Tagged: Measure of Dispersion

Measure of Dispersion or Variability

The measure of location or averages or central tendency is not sufficient to describe the characteristics of a distribution, because two or more distributions may have averages which are exactly alike, even though the distributions are dissimilar in other aspects, and on the other hand, measure of central tendency represents the typical value of the data set. To give a sensible description of data, a numerical quantity called the measure of dispersion/ variability or scatter that describe the spread of the values in a set of data have two types of measures of dispersion or variability:

  1. Absolute Measures
  2. Relative Measures

A measure of central tendency together with a measure of dispersion gives adequate description of data as compared to use of measure of location only, because the averages or measures of central tendency only describes the balancing point of the data set, it does not provide any information about the degree to which the data tend to spread or scatter about the average value. So Measure of dispersion is an indication of the characteristic of the central tendency measure. The smaller the variability of a given set, the more the values of the measure of averages will be representative of the data set.

  1. Absolute Measures
    Absolute measures defined in such a way that they have units such as meters, grams, etc. same as those of the original measurements. Absolute measures cannot be used to compare the variation/spread of two or more sets of data.
    Most Common absolute measures of variability are:  
    • Range
    • Semi-Interquartile Range or Quartile Deviation
    • Mean Deviation
    • Variance
    • Standard Deviation
  2. Relative Measures
    The relative measures have no units as these are ratios, coefficients, or percentages. Relative measures are independent of units of measurements and are useful for comparing data of different natures.  
    • Coefficient of Variation
    • Coefficient of Mean Deviation
    • Coefficient of Quartile Deviation
    • Coefficient of Standard Deviation

Different terms are used for the measure of dispersion or variability such as variability, spread, scatterness, the measure of uncertainty, deviation, etc.


The Moments in Statistics

The measure of central tendency (location) and measure of dispersion (variation) both are useful to describe a data set but both of them fail to tell anything about the shape of the distribution. We need some other certain measure called the moments to identify the shape of the distribution known as skewness and kurtosis.

Moments about Mean

The moments about mean are the mean of deviations from the mean after raising them to integer powers. The rth population moment about mean is denoted by $\mu_r$ is

\[\mu_r=\frac{\sum^{N}_{i=1}(y_i – \bar{y} )^r}{N}\]

where r=1, 2, …

Corresponding sample moment denoted by mr is

\[\mu_r=\frac{\sum^{n}_{i=1}(y_i – \bar{y} )^r}{n}\]

Note that if r=1 i.e. the first moment is zero as $\mu_1=\frac{\sum^{n}_{i=1}(y_i – \bar{y} )^1}{n}=0$. So first moment is always zero.

If r=2 then the second moment is variance i.e. \[\mu_2=\frac{\sum^{n}_{i=1}(y_i – \bar{y} )^2}{n}\]

Similarly the 3rd and 4th moments are

\[\mu_3=\frac{\sum^{n}_{i=1}(y_i – \bar{y} )^3}{n}\]

\[\mu_4=\frac{\sum^{n}_{i=1}(y_i – \bar{y} )^4}{n}\]

For grouped data the rth sample moment  about sample mean $\bar{y}$ is

\[\mu_r=\frac{\sum^{n}_{i=1}f_i(y_i – \bar{y} )^r}{\sum^{n}_{i=1}f_i}\]

where $\sum^{n}_{i=1}f_i=n$

Moments about Arbitrary Value

The rth sample sample moment about any arbitrary origin “a” denoted by $m’_r$ is
\[m’_r = \frac{\sum^{n}_{i=1}(y_i – a)^2}{n} = \frac{\sum^{n}_{i=1}D^r_i}{n}\]
where $D_i=(y_i -a)$ and r = 1, 2, ….

m’_1&=&\frac{\sum^{n}_{i=1}(y_i – a)}{n}=\frac{\sum^{n}_{i=1}D_i}{n}\\
m’_2&=&\frac{\sum^{n}_{i=1}(y_i – a)^2}{n}=\frac{\sum^{n}_{i=1}D_i ^2}{n}\\
m’_3&=&\frac{\sum^{n}_{i=1}(y_i – a)^3}{n}=\frac{\sum^{n}_{i=1}D_i ^3}{n}\\
m’_4&=&\frac{\sum^{n}_{i=1}(y_i – a)^4}{n}=\frac{\sum^{n}_{i=1}D_i ^4}{n}

The rth sample moment for grouped data about any arbitrary origin “a” is

\[m’_r=\frac{\sum^{n}_{i=1}f_i(y_i – a)^r}{\sum^{n}_{i=1}f} = \frac{\sum f_i D_i ^r}{\sum f}\]

The moment about the mean are usually called central moments and the moments about any arbitrary origin “a” are called non-central moments or raw moments.

One can calculate the moments about mean from the following relations by calculating the moments about arbitrary value

m_1&=& m’_1 – (m’_1) = 0 \\
m_2 &=& m’_2 – (m’_1)^2\\
m_3 &=& m’_3 – 3m’_2m’_1 +2(m’_1)^3\\
m_4 &=& m’_4 -4 m’_3m’_1 +6m’_2(m’_1)^2 -3(m’_1)^4

Moments about Zero

If variable y assumes n values $y_1, y_2, \cdots, y_n$ then rth moment about zero can be obtained by taking a=0 so moment about arbitrary value will be
\[m’_r = \frac{\sum y^r}{n}\]

where r = 1, 2, 3, ….

m’_1&=&\frac{\sum y^1}{n}\\
m’_2 &=&\frac{\sum y^2}{n}\\
m’_3 &=&\frac{\sum y^3}{n}\\
m’_4 &=&\frac{\sum y^4}{n}\\

The third moment is used to define the skewness of a distribution
\[{\rm Skewness} = \frac{\sum^{i=1}_{n} (y_i – \bar{y})^3}{ns^3}\]

If distribution is symmetric then the skewness will be zero. Skewness will be positive if there is a long tail in the positive direction and skewness will be negative if there is a long tail in the negative direction.

The fourth moment is used to define the kurtosis of a distribution

\[{\rm Kurtosis} = \frac{\sum^{i=1}_{n} (y_i -\bar{y})^4}{ns^4}\]

Skewness Introduction, formula, Interpretation

Skewness is the degree of asymmetry or departure from the symmetry of the distribution of a real-valued random variable.

Positive Skewed
If the frequency curve of distribution has a longer tail to the right of the central maximum than to the left, the distribution is said to be skewed to the right or to have positively skewed. In a positively skewed distribution, the mean is greater than the median and median is greater than the mode i.e. $Mean > Median > Mode$.

Negative Skewed
If the frequency curve has a longer tail to the left of the central maximum than to the right, the distribution is said to be skewed to the left or to have negatively skewed. In a negatively skewed distribution, the mode is greater than median and median is greater than mean i.e. $Mode > Median > Mean$.


In a symmetrical distribution, the mean, median and mode coincide. In a skewed distribution, these values are pulled apart.

Pearson’s Coefficient of Skewness
Karl Pearson, (1857-1936) introduced a coefficient to measure the degree of skewness of distribution or curve, which is denoted by $S_k$ and define by

S_k &=& \frac{Mean – Mode}{Standard Deviation}\\
S_k &=& \frac{3(Mean – Median)}{Standard Deviation}\\
Usually, this coefficient varies between –3 (for negative) to +3 (for positive) and the sign indicates the direction of skewness.

Bowley’s Coefficient of Skewness or Quartile Coefficient
Arthur Lyon Bowley (1869-1957) proposed a measure of skewness based on the median and the two quartiles.

\[S_k=\frac{Q_1+Q_3-2Median}{Q_3 – Q_1}\]
Its values lie between 0 and ±1.

Moment Coefficient of Skewness
This measure of skewness is the third moment expressed in standard units (or the moment ratio) thus given by

\[S_k=\frac{\mu_3}{\sigma^3} \]
Its values lie between -2 and +2.

If $S_k$ is greater than zero, the distribution or curve is said to be positively skewed. If $S_k$ is less than zero the distribution or curve is said to be negatively skewed. If $S_k$ is zero the distribution or curve is said to be symmetrical.

The skewness of the distribution of a real-valued random variable can easily be seen by drawing a histogram or frequency curve.

The skewness may be very extreme and in such a case these are called J-shaped distributions.


J-Shaped Distribution

Measure of Kurtosis

In statistics, a measure of kurtosis is a measure of the “tailedness” of the probability distribution of a real-valued random variable. The standard measure of kurtosis is based on a scaled version of the fourth moment of the data or population. Therefore, the measure of kurtosis is related to the tails of the distribution, not its peak.

Sometimes, the Measure of Kurtosis is characterized as a measure of peakedness that is mistaken. A distribution having a relatively high peak is called leptokurtic. A distribution that is flat-topped is called platykurtic. The normal distribution which is neither very peaked nor very flat-topped is also called mesokurtic.  The histogram in some cases can be used as an effective graphical technique for showing both the skewness and kurtosis of the data set.

Kurtosis Pict

Data sets with high kurtosis tend to have a distinct peak near the mean, decline rather rapidly, and have heavy tails. Data sets with low kurtosis tend to have a flat top near the mean rather than a sharp peak.

Moment ratio and Percentile Coefficient of kurtosis are used to measure the kurtosis

Moment Coefficient of Kurtosis= $b_2 = \frac{m_4}{S^2} = \frac{m_4}{m^{2}_{2}}$

Percentile Coefficient of Kurtosis = $k=\frac{Q.D}{P_{90}-P_{10}}$
where Q.D = $\frac{1}{2}(Q_3 – Q_1)$ is the semi-interquartile range. For normal distribution this has the value 0.263.

Dr. Wheeler defines kurtosis as:

The kurtosis parameter is a measure of the combined weight of the tails relative to the rest of the distribution.

So, kurtosis is all about the tails of the distribution – not the peakedness or flatness.

A normal random variable has a kurtosis of 3 irrespective of its mean or standard deviation. If a random variable’s kurtosis is greater than 3, it is said to be Leptokurtic. If its kurtosis is less than 3, it is said to be Platykurtic.

A large value of kurtosis indicates a more serious outlier issue and hence may lead the researcher to choose alternative statistical methods.

Some Examples of Kurtosis

  • In finance, risk and insurance are examples of needing to focus on the tail of the distribution and not assuming normality.
  • Kurtosis helps in determining whether resource used within an ecological guild is truly neutral or which it differs among species.
  • The accuracy of the variance as an estimate of the population $\sigma^2$ depends heavily on kurtosis.

For further reading see https://itfeature.com/statistics/measure-of-dispersion/moments

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