Economics: Basic Statistics used in Economics | Measure of Dispersion

Measures of Dispersion

The central values i.e. mean, median and mode are the representatives of the whole data. While drawing conclusions and inferences to forecast and compare two or more than two data or series. We use central values and say that the observation having higher greater central values are better than the others having comparatively small central values. That is to say, the larger are the central values, the better is the statistical distribution. But sometimes, the central values may mislead us that is to say our conclusions and forecast may be wrong if we make them just on the basis of central values.

According to Spiegel,” The degree to which numerical data tend to spread about an average value is called the variation or dispersion of data.”

According to Brooks and Dick,” Dispersion or spread is the scatter or variation of the variables about a central value.”

Range

The range is the first measure of measuring the scattered of variability of the items from their central values on either side of them. It is the difference between the largest and smallest items of the data.

In symbols,

Range = L - S

Where,

L = Largest item of data

S = smallest item of data

Merits of Range

• It is easy to calculate and easy to understand.
• It is rigidly defined.
• It is most efficient.

Demerits of Range

• It does not include all the items i.e it is not based on all the observations.
• It is heavily affected by the largest and smallest values which are known as the extreme values.
• It cannot be calculated in case of open end classes as classes do not have defined ends.

Semi-inter quartile Range or Quartile Deviation

The difference of the upper quartile Q₃ and the lower quartile Q1 i.e (Q_3- Q1) is known as the interquartile range. The half of interquartile range is known as the semi-interquartile range which is also known as quartile deviation.

In symbols,

Quartile deviation = \(\frac{Q_3\:-\: Q_1}{2}\)

Where Q₁ and Q₃ have the usual meanings. This is the absolute measure of quartile deviation. The relative measure of quartile deviation, known as the coefficient of quartile deviation is defined below:

Coefficient of quartile deviation(Q.D)=\(\frac{Q_3\:-\: Q_1}{Q_3\:+\:Q_1}\)

Merits of Quartile Deviation

• It is simple and easy to calculate .
• It is a better measure of dispersion than the range because it involves more items than the range.
• It is more suitable for the open end classes.

Demerits of Quartile Deviation

• The main demerits of quartile deviation are that it does not involve al the items.
• It is also affected by fluctuation in sampling.
• It gives no information about largest and smallest items.

Mean Deviation

The arithmetic average of the deviations of the items taken from their arithmetic mean, median or mode considering only the positive signs of the deviations is called the means of deviation. The mean deviation is written M.D in short.

Individual Series

let \(\overline{x}\)…, m_d and m_o be the arithmetic mean, and mode of the given data.

Then,

1. Mean deviation from mean =\(\frac{\sum|x\:-\:\overline{x}|}{n}\)

2. Mean deviation from median =\(\frac{\sum|x\:-\:m_d|}{n}\)

3. Mean deviation from mode =\(\frac{\sum|x\:-\:m_o|}{n}\)

Where 'n' is the number of items.

Discrete Series

In case of discrete series mean deviation (M.D) is calculated as below:

1. M.D from mean =\(\frac{\sum f|x\:-\:\overline{x}|}{N}\)

2. M.D from median=\(\frac{\sum f|x\:-\:m_d|}{N}\)

3. M.D. from mode=\(\frac{\sum f|x\:-\:m_o|}{N}\)

Where, N =Σf = total frequency

Continuous series

In a case of continuous series, the above formulas of discrete series apply very well except that 'x' is the mid value of the class interval.

Relative Measures of Mean Deviation The above measures of mean deviation are absolute measures as all these contain the original units of the items. To make them free from units, we calculate their relative measured which are defined below. The relative measures of M.D are called its coefficients.

1. Coefficient of M.D from mean =\(\frac{M.D \:from\: mean}{mean}\)

2. Coefficient of M.D from median =\(\frac{M.D\: from \:median}{median}\)

3. Coefficient of M.D from mode=\(\frac{M.D\: from \:mode}{mode}\)

Merits of Mean Deviation

• It is easy to understand and easy to calculate.
• It is rigidity defined.
• It is based on all times.

Demerits of Mean Deviation

• Its main demerit is that its neglects the negative sign while taking deviations of the items from the central values.
• Mean deviation from mode cannot be calculated when it is ill defined.

Standard Deviation

The positive square root of the mean of the squares of the deviations of the items taken from their arithmetic mean is called standard deviation. It is written as S.D in short and denoted byσ. Standard deviation is the best measure of dispersion because:

• It is based on all items.
• Deviations are taken from a mean.
• Positive and negative signs are taken into account i.e they are not neglected.

Among all the measures of dispersion, the standard deviation is considered superior because it possesses all the requisites at a good measure of dispersion.

Formula to Compute S.D or σ

a. Individual Series

In case of an individual series, standard deviation is calculated by using the following formula:

σ = \(\sqrt{\frac{\sum(x\:-\:\overline{x})^2}{n}}\) ..........(a)

Also, σ = \(\sqrt{\frac{\sum x^2}{n}-(\frac{\sum x}{n})^2}\) ................(b)

The formulas (a) and(b) are called the direct methods of computing S.D. These direct methods become very lengthy and tedious when the items are fractional. In such cases, we use the short cut methods to compute the standard deviation. The short cut method is also known as the deviation method. The deviation method is given by

σ = n\(\sqrt{\frac{\sum d^2}{n}-(\frac{\sum d}{n})^2}\)

Whereσ is standard deviation n is the number of items and d= (x-a) is the deviation taken from the assumed mean 'a'. If there is a common factor h between the observations or items, we use the method.

σ = n\(\sqrt{\frac{\sum d '^2}{n}-(\frac{\sum d '}{n})^2}\) *h

To compute the standard deviation, This method is known as the step deviation. Here, d' =\(\frac{x\:-\:a}{h}\) where 'a' is the assumed mean and h is the common factor. This is the shortest method of calculating the standard deviation.

b. Discrete Series

In case of discrete,we use the following formulas to calculate the standard deviation:

σ= \(\sqrt{\frac{\sum f(x\:-\:\overline{x})^2}{N}}\)............. (i)

Where,

N=Σf, x= Variate values

\(\overline{x}\)= arithmetic mean of the items.

The formula (i) can also be written as:

σ =\(\sqrt{\frac{\sum f x^2}{N}-(\frac{\sum f x}{N})^2}\)......... (ii)

These formulas are used to calculate in a direct way. The short-cut method or deviation method used to calculate the standard deviation in this is as below:

σ =\(\sqrt{\frac{\sum f d^2}{N}-(\frac{\sum f d}{N})^2}\)............ (iii)

Where d=x=a and a is the assumed mean. This method is also known as the change of origin method. There is another short and more convenience method of calculating the S.D which is popularly known as step deviation method and also known as the change of origin and scale method. In this method

σ =\(\sqrt{\frac{\sum f d '^2}{N}-(\frac{\sum f d '}{N})^2}\) *h

Where,

d'=\(\frac{x\:-\:a}{h}\),

h= common factor,

a= assumed mean

N= Σf

c. Continuous Series

In a case of the continuous series, the formulas are same as in the case of discrete but here 'x' is the mid value of the class interval and h is the class width. The formulas are

(i) Direct method (σ)= \(\sqrt{\frac{\sum f(x\:-\:\overline{x})^2}{N}}\)

(ii) Direct method (σ) =\(\sqrt{\frac{\sum f x^2}{N}-(\frac{\sum f x}{N})^2}\)

(iii) Deviation method (σ) =\(\sqrt{\frac{\sum f d^2}{N}-(\frac{\sum f d}{N})^2}\)

(iv) Step deviation method (σ) =\(\sqrt{\frac{\sum f d '^2}{N}-(\frac{\sum f d '}{N})^2}\) *h

Where the symbols have useful meaning. All the above measures are the absolute measures of standard deviation. Its relatives measures are known as the coefficient of standard deviation which is given by

Coefficient of S.D =\(\frac{S.D}{mean}\)=\(\frac{σ}{x}\)