Question

The median of the observations 22,24,33,37, $ x+1 $ , $ x+3 $ , 46,47,57,58 in ascending order is 42. What are the values of 5th and 6th observations respectively?\[\]
A.42,43\[\]
B.41,43\[\]
C.43,46\[\]
D.40,40\[\]

Answer 1

Hint: If the number of data is odd then the middle term is the median. If the number of data is even then the median is half of the sum of two middle terms. The number of observations is even. So half of the sum of two middle terms is 42. Solve the equation when after putting the values. \[\]

Complete step-by-step answer:
Median is a measure of central tendency of a set of data points (observations) specifically used to separate the data of the lower half from the upper half. Median can be obtained when we arrange the data either in ascending or descending order. If the number of data is odd then the middle term is the median. If the number of data is even then the median is half of the sum of two middle terms.
Mathematically, if there are $ n $ number of data points say $ {{x}_{1}},{{x}_{2}},...,{{x}_{n}} $ in ascending or descending order then if $ n $ is odd the median will be $ {{x}_{\dfrac{n+1}{2}}} $ and if $ n $ is odd then the median will be $ \dfrac{1}{2}\left( {{x}_{\dfrac{n}{2}}}+{{x}_{\dfrac{n}{2}+1}} \right) $ .\[\]
The given set of observations is 22,24,33,37, $ x+1 $ , $ x+3 $ , 46,47,57,58 which is arranged in ascending order . \[\]
We see that the number of observations is $ n=10 $ . So we median will be half sum of $ \dfrac{n}{2}={{5}^{\text{th}}} $ and $ \dfrac{n}{2}+1={{6}^{\text{th}}} $ term of the observation. We have the $ {{5}^{\text{th}}} $ term as $ x+1 $ and the $ {{6}^{\text{th}}} $ term as $ x+3 $ . The median of the given set of observations is given as 42. So
\[\begin{align}
  & \dfrac{\left( x+1 \right)+\left( x+3 \right)}{2}=42 \\
 & \Rightarrow \dfrac{2x+4}{2}=42 \\
 & \Rightarrow \dfrac{2\left( x+2 \right)}{2}=42 \\
 & \Rightarrow x+2=42 \\
 & \Rightarrow x=40 \\
\end{align}\]
So the $ {{5}^{\text{th}}} $ term is $ x+1=41 $ and the $ {{6}^{\text{th}}} $ term is $ x+3=43.

So, the correct answer is “Option B”.

Note: We need to be careful of the confusion between mean, median and mode. The mean is the sum of values of a data set divided by the number of values, median is the middle value separating the greater and lesser halves of the observation and mode is the largest value of in the observation.