Measure of Dispersion

Measures of Dispersion: Variance

Variance is one of the most important measures of dispersion of a distribution of a random variable. The term variance was introduced by R. A. Fisher in 1918. The variance of a set of observations (data set) is defined as the mean of the squares of deviations of all the observations from their mean. When it is computed for the entire population, the variance is called the population variance, usually denoted by $\sigma^2$, while for sample data, it is called sample variance and denoted by $S^2$ to distinguish between population variance and sample variance. Variance is also denoted by $Var(X)$ when we speak about the variance of a random variable. The symbolic definition of population and sample variance is

$\sigma^2=\frac{\sum (X_i – \mu)^2}{N}; \quad \text{for population data}$

$\sigma^2=\frac{\sum (X_i – \overline{X})^2}{n-1}; \quad \text{for sample data}$

It should be noted that the variance is in the square of units in which the observations are expressed and the variance is a large number compared to the observations themselves. The variance because of its nice mathematical properties, assumes an extremely important role in statistical theory.

Variance can be computed if we have standard deviation as the variance is the square of standard deviation i.e. Variance = (Standard Deviation)$^2$.

Variance can be used to compare dispersion in two or more sets of observations. Variance can never be negative since every term in the variance is the squared quantity, either positive or zero.
To calculate the standard deviation one has to follow these steps:

  1. First, find the mean of the data.
  2. Take the difference of each observation from the mean of the given data set. The sum of these differences should be zero or near zero it may be due to the rounding of numbers.
  3. Square the values obtained in step 1, which should be greater than or equal to zero, i.e. should be a positive quantity.
  4. Sum all the squared quantities obtained in step 2. We call it the sum of squares of differences.
  5. Divide this sum of squares of differences by the total number of observations if we have to calculate population standard deviation ($\sigma$). For sample standard deviation (S) divide the sum of squares of differences by the total number of observations minus one i.e. degree of freedom.
    Find the square root of the quantity obtained in step 4. The resultant quantity will be the standard deviation for the given data set.
Measures of Dispersion

The major characteristics of the variances are:
a)    All of the observations are used in the calculations
b)    Variance is not unduly influenced by extreme observations
c)    The variance is not in the same units as the observation, the variance is in the square of units in which the observations are expressed.

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Skewness and Measures of Skewness

If the curve is symmetrical, a deviation below the mean exactly equals the corresponding deviation above the mean. This is called symmetry. Here, we will discuss Skewness and Measures of Skewness.

Skewness is the degree of asymmetry or departure from the symmetry of a distribution. Positive Skewness means when the tail on the right side of the distribution is longer or fatter. The mean and median will be greater than the mode. Negative Skewness is when the tail of the left side of the distribution is longer or fatter than the tail on the right side.

Skewness and Measures of Skewness

Measures of Skewness

Karl Pearson Measures of Relative Skewness
In a symmetrical distribution, the mean, median, and mode coincide. In skewed distributions, these values are pulled apart; the mean tends to be on the same side of the mode as the longer tail. Thus, a measure of the asymmetry is supplied by the difference ($mean – mode$). This can be made dimensionless by dividing by a measure of dispersion (such as SD). The Karl Pearson measure of relative skewness is
$$\text{SK} = \frac{\text{Mean}-\text{mode}}{SD} =\frac{\overline{x}-\text{mode}}{s}$$
The value of skewness may be either positive or negative.

The empirical formula for skewness (called the second coefficient of skewness) is

$$
\text{SK} = \frac{3(\text{mean}-\text{median})}{SD}=\frac{3(\tilde{X}-\text{median})}{s}
$$

Bowley Measures of Skewness

In a symmetrical distribution, the quartiles are equidistant from the median ($Q_2-Q_1 = Q_3-Q_2$). If the distribution is not symmetrical, the quartiles will not be equidistant from the median (unless the entire asymmetry is located in the extreme quarters of the data). The Bowley suggested measure of skewness is

$$\text{Quartile Coefficient of SK} = \frac{Q_(2-Q_2)-(Q_2-Q_1)}{Q_3-Q_1}=\frac{Q_2-2Q_2+Q_1}{Q_3-Q_1}$$

This measure is always zero when the quartiles are equidistant from the median and is positive when the upper quartile is farther from the median than the lower quartile. This measure of skewness varies between $+1$ and $-1$.

Moment Coefficient of Skewness

In any symmetrical curve, the sum of odd powers of deviations from the mean will be equal to zero. That is, $m_3=m_5=m_7=\cdots=0$. However, it is not true for asymmetrical distributions. For this reason, a measure of skewness is devised based on $m_3$. That is

\begin{align}
\text{Moment of Coefficient of SK}&= a_3=\frac{m_3}{s^3}=\frac{m_3}{\sqrt{m_2^3}}\\
&=b_1=\frac{m_3^2}{m_2^3}
\end{align}

For perfectly symmetrical curves (normal curves), $a_3$ and $b_1$ are zero.

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Standard Deviation: A Measure of Dispersion

The standard deviation is a widely used concept in statistics and it tells how much variation (spread or dispersion) is in the data set. It can be defined as the positive square root of the mean (average) of the squared deviations of the values from their mean.
To calculate the standard deviation one has to follow these steps:

Calculation of Standard Deviation

  1. First, find the mean of the data.
  2. Take the difference of each data point from the mean of the given data set (which is computed in step 1). Note that, the sum of these differences must be equal to zero or near to zero due to rounding of numbers.
  3. Now compute the square of the differences obtained in Step 2, it would be greater than zero, and it will be a positive quantity.
  4. Now add up all the squared quantities obtained in step 3. We call it the sum of squares of differences.
  5. Divide this sum of squares of differences (obtained in step 4) by the total number of observations (available in data) if we have to calculate population standard deviation ($\sigma$). If you want t to compute sample standard deviation ($S$) then divide the sum of squares of differences (obtained in step 4) by the total number of observations minus one ($n-1$) i.e. the degree of freedom. Note that $n$ is the number of observations available in the data set.
  6. Find the square root (also known as under root) of the quantity obtained in step 5. The resultant quantity in this way is known as the standard deviation (SD) for the given data set.

The sample SD of a set of $n$ observation, $X_1, X_2, \cdots, X_n$ denoted by $S$ is

\begin{aligned}
\sigma &=\sqrt{\frac{\sum_{i=1}^n (X_i-\overline{X})^2}{n}}; Population\, SD\\
S&=\sqrt{ \frac{\sum_{i=1}^n (X_i-\overline{X})^2}{n-1}}; Sample\, SD
\end{aligned}

The standard deviation can be computed from variance too.

The real meaning of the standard deviation is that for a given data set 68% of the data values will lie within the range $\overline{X} \pm \sigma$ i.e. within one standard deviation from the mean or simply within one $\sigma$. Similarly, 95% of the data values will lie within the range $\overline{X} \pm 2 \sigma$ and 99% within $\overline{X} \pm 3 \sigma$.

Standard Deviation

Examples

A large value of SD indicates more spread in the data set which can be interpreted as the inconsistent behaviour of the data collected. It means that the data points tend to be away from the mean value. For the case of smaller standard deviation, data points tend to be close (very close) to the mean indicating the consistent behavior of the data set.
The standard deviation and variance are used to measure the risk of a particular investment in finance. The mean of 15% and standard deviation of 2% indicates that it is expected to earn a 15% return on investment and we have a 68% chance that the return will be between 13% and 17%. Similarly, there is a 95% chance that the return on the investment will yield an 11% to 19% return.

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