Measures of Dispersion (Variability)

Measures
of
Dispersion (Variability)
Dr. Anil V Dusane
Sir Parashurambhau College
Pune, India
anildusane@gmail.com
www.careerguru.co.com
1

Measures of dispersion
• A measure of central tendency tells only about the general level of magnitudes
of the distribution but fails to give any idea of the variability of the
observations.
• To know the extent of the spread about these variations one has to resort to
another measure called Dispersion.
• The dispersion finds how individual values are dispersed away from the
average.
• Definition Dispersion (variability): It the extent to which the values of a data
set differ from its computed mean.
• In general, greater the spread from the mean, greater is the variability.
2

Measures of dispersion (variability)
• There are different measures of variability (dispersion) 1. 1.
1. Range
2. Mean deviation
3. Variance
4. Standard deviation
5. Standard error
• These measures are used to describe the structure of a
frequency distribution.
• These measures are extremely useful in comparing two or
more series with respect to their variability.
3

Range
• It is simply the difference between the highest and the lowest
values.
• Range = Highest value – Lowest value.
• Merits:
1.The range is one of the simplest measures of variability to calculate.
2.Easy to calculate and understand.
• Demerits:
• 1. It ignores a good deal of information about the spread and
variability of the set of observations.
4

• Mean Deviation: It is simply the average deviation of all the values from
the mean.
• Deviation is always expressed by an equation  = X -X,
• Here X is the observation where as X is the mean.
• The deviations above the mean are negative and below mean are positive.
Mean deviation is calculated by using following formula:
• Mean deviation = ( X -X) /n
• Mean deviation can be calculated either from the arithmetic mean or
median.
5

Mean Deviation(Md)
Merits of Mean deviation:
1. It is easy to calculate.
2. It is based on all the observations.
3. It is less affected by extreme values.
4. It is a better measure for comparison.
5. It is flexible as it can be calculated from any measure of central tendency.
Demerits:
1. It can not be used for further mathematical processing.
2. It can not be calculated in open-end classes.
6

Mean Deviation for series of individual observations
• It involves the following steps:
1.Calculate the arithmetic mean.
2. Take deviations of items from the mean ( ).
3. Take summation of .
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Mean Deviation for Discrete series
• Important steps :
1. Calculate the arithmetic mean (X ).
2. Take deviations from the mean.
3. Multiply deviations with respective frequency f[dX].
4. Divide the total by the no. Of observations.
5. Calculate mean deviation by using following formula:
Mean deviation = f[dX]/f OR f (X-X)/f
Significance: Mean deviation is the easier way measuring dispersion however it
is having less statistical importance.
8

Variance
• It is an important measure in quantitative analysis of data in biological and
agricultural science.
• It is defined as the mean of squares of deviations.
• Variance is denoted by s2.
• To calculate the variance,
• i) Calculate mean deviation of the variables from the mean
• ii) square them
• iii) sum up
• iv) Sum of the squares of deviations is divided by the number of observations to get
the variance.
• Formula for calculating the variance of a grouped data is as below.
• s2=f [dX] 2/(n-1)
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Example of varianceCalculate the variance from the given data:
• Solution:
• X f fX  X d =X -X d2 f d2
• 48 8 384 52.75 -4.75 22.56 180.48
• 50 32 1600 52.75 -2.75 7.56 241.92
• 52 75 3900 52.75 -0.75 0.56 42.0
• 54 52 2808 52.75 1.25 1.56 81.10
• 56 28 1568 52.75 3.25 10.56 295.68
• 58 5 290 52.75 5.25 27.56 137.80
• f =200 fX = 10,550  f d2 = 979.90
• Arithmetic mean = fX /f = 10,550 / 200 = 52.75
• Variance ( s2 ) = f [ dX ] 2 / (n-1)
• = 979.90/200-1
• = 4.92
10
Height (class value X) 48 50 52 54 56 58
Frequency (f) 8 32 75 52 28 5

Limitations of variance
1.When the deviation and number of observations are more
then the variance becomes a large number. It is then
difficult to be expressed numerically.
2.The unit in which the variance is expressed is not in the
same unit of the observation. If the observations are made
in cm then the variance is expressed in sq.cm.
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Standard Deviation (S.D.)
• The concept of standard deviation was introduced by
Pearson (1893).
• It is the most used measure of dispersion.
• It measures the variability of a distribution i.e. the variability
of the sample around the mean.
• Therefore, larger the standard deviation, greater is the value
of variability.
• On the other hand when the value of standard deviation is
small there would be less variability indicating the more
homogeneity.
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Standard Deviation (S.D.)
• It is usually used to compare the dispersion of
two or more sets of observations.
• It is defined as a measure of variability that
indicates by how much all the values in a
distribution typically deviate from the mean.
• S.D. is nothing but the square root of the
variance.
• S.D. of sample is denoted by s and that of
population is by  (lower case of Greek sigma)
13

Merits of Standard Deviation
1. It is useful for the comparison of the degree of variation of two or more
samples or distributions.
2. Standard deviation mean gives good idea about the distribution.
3. It is used in the analysis of the statistical series such as correlation,
skewness, etc.
4. It is less affected by sampling fluctuations.
5. It is based on all the observations.
6. Standard deviation will be more representative in case of smaller
distribution.
7. It is essential for calculating Coefficient of Variation (C.V.) 14

Demerits of SD
1. Standard deviation is difficult to calculate as compared to the other
measures of dispersion such as mean deviation, range etc.
2. It is an absolute measure of dispersion (i.e. expressed in terms of original
units in which data is given) therefore it is of limited utility in comparing
the series which are expressed in different units.
3. It gives more weightage to values that are away from the mean and less to
those items which are near to mean.
4. S.D. can not be computed for open-end distribution.
15

S.D. for individual observations
• It involves the following important steps:
1.Calculate the Arithmetic mean (AM).
2.Find the deviation of individual observation from the mean (dX).
3.Square these deviations and carry out summation ( [ dX ]2 ).
4.Divide the sum by total no. of observations to get Variance.
5.Take square root to find out S.D. by using following formula:
• s= [dX] 2 / (n-1) for ungrouped data series.
• s = f [ dX ] 2 / (n-1) for discrete and grouped data series.
16

Applications of Standard Deviation
1. It is used in correlating and comparing different samples.
2.It has application in finding the suitable size of sample for
valid conclusion.
3.It is used in finding ‘standard error’ that determines whether
the difference between means of two similar samples is due to
the chance or it is real.
4.The value of mean and SD is used to comment on the
population on the basis of observation of sample.
17

Coefficient of Variation (C.V.)
• In probability theory, the coefficient of variation (CV) is a measure of dispersion of
a probability distribution.
• CV is used when we want to compare the dispersion of two different characters in
the same population.
• This measurement is expressed in percentage. E.g. From the same population if we
like to study the length of the pod and number of seeds/pod and to compare the
dispersion of both the characters we must calculate the CV.
• Coefficient of variation = standard deviation/Mean x 100
• It is defined as the ratio of the standard deviation to the mean.
• The coefficient of variation should only be computed for data measured on a ratio
scale.
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Merits of CV
1. It is the best measure to compare variability of two series or sets of
observation.
2. The coefficient of variation is useful because the standard deviation
of data must always be understood in the context of the mean of the
data.
3. The coefficient of variation is a dimensionless number. So when
comparing between data sets with different units or widely different
means, one should use the coefficient of variation for comparison
instead of the standard deviation.
19

Demerits of CV
• One must know standard deviation before calculating C.V.
• When the mean value is near zero, the coefficient of
variation is sensitive to small changes in the mean, limiting
its usefulness.
• Unlike the standard deviation, it cannot be used to
construct confidence intervals for the mean.
20

Applications of CV
• This is useful, for instance, in the construction of hypothesis
tests or confidence intervals.
• CV is useful to get an idea or compare two different
populations w.r. to the dispersion of a character.
• More the CV more is the inconsistency about the dispersion
of character.
21

Standard Error (SE)
• It is the mean of the means.
• The standard deviation of the means is termed as the standard error.
• It is found that when the sample size is large the standard error is
small and vice-versa.
• S.E. (X) = s /n
• Where s = Standard deviation of the sample.
• N = total no. of observations.
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Importance of S.E.
 The magnitude of S.E. confirms how precise is the mean of the
sample is.
 It is more advantageous than the S.D. as it is not influenced by the
extreme values.
 It is commonly used in the testing of the hypothesis.
 It can also be used when the ratio between the two classes is to be
tested.
 If the ratio of deviation (the difference between expected and
obtained values)./ S.E. less than 1.96, the obtained results are said to
be a good fit.
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Measures of Dispersion (Variability)

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