Measure of Dispersion

Absolute Measure of Dispersion

An absolute Measure of Dispersion gives an idea about the amount of dispersion/ spread in a set of observations. These quantities measure the dispersion in the same units as the units of original data. The absolute measure of dispersion cannot be used to compare the variation of two or more series/ data sets. The absolute measure of dispersion does not in itself, tell whether the variation is large or small.

Absolute Measure of Dispersion

The absolute Measure of Dispersion:

  1. Range
  2. Quartile Deviation
  3. Mean Deviation
  4. Variance or Standard Deviation
Absolute Measures of Dispersion

Range

The Range is the difference between the largest value and the smallest value in the data set. For ungrouped data, let $X_0$ be the smallest value and $X_n$ be the largest  value in a data set then the range ($R$) is defined as
$R=X_n-X_0$.

For grouped data Range can be calculated in three different ways
R=Mid point of the highest class – Midpoint of the lowest class
R=Upper class limit of the highest class – Lower class limit of the lower class
R=Upper class boundary of the highest class – The lower class boundary of the lowest class

Quartile Deviation (Semi-Interquantile Range)

The Quartile deviation (an absolute measure of dispersion) is defined as the difference between the third and first quartiles, and half of this range is called the semi-interquartile range (SIQD) or simply quartile deviation (QD). $$QD=\frac{Q_3-Q_1}{2}$$

The Quartile Deviation is superior to the range as it is not affected by extremely large or small observations, anyhow it does not give any information about the position of observation lying outside the two quantities. It is not amenable to mathematical treatment and is greatly affected by sampling variability. Although Quartile Deviation is not widely used as a measure of dispersion, it is used in situations in which extreme observations are thought to be unrepresentative/ misleading. Quartile Deviation is not based on all observations therefore it is affected by extreme observations.

Note: The range “Median ± QD” contains approximately 50% of the data.

Mean Deviation (Average Deviation)

The Mean Deviation is another absolute measure of dispersion and is defined as the arithmetic mean of the deviations measured either from the mean or from the median. All these deviations are counted as positive to avoid the difficulty arising from the property that the sum of deviations of observations from their mean is zero.

$MD=\frac{\sum|X-\overline{X}|}{n}\quad$ for ungrouped data for mean
$MD=\frac{\sum f|X-\overline{X}|}{\sum f}\quad$ for grouped data for mean
$MD=\frac{\sum|X-\tilde{X}|}{n}\quad$ for ungrouped data for median
$MD=\frac{\sum f|X-\tilde{X}|}{\sum f}\quad$ for grouped data for median
Mean Deviation can be calculated about other central tendencies but it is least when deviations are taken as the median.

The Mean Deviation gives more information than the range or the Quartile Deviation as it is based on all the observed values. The Mean Deviation does not give undue weight to occasional large deviations, so it should likely be used in situations where such deviations are likely to occur.

Variance and Standard Deviation

This absolute measure of dispersion is defined as the mean of the squares of deviations of all the observations from their mean. Traditionally population variance is denoted by $\sigma^2$ (sigma square) and for sample data denoted by $S^2$ or $s^2$.

Symbolically
$\sigma^2=\frac{\sum(X_i-\mu)^2}{N}\quad$ Population Variance for ungrouped data
$S^2=\frac{\sum(X_i-\overline{X})^2}{n}\quad$ sample Variance for ungrouped data
$\sigma^2=\frac{\sum f(X_i-\mu)^2}{\sum f}\quad$ Population Variance for grouped data
$\sigma^2=\frac{\sum f (X_i-\overline{X})^2}{\sum f}\quad$ Sample Variance for grouped data

The variance is denoted by $Var(X)$ for random variable $X$. The term variance was introduced by R. A. Fisher (1890-1982) in 1918. The variance is in squares of units and the variance is a large number compared to observations themselves.
Note that there are alternative formulas to compute Variance or Standard Deviations.

The positive square root of the variance is called Standard Deviation (SD) to express the deviation in the same units as the original observation. It is a measure of the average spread about the mean and is symbolically defined as

$\sigma^2=\sqrt{\frac{\sum(X_i-\mu)^2}{N}}\quad$ Population Standard for ungrouped data
$S^2=\sqrt{\frac{\sum(X_i-\overline{X})^2}{n}}\quad$ Sample Standard Deviation for ungrouped data
$\sigma^2=\sqrt{\frac{\sum f(X_i-\mu)^2}{\sum f}}\quad$ Population Standard Deviation for grouped data
$\sigma^2=\sqrt{\frac{\sum f (X_i-\overline{X})^2}{\sum f}}\quad$ Sample Standard Deviation for grouped data
Standard Deviation is the most useful measure of dispersion and is credited with the name Standard Deviation by Karl Pearson (1857-1936).

In some text Sample, Standard Deviation is defined as $S^2=\frac{\sum (X_i-\overline{X})^2}{n-1}$ based on the argument that knowledge of any $n-1$ deviations determines the remaining deviations as the sum of n deviations must be zero. This is an unbiased estimator of the population variance $\sigma^2$. The Standard Deviation has a definite mathematical measure, it utilizes all the observed values and is amenable to mathematical treatment but affected by extreme values.

References

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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 that are exactly alike, even though the distributions are dissimilar in other aspects. On the other hand, a 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 describes the spread of the values in a set of data has two types of measures of dispersion or variability:

measures-of-dispersion
  1. Absolute Measures
  2. Relative Measures

A measure of central tendency together with a measure of dispersion gives an adequate description of data as compared to the use of a measure of location only, because the averages or measures of central tendency only describe 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, the 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.

Absolute Measure of Dispersion

Absolute measures are 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 data sets.
Most Common absolute measures of variability are:

Relative Measures of Dispersion

The relative measures have no units as these are ratios, coefficients, or percentages. Relative measures are independent of units of measurement 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.

References:
http://www2.le.ac.uk/offices/careers/ld/resources/numeracy/variability

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Moments Statistics

Moments Statistics

The measure of central tendency (location) and the 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 the mean are the mean of deviations from the mean after raising them to integer powers. The rth population moment about the mean is denoted by $\mu_r$ is

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

where $r=1,2,\cdots$

The corresponding sample moment denoted by $m_r$ 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 the 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 the 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 $r=1,2,\cdots$.

therefore
\begin{eqnarray*}
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}
\end{eqnarray*}

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 moments 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

\begin{eqnarray*}
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
\end{eqnarray*}

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 the moment about arbitrary value will be
\[m’_r = \frac{\sum y^r}{n}\]

where $r=1,2,3,\cdots$.

therefore
\begin{eqnarray*}
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}\\
\end{eqnarray*}

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 the 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

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

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