Measures of Dispersions

Measures of Dispersions:

The various measures of central tendency take into consideration one important characteristic of statistical distribution- namely, the single representative i.e., average. But to study the reliability of the central tendency, it is equally important to know how the values of the variable cluster around the point denoting the measure of central tendency. The measures of central tendency are inadequate to give a complete idea about the items; rather we would be more definite about the nature of distribution if some idea about the scatter of items about some average value is provided.

A measure of scatteredness of the item about some average is referred to as a measure of dispersion. A measure of dispersion for statistical data may also be defined as a measurement of the spread of the items about an average value. One may think it is in terms of a lack of uniformity or homogeneity in the size of the series.

The requisites of a good measure of dispersion are the same as those for a good measure of central tendency, namely:

• It should be easy to compute.
• It should be well-defined.
• It should be simple to understand.
• It should be based on all the items.
• It should not be unexpectedly affected by the presence of extreme items.
• It should have sampling stability.
• It should be capable of further algebraic treatment.

Some of the commonly used measures of dispersion are:

• Range.
• Mean Deviation.
• Variance.
• Standard Deviation.

The simplest measure of dispersion is the range of the values which may be defined as the difference between the highest and the lowest values of the variable appearing in the distribution. As it depends entirely on extreme values, it is most unstable and unreliable. The deletion or insertion of any number of intermediate values would not at all change the dispersion in this case. Owing to these backwardnesses it has a limited use.

Mean deviation and standard deviation are widely used nowadays and we shall discuss only these two types of dispersions here.

Mean Deviation (M.D.):

The mean deviation of a series of values of a variable is the arithmetic mean of all the absolute deviations from the Mean or Median. (absolute deviation means the positive value of the deviation).

Mean deviation of a set of n discrete scores x₁, x₂, x₃,…..x_n about arithmetic mean is given by-

When the values of the variables are given in the form of classes, then their respective class marks (mid-values of the classes) are to be taken as the values of the variables.

Coefficient of Mean Deviation:

It is defined to be the ratio of the mean deviation to the arithmetic mean i.e.,

It is generally used for the comparison of the variability of two or more series.

Merits of Mean Deviation:

• It is well-defined.
• It is easy to understand.
• It is easy to compute.
• It is based on all the items.
• It can be computed about any average.
• It is not much affected by the presence of extreme scores.

Demerits of Mean Deviation:

• It does not consider the signs of the individual deviation [d_i = (x_i – x̄)] of the items.
• It is not capable of further algebraic treatment so as to permit more analysis of the distribution.

Variance and Standard Deviation:

The mean square deviation from any given average A (usually calculated from mean or median) is defined as the arithmetic mean of the squares of the deviation from the average A. It is generally denoted by S². Thus, the mean square deviation from average A is given by-

If the mean square deviation is calculated from the mean, then it is called the variance and is denoted by σ². If x̄ be the mean of the set of scores x₁, x₂, x₃, ……., x_n, then-

In practical problems, the means of a set of scores may be a decimal fraction and so each of the values of x_i – x̄ would be also a decimal fraction. Finding out squares of those values would be a laborious and boring job. In order to avoid that, the above formula for finding out the variance is modified as follows.

If the scores are repeated, i.e. if the score x₁ occurs f₁ times, x₂ occurs f₂ times, and so on, then the variance of the scores is given by-

To find the variance from the frequency distribution Table, the continuous series is first converted into a discrete series by considering the values of the variable as the mid-values of the corresponding classes.

If the scores are large enough so that squaring them is not desirable, then the scores may be reduced by changing the origin. Change of origin does not alter the variance- as will be clear from the discussion made below. We write u_i = x_i – a, where a is the reduction in each score.

Thus, σ² remains unaffected by replacing x_i with u_i.

If the classes of the frequency distribution table are of equal width, then one can change the scale also (in addition to the shift of origin).

If we take d_i = (x_i – a)/h, where h is the width of the class intervals, then x_i = a + hd_i

The positive square root of the variance is called the standard deviation. It is denoted by σ.

We use this formula to compute σ directly by finding out the actual mean of the score. This method of finding out σ is known as the direct method.

In the short-cut method, we do not find out the actual mean. In this case, we use the formula

In this case, each score has been reduced to the same extent.

In addition to the change of origin if the scale is also reduced, then the method is known as the step deviation method. The formula is-

Coefficient of Standard Deviation (S.D.):

In order to compare two or more series for variability, the corresponding relative measures are used. The relative measure is known as the coefficient of standard deviation and is defined as-

Another relative measure of variability is known as the coefficient of variation defined as

Merits of Standard Deviation:

• It is well-defined.
• It is simple to understand.
• It is based on all the items.
• It has sampling stability.
• In calculating S.D. the signs of deviations of the items are also taken into consideration.
• It is capable of further algebraic treatment.

Demerits of Standard Deviation:

• It is not easy to calculate.
• As the squares of the deviations are involved in the calculation, it is highly affected by the presence of extremely high or extremely low scores.

Standard Deviation of Composite Group:

If σ₁ and σ₂ be the standard deviations of two groups containing n₁ and n₂ items respectively, x̄₁ and x̄₂ their respective arithmetic means, then x̄, the arithmetic mean of the combined group is given by-

and the standard deviation of the combined group is given by-

Example 1: Calculate the mean deviation from the mean for the following data. 39, 31, 62, 29, 33, 49, 44 Solution- The given observations are, 39, 31, 62, 29, 33, 49, 44 ∴ Sum of observations = 39 + 31+ ……… + 44 = 287 Number of observations = 7 ∴ Mean, x̄ = sum of observations/number of observations ⇒ x̄ = 287/7 = 41

⇒ Mean deviation about mean = 64/7 = 9.14

Example 2: Calculate the mean deviation about the mean for the following data.

Solution-

Example 3: Calculate the mean deviation about the Median for the following distribution.

Solution-

Example 4: Find the mean deviation about the mean for the following distribution.

Solution-

Example 5: Calculate the mean and variance of the following distribution.

Solution-

Example 6: Find the variance and standard deviation for the following data.

Solution-

Example 7: Calculate the mean and standard deviation from the following distribution and also find the coefficient of variation.

Solution-