Absolute and Relative measures of dispersion

Generally speaking, dispersion refers to the state of getting spread. In statistics, dispersion stands for the extent to which the data can vary around the average value. It helps in comprehending the distribution of data. The different measures of dispersion help in understanding the variability of data, of how homogeneous or heterogeneous the data is. It helps us comprehend if the given data is spread or squeezed. 

There are two kinds of data dispersion measures, namely, 

absolute measure of dispersion and relative measure of dispersion. 

Absolute Measure of Dispersion 

The absolute measure of dispersion consists of the same units as the original data set. This measure is used to identify the variations in terms of the average of deviations, such as standard deviation or mean deviation. It is inclusive of standard deviation, range, quartile deviation, etc. 

There are different types of absolute measures of dispersion, such as: 

• Range: Range is the difference between the maximum value present in the data set and the minimum value present. Range = Maximum value – minimum value. 

For example, the set range 1, 3, 5, 7, 9 is 9 – 1 = 8. 

• Variance: For finding variance, the first step is to subtract the mean from each value in the given set and then square each number. Add these squares and divide them by the total number of values present in the set to get the variance. 

σ2 (variance) = ∑(X−μ)2/ N

• Standard Deviation: Square root of the variance is simply known as the Standard deviation. S.D. (Standard Deviation) = √σ

• Quartiles and quartile deviation: Quartile refers to the values which divide the set into quarters. Quartile deviation can be found by dividing the distance between the third and the first quartile by 2. 

• Mean and mean deviation: Mean is the average of all numbers in a given set. Further, calculating the average deviation from the mean value is known as the mean deviation. 

Relative Measures of Dispersion 

The relative measures of dispersion are employed for comparing the distribution of two data sets or more. They are used to compare the different unit sets. These include: 

1. Coefficient of range
2. Coefficient of variation
3. Coefficient of standard deviation
4. Coefficient of quartile deviation
5. Coefficient of mean deviation 

We calculate the dispersion coefficient when two series varying widely in their averages are to be compared. It is also used to compare two series having varied measurement units. The coefficient of dispersion is denoted as C.D. 

The commonly used coefficients of dispersion have been listed below: 

• Coefficient of dispersion in terms of range : 

C.D. = (Xmax – Xmin) ⁄ (Xmax + Xmin)

• Coefficient of dispersion in terms of Quartile deviation: 

C.D. = (Q3 – Q1) ⁄ (Q3 + Q1)

• Coefficient of dispersion in terms of Standard deviation: 

C.D. = S.D. ⁄ Mean

• Coefficient of dispersion in terms of Mean deviation: 

C.D. = Mean deviation/Average

Formulas of Measures of Dispersion

Some useful formulas of measures of dispersion are: 

• Arithmetic means formula: Arithmetic mean is calculated by adding all the numbers in the data set and dividing the result by the total number of values. For example: The arithmetic mean of 1, 2, 3, 4, 5, 6, 7 is (1 +2 + 3 + 4 + 5 + 6 + 7) / 7 = 28/7 = 4

The formula represents it

 A=1ni=1nai

Where, A = Arithmetic Mean

n = number of values

ai= data set values

• Standard deviation formula: The formula for standard deviation helps find dispersed data set values. It can be described as the deviation from the average value. A lower standard deviation suggests that the values are close to their average, and similarly, a higher standard deviation suggests that the values are far from their average. 

Standard deviation can further be categorized into two types: 

• Population Standard Deviation

       =xi-μ2N

           Where σ denotes population standard deviation

• Sample Standard Deviation

s=i=1Nxi– x2N-1

Where s denotes sample standard deviation.

• Quartile formula: The role of the quartile is to divide the observations into four parts. The middle term present between the median and the first term is the lower quartile (Q1). Likewise, the value of the middle term lying between the median and the last term is called the upper quartile (Q3). The second quartile, the median, is denoted by Q2.

When the observations in the set are arranged in ascending order:

The 25th percentile is given by: Q1 = [(n + 1)/4]th term

The 50th percentile or the median is given as Q2 = [(n + 1)/2]th term 

The 75th percentile is given by: Q3 = [3(n + 1)/4]th term

• Variance formula: The variance formula is used to ascertain how far the numbers spread out. It tells how greatly the individual numbers in a given set vary. If there is a huge variation in the set of variables, then the variance number is significant, and if there is less variation, the variance number is small. The variance values come out to be zero if all the values in a set are identical. 

Variance is of two types – variance of population and variance of the sample. 

The variance formula for grouped data (population) is: σ2 = ∑ f (m − x̅)2 / n

The variance formula for grouped data (sample) is: s2 = ∑ f (m − x̅)2/ n-1

Here f denotes the class frequency, and m denotes the midpoint of the class.

• Interquartile range formula: The interquartile range formula measures the variability in a set when divided into different quartiles. The values dividing the set are the first quartile, second quartile, and third quartile. The middle term present between the median and the first term is the lower quartile (Q1). Likewise, the value of the middle term lying between the median and the last term is called the upper quartile (Q3). The second quartile, the median, is denoted by Q2

The formula for interquartile range is IQR =Q3−Q1

IQR denotes the interquartile range, Q3 denotes the third quartile, and Q1 denotes the first quartile.