Measure of Dispersion

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Introduction

Dispersion लाई फैलावट भनिन्छ । दिएको डाटाहरु यसको औसतमान बाट कति फैलिएको छ भन्ने बारेको अध्ययनलाई Dispersion भनिन्छ । दुई वटा समान औसत भएका तर फरक फरक फैलावट भएको एउटा उदाहरण तल दिइएको छ ।
डाटा १: 5, 5, 5, 5, 5 whose mean=5
डाटा २: 1, 5, 6, 6, 7 whose mean=5
यहा दुबै डाटाको फैलावट फरक छ।
तसर्थ यस्तो अवस्थामा हामीलाई Measure of Dispersion को आवश्यकता पर्दछ। सामान्यतया Measure of Dispersion दुई प्रकारका हुन्छन ।

  1. फैलावको पूर्ण मापन (Absolute measure of Dispersion)
  2. फैलावको सापेक्षित मापन (Relative measure of Dispersion)
NOTE
  1. The measure of dispersion cannot be negative, it is always positive or zero.
    फैलावटको मापन Negative हुदैन, तर यसको मापन positive वा zero हुनसक्छ ।
  2. The measure of dispersion is not affected by origin.
    फैलावटको मापनलाई तथ्याङको डाटासेटमा हुने जोड वा घटाउले असर गर्दैन ।
    जस्तैः
    a. 2, 3,4,5,6,7,8
    b. 2+5, 3+5,4+5,5+5,6+5,7+5,8+5
    c. 2-13, 3-13,4-13,5-13,6-13,7-13,8-13
    उतरः a,b,c तिनवटै डाटासेटको measure of dispersion बराबर हुन्छ।
    वा Dispersion(X)= Dispersion (X+5)= Dispersion (X-13)
  3. The measure of dispersion is affected by scale
    फैलावटको मापनलाई तथ्याङको डाटासेटमा हुने गुणन वा भागले असर गर्दछ ।
    जस्तैः
    a. 2, 3,4,5,6,7,8
    b. 2x5, 3x5, 4x5, 5x5, 6x5, 7x5, 8x5
    c. 2/13, 3/13,4/13,5/13,6/13,7/13,8/13
    उतरः a,b,c तिनवटै डाटासेटको measure of dispersion फरक हुन्छ।
  4. The measure of dispersion of constant data is zero
    अचर हुने डाटासेटको फैलावटको मापनलाई zero हुन्छ।
    जस्तैः
    4,4,4,4,4,4,4
    उतरः यस डाटासेटको measure of dispersion zero हुन्छ।

Absolute Dispersion

Quantities that measure the dispersion in the same units as the units of data are called absolute measures of dispersion. Absolute measures of dispersions cannot be used to compare the variation of two or more data set measured in different units. Following are examples of Absolute measures of dispersions.

  1. Range
  2. Quartile Deviation
  3. Mean Deviation
  4. Standard Deviation
  5. Variance
  6. Skewness
  7. Kurtosis

Range

In statistics range is a measure of dispersion to calculate a single best value utilizing extreme data points. It is the index between largest value and smallest value within data set. It is denoted by \(R\) and defined by
\(R = L- S\)
where
\(L\) = largest value in data set and \(S\)= smallest value in data set

Example 1

Find the range of the following data:
\(3, 7, 5, 13, 20, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29\)
Given data set are
\(3, 7, 5, 13, 20, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29\)
Here,
largest data is \(L=56\)
smallest data is \(S=3\)
Thus the range is
\(R=L-S=56-3=53\)

Quartile Deviation (QD)

The presence of extremely high or low value in distribution can reduce the utility of range as a measure of dispersion. Thus, you may need a measure that is not completely affected by the outliers. In such a situation, if the entire data is divided into four equal parts, each containing 25% , we get the values of Quartiles

The upper and lower quartiles (Q3 and Q1, respectively) are used to calculate the Quartile Deviation (semi-interquartile range) which is \(\frac{Q_3-Q_1}{2}\). The semi-interquartile range is based upon the middle 50% of the values in a distribution and is, therefore, not affected by extreme values.

Meaning of QD

Let us suppose that QD of a group of students score = 11
It means,
If the entire group is divided into two equal halves and the median calculated for each half, the median of upper group of students and the median of lower group of students, these medians differ from the median of the entire group by 11 on an average.

For example, suppose Median score of all students is 33, now, if all students are divided into two equal groups of better (Md=44) and poor (Md=22) students. Quartile deviation tell us the average difference between medians of these two groups belonging to better and poor students, from the median of the entire group

Relative Measure of Dispersion

Quantities that measure the dispersion in the form of ratio, percentage or coefficient are called relative measures of dispersion. These quantities have no unit of measurement and dimension and are used to compare the dispersion in two or more data sets measured in different units. Commonly used measure of relative dispersions are:

  1. Coefficient of Range
  2. Coefficient of Quartile Deviation
  3. Coefficient of Mean Deviation
  4. Coefficient of Variance

Coefficient of Variation

The coefficient of variation (CV) is a statistical measure of the dispersion of data points in a data series around the mean. The coefficient of variation represents the ratio of the standard deviation to the mean, and it is a useful statistic for comparing the degree of variation from one data series to another, even if the means are drastically different from one another.
\( CV= \frac{ \text{standard deviation} }{ \text{mean} } \times 100\% \)

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