Question

The median of the observations 30, 91, 0, 64, 42, 80, 30, 5, 117, 71 is

Answer 1

Mean is the middle value when a set of data values are arranged in the order from lowest to highest. Median is a statistical value in statistics and probability theory. If the number of observations in a given set of data are odd, then the mean is the middle value of the observations. But if the number of observations are even in the data set, then the median is the average of the two observations in the middle.

Complete step-by-step solution:
Given the set of observations, while these are not arranged in order, i.e, they are not in the order from lowest to highest.
$\therefore$Arranging the set of observations in the order of lowest to highest, is the same as arranging them in the increasing order.
The increasing order of the set of observations is given by:
$\Rightarrow 0,5,30,30,42,64,71,80,91,117$
Now the median of the above set of observations is the middle value of the set, as it is arranged in increasing order already.
But here the total no. of observations is 10, which is an even number.
$\therefore$The median is the average of the two middle most values in the observations.
$\Rightarrow$The $1^{st}$ observation is 0.
$\Rightarrow$The $2^{nd}$ observation is 5.
$\Rightarrow$The $3^{rd}$ observation is 30.
$\Rightarrow$The $4^{th}$ observation is 30.
$\Rightarrow$The $5^{th}$ observation is 42.
$\Rightarrow$The $6^{th}$ observation is 64.
$\Rightarrow$The $7^{th}$ observation is 71.
$\Rightarrow$The $8^{th}$ observation is 80.
$\Rightarrow$The $9^{th}$ observation is 91.
$\Rightarrow$The $10^{th}$ observation is 117.
Here the middle most values in the set of observations are $5^{th}$ observation and $6^{th}$ observation.
$\therefore$The median here is the average of $5^{th}$ and $6^{th}$ observations.
5th observation is 42, whereas 6th observation is 64.
$\Rightarrow$The median is the average of 42 and 64, given by:
$\Rightarrow \dfrac{{42 + 64}}{2} = \dfrac{{106}}{2}$
$\Rightarrow \dfrac{{106}}{2} = 53$
$\therefore$Median = 53.

The median of the observations is 53.

Note: While finding the median of the given data, it is very important to remember that first the given set of observations has to be arranged in increasing order, and only then we can proceed to find the median of the data.