Question

The mean of five numbers is $18$. If one number is excluded, then their mean is $16$, the excluded number is
A) $24$
B) $26$
C) $28$
D) $25$

Answer 1

In this problem we are going to find an excluded number with the use of the mean values. Here it is given that the mean of five numbers and one number will be excluded so there must be some difference in mean. Also we are given the mean value after the exclusion of one number. Now students aim is to find the excluded number.

Formula used: Mean value = $\dfrac{{{\text{sum of observation}}}}{{{\text{number of observation}}}}$
That is $\overline X$=$\dfrac{1}{n}\left( {\sum {{X_i}} } \right)$ $- - - - - \left( 1 \right)$

Complete step-by-step solution:
It is given that, the mean of $5$ numbers is $18$
Here the number of observations is $5$ and mean value is $18$.
That is, $n = 5$ and $\overline X = 18$
Now students aim to find the sum of observation.
That is to claim ${\text{ }}\sum {{X_i}}$
And $\sum {{X_i} = {X_1} + {X_2} + {X_3} + {X_4} + {X_5}}$
Now substitute the given values in equation $\left( 1 \right)$
We get $18 = \dfrac{1}{5}\sum {{X_i}}$
Cross multiply the equation,
$\Rightarrow$ $18$×$5$ $= \sum {{X_i}}$
$\Rightarrow$ $90 = \sum {{X_i}}$
$\Rightarrow$ ${X_1}$+${X_2}$+${X_3}$+${X_4}$+${X_5}$=$90$ $- - - - - \left( 2 \right)$
That is, sum of observation $= 90$
Let the number has been excluded be ${X_5}$
After the exclusion of one number, then their mean is $16$
Here, $n = 4$ and $\overline X = 16$
$\therefore$Equation $\left( 2 \right)$becomes, ${X_1}$+${X_2}$+${X_3}$+${X_4}$=$90$−${X_5}$
Now the mean value of $4$ numbers is equal to $\dfrac{{90 - {X_5}}}{4}$
If one number is excluded, then their mean is $16$ is given
Therefore, $16 = \dfrac{{90 - {X_5}}}{4}$
Cross multiply the equation, we get
$\Rightarrow$ $16$×$4$=$90$−${X_5}$
$\Rightarrow$ $64$=$90$−${X_5}$
$\Rightarrow$ ${X_5}$=$90$−$64$
$\Rightarrow$ ${X_5}$=$26$
Therefore the excluded number is $26$

The answer is option $\left( B \right)$

Additional information: There are several kinds of mean in mathematics, especially in statistics. For a data set, the arithmetic mean, also called the expected value or average, is the central value of a discrete set of numbers, specifically the sum of the values divided by the number of values.

Note: The simple way to find mean value is just like an average.
Alternative method: Mean of $5$ numbers =$18$
$\Rightarrow$ Sum of those $5$numbers =$18 \times 5$=$90$
Let x be the excluded number
New mean = $16 =$ $\dfrac{{90 - x}}{4}$ $- - - - - \left( * \right)$
Solving equation $\left( * \right)$, we get
$90 - x = 64$
$\therefore {\text{ x = 26}}$