Measures of Dispersion
MEASURES OF DISPERSION
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Introduction
Dispersion is
the state of getting dispersed or spread. Statistical dispersion means the
extent to which numerical data is likely to vary about an average value. In
other words, dispersion helps to understand the distribution of the data.
Measures of Dispersion
In statistics,
the measures of dispersion help to interpret the variability of data i.e. to
know how much homogenous or heterogeneous the data is. In simple terms, it
shows how squeezed or scattered the variable is.
Types of Measures
of Dispersion
There are two
main types of dispersion methods in statistics which are:
- Absolute
Measure of Dispersion
- Relative
Measure of Dispersion
Absolute Measure of Dispersion
An absolute
measure of dispersion contains the same unit as the original data set. The
absolute dispersion method expresses the variations in terms of the average of
deviations of observations like standard or means deviations. It includes
range, standard
deviation, quartile deviation, etc.
The types of
absolute measures of dispersion are:
- Range: It is
simply the difference between the maximum value and the minimum value
given in a data set. Example: 1, 3,5, 6, 7 => Range = 7 -1= 6
- Variance: Deduct
the mean from each data in the set, square each of them and add each
square and finally divide them by the total no of values in the data set
to get the variance. Variance (σ2)
= ∑(X−μ)2/N
- Standard
Deviation: The
square root of the variance is known as the standard deviation i.e. S.D. =
√σ.
- Quartiles
and Quartile Deviation: The quartiles are values that
divide a list of numbers into quarters. The quartile deviation is half of
the distance between the third and the first quartile.
- Mean and
Mean Deviation: The average of numbers is known as the
mean and the arithmetic mean of the absolute deviations of the
observations from a measure of central tendency is known as the mean
deviation (also called mean absolute deviation).
Relative Measure
of Dispersion
The relative
measures of dispersion are used to compare the distribution of two or more data
sets. This measure compares values without units. Common relative dispersion
methods include:
- Co-efficient
of Range
- Co-efficient
of Variation
- Co-efficient
of Standard Deviation
- Co-efficient
of Quartile Deviation
- Co-efficient
of Mean Deviation
Co-efficient of
Dispersion
The
coefficients of dispersion are calculated (along with the measure of
dispersion) when two series are compared, that differ widely in their averages.
The dispersion coefficient is also used when two series with different
measurement units are compared. It is denoted as C.D.
The common
coefficients of dispersion are:
|
C.D. in terms of |
Coefficient of dispersion |
|
Range |
C.D.
= (Xmax – Xmin) ⁄ (Xmax + Xmin) |
|
Quartile
Deviation |
C.D.
= (Q3 – Q1) ⁄ (Q3 + Q1) |
|
Standard
Deviation (S.D.) |
C.D.
= S.D. ⁄ Mean |
|
Mean
Deviation |
C.D.
= Mean deviation/Average |
Solved Example
Find the Variance and Standard
Deviation of the Following Numbers: 1, 3, 5, 5, 6, 7, 9, 10.
Solution:
The mean = (1+
3+ 5+ 5+ 6+ 7+ 9+ 10)/8 = 46/ 8 = 5.75
Step 1: Subtract the mean value from
individual value
(1 – 5.75), (3
– 5.75), (5 – 5.75), (5 – 5.75), (6 – 5.75), (7 – 5.75), (9 – 5.75), (10 –
5.75)
= -4.75,
-2.75, -0.75, -0.75, 0.25, 1.25, 3.25, 4.25
Step 2: Squaring the above values we
get, 22.563, 7.563, 0.563, 0.563, 0.063, 1.563, 10.563, 18.063
Step 3: 22.563 + 7.563 + 0.563 + 0.563 +
0.063 + 1.563 + 10.563 + 18.063
= 61.504
Step 4: n = 8, therefore variance (σ2) = 61.504/ 8 = 7.69
Now, Standard deviation (σ) = 2.77
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