and different items are converted into percentage of the cumulative sum.
2. Cumulative sum of frequency is assumed to be 100 and different Frequencies are converted into percentage of sum of frequency.
3. Cumulative frequencies are plotted on x-axis and cumulative items are plotted on y-axis of graph.
4. On both axis values are plotted from 0-100.
5. A diagonal line(0 on X-axis and 100 on Y-axis are joined by a line). It is called line of equal distribution.
6. Actual data are plotted by joining different points. This is Lorenz Curve.
7. Lesser distance between the line of equal distribution and line of actual distribution shows lesser dispersion and so on.