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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.”
The range is the first measure of measuring the scattered of a 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
Demerits of Range
The difference of the upper quartile Q_{3} 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 Q_{1 }and Q_{3} 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
Demerits of Quartile 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.
let \(\overline{x}\)…, m_{d }and m_{o }be the arithmetic mean, and mode of the given data.
Then,
Where 'n' is the number of items.
In case of discrete series mean deviation (M.D) is calculated as below:
Where, N =Σf = total frequency
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
Demerits of Mean 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:
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
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.
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
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 =\(\frac{S.D}{mean}\)=\(\frac{σ}{x}\)
References:
Adhikari, Ramesh Prasad, Economics-XI, Asmita Pustak Prakashan, Kathmandu
Kanel, Navaraj et.al., Principles of Economics-XI, Buddha Prakashan, Kathmandu
Kharel, Khom Raj et.al., Economics In English Medium-XI, Sukunda Pustak Bhawan, Kathmandu
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