Question

The list of numbers 41, 35, 30, X, Y, 15 has a median of 25. The mode of the list of numbers is 15. To the nearest whole number, what is the mean of the list?

(a) 20
(b) 25
(c) 26
(d) 27
(e) 30

Answer 1

Hint: Using the formula of median, find the value of X. Here numbers are even, so use the formula for that. Mode is 15, the Y is the same as the mode. Thus take mean or average of the 6 numbers in the list.

Complete step-by-step answer:
Before we get on with this question, let us understand what is mean, median and mode.

The mean is the average you used here you add up all the numbers and then divide by the number of numbers.

The median is the middle value in the list of numbers. To find the median, your number has to be listed in numerical order from smallest to largest, so you may have to rewrite your list before you can find the median.

The mode is the value that occurs most often. If no number in the list is repeated, then there is no mode for the list.

We have been given a list of 6 numbers i.e. the number of terms are even.

41, 35, 30, X, Y, 15, here n = 6.

First we need to find the value of X and Y with the help of the formula of the median. We have been given the median of the list of numbers as 25.

\[\therefore \] Median = \[{{\dfrac{n}{2}}^{th}}\] observation + \[\left( {{\dfrac{n}{2}}^{th}}+1

\right)\] observation / 2

                    =\[\left( {{\dfrac{6}{2}}^{th}} \right)\] observation + \[{{\left( \dfrac{6}{2}+1

\right)}^{th}}\] observation / 2

                    = \[{{3}^{rd}}\] observation + \[{{4}^{th}}\] observation / 2

From the list the \[{{3}^{rd}}\]and \[{{4}^{th}}\] observations are 30 and X.

Median \[=\dfrac{30+X}{2}\]

\[\begin{align}

  & \therefore 25=\dfrac{30+X}{2} \\

 & X+30=50 \\

 & \therefore X=50-30=20 \\

\end{align}\]

Thus we got the \[{{4}^{th}}\] term on the list as, X = 20.

Given mode = 15, which means that mode is a frequent occurring observation as we have told.

\[\therefore Y=15\]

Thus the list becomes 41, 35, 30, 20, 15, 15.

Now let us find the Mean,

Mean = Sum of all the numbers / Number of total numbers = \[\dfrac{41+35+30+20+15+15}{6}\]

Mean = \[\dfrac{156}{6}=26\]

Thus we got the mean of the list as 26.

\[\therefore \] Option (c) is the correct answer.


Note: Depending on the list of numbers, the above data may vary. Thus it is important that you know the basic concepts of mean, median and mode before solving these kinds of questions.

If the list of number is odd, then median is given as,

Median = number of data points + 1 / 2

Or you can also go from both ends until you meet the middle term, if the list is short.