Question

: which of the following are dimensionless
A. $\text{S}\text{.D}$
B. $\text{M}\text{.D}$
C. Variance
D. Coefficient of variation.

Answer 1

Hint: In this problem we need to find the dimensionless variable among the given variables. So we will first consider each variable and write the definition of the variable. From the definition we will analyze the dimension of the variable. After having the dimensions of all the variables we will check for the dimensionless variable.

Complete step by step answer:
Given variables are $\text{S}\text{.D}$ , $\text{M}\text{.D}$ , variance and coefficient of variation.
Consider the variable $\text{S}\text{.D}$. The Standard deviation of the data tells us how much the members of the data are differing from the mean value of the data. We know that mean is a dimensioned variable. So the difference between mean and an observation always has a dimension.
Consider the variable $\text{M}\text{.D}$, the mean deviation of the data tells us how much the members of the data differ from the mean or median of the data. So the mean deviation is also a dimensioned number.
Consider the variable variance, The variance is a measure of variability. It is calculated by taking the average of squared deviations from the mean. Variance tells you the degree of spread in your data set. So the variance is also a dimensioned variable.
Consider the variable coefficient of variation, The coefficient of variation is defined as the ratio of the standard deviation to the mean. So the dimensions get cancelled to each other which results to a dimensionless number.
Hence the coefficient of variance is the dimensionless number.

So, the correct answer is “Option D”.

Note: The topic related to coefficient of variance plays a key role in competitive exams. So some additional information about the coefficient of variance is given here. It shows the extent of variability in relation to the mean of the population. The coefficient of variation is useful because the standard deviation of data must always be understood in the context of the mean of the data. In contrast, the actual value of the CV is independent of the unit in which the measurement has been taken, so it is a dimensionless number.