Question

Which of the following is the most commonly used measure of central tendency?
A. AM
B. Median
C. Mode
D. GM and HM

Answer 1

Hint: To find the most commonly used measure of central tendency, we will have to see what all are included in the measure of central tendency. Mean, median and mode are the commonly used measure of central tendency. Arithmetic mean represents a number that is obtained by dividing the sum of the elements of a set by the total number of values in the set. Geometric mean is the mean value or central term in the set of numbers in geometric progression. Harmonic mean or HM is calculated by dividing the number of values in the sequence by the sum of the reciprocals of the terms in the sequence. Of these 3 meanings, AM is mostly used. Now, we have to find the most commonly used from AM, median and mode. We know that median is the middle value in the data set while mode is the value which is repeated maximum number of times in a given set of observations. Most commonly used one will include all the observations in a given data and has more applications.

Complete step-by-step answer:
We need to find the most commonly used measure of central tendency.
Mean, median and mode are the commonly used measure of central tendency. Let us see the definition and formula of each of this.
Mean is also known as Arithmetic Mean or AM. Arithmetic mean represents a number that is obtained by dividing the sum of the elements of a set by the total number of values in the set. We can write mean as
$\text{AM}=\dfrac{\text{Sum of elements}}{\text{Total number of elements}}$

Now, let us see what Geometric mean is. Geometric mean is the ${{n}^{th}}$ root of the product of n values in a data. It is the mean value or central term in the set of numbers in geometric progression. It can be written as
\[GM=\sqrt[n]{{{a}_{1}}\times {{a}_{2}}\times {{a}_{3}}\times ...\times {{a}_{n}}}\] , where ${{a}_{1}},{{a}_{2}},{{a}_{3}}...,{{a}_{n}}$ are the terms of a series.
Let us recollect what Harmonic Mean is.
Harmonic mean or HM is calculated by dividing the number of values in the sequence by the sum of the reciprocals of the terms in the sequence. We can express this as
$HM=\dfrac{n}{\dfrac{1}{{{a}_{1}}}+\dfrac{1}{{{a}_{2}}}+...+\dfrac{1}{{{a}_{n}}}}$ , where ${{a}_{1}},{{a}_{2}},{{a}_{3}}...,{{a}_{n}}$ are the terms of a sequence.
From these 3 types of mean, if we have to find a mean, we will be using mostly AM.
Now, let us see the definition for median.
Median is the middle value in the data set. We will be using the formula
$\text{median}={{\left[ \dfrac{\left( n+1 \right)}{2} \right]}^{th}}\text{ term}$ if n is odd.
$\text{median}=\dfrac{{{\left( \dfrac{n}{2} \right)}^{th}}\text{ term}+{{\left( \dfrac{n}{2}+1 \right)}^{th}}\text{ term}}{2}$ , if n is even.
Lastly, let us recollect the definition for mode.
Mode is the value which is repeated maximum number of times in a given set of observations.

Of AM, median and mode, the most commonly used measure of central tendency is AM because it includes all the observations in a given data. It also has more applications than others.


So, the correct answer is “Option A”.

Note: Median and mode moves to the second commonly used while GM and HM are the least. Average is also used most commonly in statistics but when measure of central tendency is considered, this is cancelled out.