Question

Which one of the following is a measure of dispersion?
A. Mean
B. Median
C. Mode
D. Standard deviation

Answer 1

Hint: Dispersion in statistics means the extent to which a numerical data is likely to vary about an average value. In other words, we can say that dispersion helps to understand the distribution of the data. The measures of dispersion helps us to interpret the variability of data i.e. to know how homogeneous or heterogeneous the data is. It shows us how squeezed or scattered the variable is. There are two types, that is, Absolute Measure of Dispersion and Relative Measure of Dispersion. Absolute Measure of Dispersion includes range, variance, standard deviation, quartiles and quartile deviation, and mean and mean deviation. When we look into Relative Measure of Dispersion, we can see coefficient of range, coefficient of variation, coefficient of standard deviation, coefficient of quartile deviation and coefficient of mean deviation.

Complete step-by-step answer:
We have to find the measure of dispersion from the given options. Let’s see what dispersion means.
Dispersion is the extent to which the distribution is stretched, squeezed or spread. Dispersion in statistics means the extent to which a numerical data is likely to vary about an average value. In other words, we can say that dispersion helps to understand the distribution of the data.
Now, let us see what measure of dispersion is. The measures of central tendency are not appropriate to describe data. Two data sets can have the same meaning but they can be completely different. Hence, if we want to describe data, we have to know the extent of variability. This is given by the measures of dispersion.
We can also say that the measures of dispersion helps us to interpret the variability of data i.e. to know how homogeneous or heterogeneous the data is. It shows us how squeezed or scattered the variable is.
Let us understand the types of measures of dispersion. There are two types, that is, Absolute Measure of Dispersion and Relative Measure of Dispersion.
Let us see the types in Absolute Measure of Dispersion. These include range, variance, standard deviation, quartiles and quartile deviation, and mean and mean deviation. When we look into Relative Measure of Dispersion, we can see coefficient of range, coefficient of variation, coefficient of standard deviation, coefficient of quartile deviation and coefficient of mean deviation.
Now, let us observe the given options. From the above mentioned measure of dispersion, we can see that mean, median and mode are not measures of dispersion but standard deviation is. Mean, median and more are central tendencies not measures of dispersion.
Standard deviation (SD) is a measure of spread of data about the mean. It is the square root of the sum of squared deviation from the mean divided by the number of observations. We can represent SD as
\[\sigma =\sqrt{\dfrac{\sum{{{\left( {{x}_{i}}-\mu \right)}^{2}}}}{N}}\]
Where $N$ is the size of the population, ${{x}_{i}}$ is each value from the population and $\mu $ is the population mean.

So, the correct answer is “Option D”.

Note: You may consider mean to be a measure of dispersion since the absolute measure of dispersion consists of mean and mean deviation. It doesn’t mean that “mean” is a measure of dispersion. We can see that standard deviation is the square root of variance.