Download Measures of Dispersion, Skewness and Kurtosis

Important Summary Statistics
(Measures of Dispersion, Skewness and Kurtosis)
Dr. Dennis S. Mapa
Prof. Michael Dominic del Mundo
UP School of Statistics
Measures of Dispersion

Measures of dispersion indicate the extent
to which individual items in a series are
scattered about an average.

Used as a measure of reliability of the
average value.
General Classifications of Measures of
Dispersion

Measures of Absolute Dispersion
- used to describe the variability of a data set

Measures of Relative Dispersion
- used to compare two or more data sets with
different means and different units of
measurement
Variance and Standard Deviation

The variance and standard deviation are
measures of dispersion of data with respect
to the mean.

For a finite population of size N, the
population variance is
N
2 
2


X


 i
i 1
N
Variance and Standard Deviation
The population standard deviation is defined as the
positive square root of the variance,
N

2


X


 i
i 1
N
The standard deviation is often referred to the
measure of “volatility.”
Variance and Standard Deviation
For a sample of size n, the sample variance is
defined as,
n
s2 
 X
i 1
i

2
X
1
and the sample standardndeviation
is defined
as,
 X
n
s
i 1
i
X
n 1

2
Variance and Standard Deviation

If there is a large amount of variation in the
data set, the data values will be far from the
mean. In this case, the standard deviation will
be large.

If, on the other hand, there is only a small
amount of variation in the data set, the data
values will be close to the mean. Hence, the
standard deviation will be small.
Characteristics of the Standard Deviation

Just like the mean, it is affected by
the value of every observation.

It may be distorted by few extreme
values.

It is always positive.
Measures of Relative Dispersion
 Measures of relative dispersion are unit
less and are used to compare the scatter
of one distribution with the scatter of
another distribution.
Coefficient of Variation

Commonly
dispersion.
used
measure
of
relative

The coefficient of variation utilizes two
measures: the mean and the standard
deviation.

It is expressed as a percentage, removing the
unit of measurement, thus, allowing
comparison of two or more data sets.
Coefficient of Variation
The formula of the coefficient of variation is
given as,

CV  x 100%

The sample counterpart is defined as,
s
CV  x 100%
X
Standard Score
The standard score measures how many standard
deviations an observation is above or below the mean.
It is computed as,
Z
X 

and the sample counterpart is,
X X
Z
s
Standard Score
 The standard score is not a measure of relative
dispersion per se but is somewhat related.
 It is useful for comparing two values from different
series specially when these two series differ with
respect to the mean or standard deviation or both
are expressed in different units.
Measure of Skewness
A measure of skewness shows the
degree of asymmetry, or departure from
symmetry of a distribution. It indicates
not only the amount of skewness but also
the direction (skewed to the left or
skewed to the right).
Positive Skewness or Skewed to the Right
distribution tapers more to the right than
to the left
longer tail to the right
more concentration of values below than
above the mean
Positive Skewness
_______________________________________________
Income
frequency distribution of income
Negative Skewness or Skewed to the left
 distribution tapers more to the left than
to the right
 longer tail to the left
 more concentration of values above
than below the mean
Negative Skewness
________________________________________________
Negative Skewness
rarely do we find curves that are skewed
to the left, and even more rarely do we
find data characteristically skewed to the
left
 however, recent studies have shown
that asset returns (particularly equities)
follow a skewed to the left distribution
during times of “crisis”
Measure of Skewness
Pearson’s Coefficient of Skewness
SK 
3( x  Md )
s
SK=0 implies symmetry
SK>0 implies positive skewness
SK<0 implies negative skewness
Kurtosis
The skewness coefficient enables the analyst to
distinguish between a symmetric distribution and a nonsymmetric distribution but that still leaves us with the
problem of distinguishing between two symmetric
distributions with different shapes.
Kurtosis
 The two graphs are both symmetric with respect to
their mean, but differ with respect to their peaks and
tails.
 The distribution on the right has greater kurtosis –
more peaked, less flat – but it is possible that it has
the same standard deviation as the graph on the left,
which is more spread out but is thinner at the tails.
 Kurtosis is a measure that distinguishes distributions
by measuring peakedness in relation to tails.
Measuring Kurtosis
Kurtosis is a standardized version of
the fourth moment defined as,
n
K
(X  X )
i 1
i
nS
4
4
Measuring Kurtosis
K  3 mesokurtic distribution
K  3 leptokurtic distribution
K  3 platykurtic distribution
Measuring Kurtosis
 A platykurtic distribution is one with a
flatter peak.
 A mesokurtic distribution is one with
neither
pointed
peak
nor
flat peak.
 A leptokurtic distribution is one with
more pointed peak.
Measuring Kurtosis
 Intuitively, we can think of the kurtosis
coefficient as a measure which indicates
whether a symmetric distribution when
compared with the Normal distribution (with
kurtosis equal to 3) has thicker tails and more
pointed peaks or not.