Question

Sam's test scores are history 76, geography 74, math 92, english 81 and chemistry 80. If the average (arithmetic mean) score is M and the median score is m, what is the value of M-m?
\[\begin{align}
  & A.\text{ }0.4 \\
 & B.\text{ }0.5 \\
 & C.\text{ }0.6 \\
 & D.\text{ }0.8 \\
\end{align}\]

Answer 1

Hint: In this question, we are given test scores of five subjects and we have to find the difference between the mean and median of the data. For this, we will calculate the mean and median of the data separately. For finding the mean, we will add the scores and divide it by the number of subjects. For finding the median, we will rearrange the data in ascending order and then find the middle term. Middle term can be found by taking ${{\left( \dfrac{n+1}{2} \right)}^{th}}$ term from data where n is the total number of terms (n is odd).

Complete step-by-step solution:
We are given scores as history 76, geography 74, math 92, English 81, chemistry 80. Since we are concerned with numbers only, so our data becomes 76, 74, 92, 81, 80.
Firstly, let us calculate the mean of the data. As we know, mean is given by dividing the sum of all terms by the number of terms, so we can see from data that the number of terms are 5. Sum of the term is equal to $\text{76}+\text{74}+\text{92}+\text{81}+\text{8}0=\text{4}0\text{3}$. So,
\[\text{Mean}=\dfrac{\text{Sum of all terms}}{\text{Number of terms}}\]
Hence, \[\text{Mean}=\dfrac{\text{403}}{\text{5}}=80.6\]
Since, mean is represented by M, 80 M for this data becomes equal to 80.6
\[M=80.6\cdots \cdots \cdots \cdots \cdots \left( 1 \right)\]
Now, we have to find the median of the data.
For this, let us rearrange the data first. Rearranging data in ascending order, we get:
74, 76, 80, 81, 92.
As we know, the number of terms, n = 5. So, ${{\left( \dfrac{n+1}{2} \right)}^{th}}$ term will be ${{\left( \dfrac{6}{2} \right)}^{th}}$ terms.
Hence, ${{3}^{rd}}$ term will be our median. From data, we can see that 80 is the ${{3}^{rd}}$ term. Hence, the median of the data is 80.
Since, the median is represented by m, so m for this data becomes equal to 80.
\[m=80\cdots \cdots \cdots \cdots \cdots \left( 2 \right)\]
Now, we have to find M-m. So taking difference of (1) and (2) we get:
\[\begin{align}
  & M-m=80.6-80 \\
 & M-m=0.6 \\
\end{align}\]
Here 0.6 is the required answer. Hence, option C is the correct answer.

Note: Students should not forget to rearrange the data before finding median. Students should note that, while finding median for odd number of terms, we take ${{\left( \dfrac{n+1}{2} \right)}^{th}}$ term and when number of terms are even, we take average of ${{\left( \dfrac{n}{2} \right)}^{th}}\text{ and }{{\left( \dfrac{n+1}{2} \right)}^{th}}$ term to find our answer.