Question

What is the difference between the mean, median, and mode?

Answer 1

Hint: We need to find the difference between the mean, median, and mode. We start to solve the given question by defining the terms mean, median, and mode. Then, we find the differences between the mean, median, and mode through examples to get the desired result.

Complete step by step answer:
We are asked to find out the differences between the mean, median, and mode. We will be solving the given question by defining the terms and finding out the differences between them through examples.
The concept of mean is widely used in mathematics and statistics. It is the average of two or more numbers.
The mean is also referred to as arithmetic mean and is given as follows,
$\Rightarrow mean={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{a}_i$
Here,
n is the number of values
$a_i$ is the data set values.
In simple terms, It is defined as the ratio of the sum of all the values in a set of numbers to the total number of values.
Writing the above lines in the form of the equation, we get,
$\Rightarrow Mean\text{ = }{\displaystyle \frac{\text{sum of all the terms}}{\text{total number of terms}}}$
The median is the middlemost number of the data set. It is the center value in a sorted list of numbers. The median is the mid-value in the sorted list of numbers.
We must arrange the given numbers in ascending order before computing the median for the data. The formula of median depends upon the total number of observations(n).
If n is odd, then $median={\left({\displaystyle \frac{n+1}{2}}\right)}^{th}term$
If n is even, then $median={\displaystyle \frac{({\left({\displaystyle \frac{n}{2}}\right)}^{th}term+{(\left({\displaystyle \frac{n}{2}}\right)+1)}^{th}term)}{2}}$
The mode is the value that has a higher frequency in the data set. In simple terms, it is the value that appears more often in the data set.
Let us understand the concept of mean, median, and mode through an example.
Example:
Find the mean, median, and mode of the 1, 4, 6, 4, 5.
In our case,
$\Rightarrow total\text{ no of observations = 5}$
We know that the mean the ratio of the sum of all the values in a set of numbers to the total number of values.
Writing the above in the form of the equation, we get,
$\Rightarrow Mean\text{ = }{\displaystyle \frac{\text{sum of all the terms}}{\text{total number of terms}}}$
Substituting the values in the above equation, we get,
$\Rightarrow Mean\text{ = }{\displaystyle \frac{1+4+6+4+5}{5}}$
Simplifying the above equation, we get,
$\Rightarrow Mean\text{ = }{\displaystyle \frac{20}{5}}$
Canceling the common terms, we get,
$\therefore Mean\text{ = 4}$
Arranging the observations in the ascending order, we get,
$\Rightarrow 1,4,4,5,6$
In our case, the total number of observations is odd. The formula of the median is given as follows,
$\Rightarrow median={\left({\displaystyle \frac{n+1}{2}}\right)}^{th}term$
Substituting the value n = 5 in the above equation, we get,
$\Rightarrow median={\left({\displaystyle \frac{5+1}{2}}\right)}^{th}term$
Simplifying the above equation, we get,
$\Rightarrow median={\left({\displaystyle \frac{6}{2}}\right)}^{th}term$
Canceling the common factors, we get.
$\Rightarrow median=3^{rd}term$
From the above,
$\therefore median=4$
In the given data set, the number 4 has the highest frequency in the set. So,
$\therefore mode=4$

Note: The mean, median, and mode are used to measure the central location of the data set. In statistics, the relationship between the mean, mode, and median is given by $Mode=3Median-2Mean$ .