We know that averages are representatives of a frequency distribution but they fail to give a complete picture of the distribution. They do not tell anything about the scatterness of observations within the distribution.
Suppose that we have the distribution of the yields (kg per plot) of two paddy varieties from 5 plots each.
The distribution may be as follows:
Variety I 45 42 42 41 40
Variety II 54 48 42 33 30
It can be seen that the mean yield for both varieties is 42 kg. But we can not say that the performance of the two varieties is same. There is greater uniformity of yields in the first variety whereas there is more variability in the yields of the second variety. The first variety may be preferred since it is more consistent in yield performance. From the above example, it is obvious that a measure of central tendency alone is not sufficient to describe a frequency distribution. In addition to it we should have a measure of scatterness of observations. The scatterness or variation of observations from their average is called the dispersion. There are different measures of dispersion like the range, the quartile deviation, the mean deviation and the standard deviation.
Range
The simplest measure of dispersion is the range. The range is the difference between the minimum and maximum values in a group of observations for example, suppose that the yields (kg per plot) of a variety from five plots are 8, 9, 8, 10 and 11. The range is (11 - 8) = 3 kg. In practice the range is indicated as 8 - 11 kg.
Range takes only the maximum and minimum values into account and not all the values. Hence it is a very unstable or unreliable indicator of the amount of deviation. It is affected by extreme values. In the above example, if we have 15 instead of figure 11, the range will be (8 - 15) = 7 kg. In order to avoid these difficulties another measure of dispersion called quartile deviation is preferred.
Quartile Deviation
We can delete the values below the first quartile and the values above the third quartile. It is assumed that the unusually extreme values are eliminated by this way. We can then take the mean of the deviations of the two quartiles from the second quartile (median). That is,
This quantity is known as the quartile deviation (Q.D.).
The quartile deviation is more stable than the range as it depends on two intermediate values. This is not affected by extreme values since the extreme values are already removed. However, quartile deviation also fails to take the values of all deviations
Mean
Deviation
Mean deviation is the mean
of the deviations of individual values from their average. The average may be
either mean or median. For raw data the mean deviation from the median is the
least. Therefore, median is considered to be most suitable for raw data. But
usually the mean is used to find out the mean deviation. The mean deviation is
given by
M.D. =
for raw data and M.D.
= 
for grouped data
All positive and negative
differences are treated as positive values. Hence we use the modulus symbol | |. We have to read as “modulus
”. If we take 
as such, the sum of
the deviations,
will be 0. Hence, if the signs are not eliminated the mean
deviation will always be 0, which is not correct.
The steps of computation are
as follows :
Step
1: If the classes are not continuous we have to make them continuous.
Step 2: Find out the mid
values of the classes (mid - X = x).
Step 3: Compute the mean.
Step 4: Find out for all values of x.
Step 5: Multiply eachby the corresponding
frequencies.
Step 6: Use the formula.
The mean deviation takes all
the values into consideration. It is fairly stable compared to range or
quartile deviation. Since, the mean deviation ignores signs of deviations, it is
not possible to use it for further statistical analysis and it is not stable as
standard deviation which is defined as:
Standard
Deviation
Ignoring the signs of the
deviations is mathematically not correct. We may square the deviation to make a
negative value as positive. After calculating the average squared deviations,
it can be expressed in original units by taking its square root. This type of
the measure of variation is known as Standard Deviation.
The standard deviation is
defined as the square root of the mean of the squared deviations of individual
values from their mean. Symbolically,
Standard Deviation
(S.D.) or 
This is called standard
deviation because of the fact that it indicates a sort of group standard spread
of values around their mean. For grouped data it is given as
Standard Deviation (S.D.) or 
The sample standard
deviation should be an unbiased estimate of the population standard deviation
because we use sample standard deviation to estimate the population standard
deviation. For this we substitute n - 1 for n in the formula. Thus, the sample
standard deviation is written as
For grouped data it is given
by
where,
C = class
interval
d = (x - A)
/ C as given under mean.
The square of the
standard deviation is known as the variance. In the analysis of variance
technique, the term
is called the sum of
squares, and the variance is called the mean square. The standard deviation is
denoted by s in case of sample, and by s (read ‘sigma’) in case of population.
The standard deviation is
the most widely used measure of dispersion. It takes all the items into
consideration. It is more stable compared to other measures. However, it will
be inflated by extreme items as is the mean.
The standard deviation has
some additional special characteristics. It is not affected by adding or
subtracting a constant value to each observed value. It is affected by
multiplying or dividing each observation by a constant. When the observations
are multiplied by a constant, the resulting standard deviation will be
equivalent to the product of the actual standard deviation and the constant.
(Note that division of all observations by a constant, C is equivalent to
multiplication by its reciprocal, 1/C. Subtracting a constant C is equivalent
of adding a constant, - C.)
The standard deviations can
be pooled. If the sum of squares for the first distribution with n1
observations is SS1, and the sum of squares for the second
distribution with n2 observations is SS2, then the pooled standard deviation is given
by,
Measures
of Relative Dispersion
Suppose that the two
distributions to be compared are expressed in the same units and their means
are equal or nearly equal. Then their variability can be compared directly by
using their standard deviations. However, if their means are widely different
or if they are expressed in different units of measurement, we can not use the
standard deviations as such for comparing their variability. We have to use the
relative measures of dispersion in such situations.
There are relative
dispersion in relation to range, the quartile deviation, the mean deviation,
and the standard deviation. Of these, the coefficient of variation which is
related to the standard deviation is important. The coefficient of variation is
given by,
C.V. = (S.D. / Mean) x 100
The C.V. is a unit-free
measure. It is always expressed as percentage. The C.V. will be small if the
variation is small of the two groups, the one with less C.V. is said to be more
consistent.
The coefficient of variation
is unreliable if the mean is near zero. Also it is unstable if the measurement
scale used is not ratio scale. The C.V. is informative if it is given along
with the mean and standard deviation. Otherwise, it may be misleading.
