Question

Find the median of the following data: 15, 6, 16, 8, 22, 21, 9, 18, 25.

Answer 1

Hint: We first define what median is. Then we discuss different cases for different types of data. We also discuss how to find the median in those different cases. We use a formula of the median for discrete data to find the solution.

Complete step-by-step solution:
The median of a variable is defined at the middlemost value when its values are arranged in ascending or descending order of magnitude. The value divides the whole set into two parts such that half of the observations are less than or equal to it and half are more than or equal to it.
We can divide any given observations into two parts. One being continuous grouped data and the other one discrete data. In both the cases the middlemost value (for a series ${{\left( \dfrac{n+1}{2} \right)}^{th}}$) represents the median of the given observations.
Now for discrete data we first arrange the given data in order, preferably in ascending order. if n is even then there would be two values as middlemost. The values being ${{\left( \dfrac{n}{2} \right)}^{th}}$ and ${{\left( \dfrac{n}{2}+1 \right)}^{th}}$. We take the average of those two values to find the median.
If the value n represents odd then we need only ${{\left( \dfrac{n+1}{2} \right)}^{th}}$.
For our given problem the given data is 15, 6, 16, 8, 22, 21, 9, 18, 25.
There are nine given observations. These are all discrete data.
We arrange them in ascending order and get 6, 8, 9, 15, 16, 18, 21, 22, 25.
The value of n being odd we take the ${{\left( \dfrac{n+1}{2} \right)}^{th}}$ value of the series as the median.
So, ${{\left( \dfrac{n+1}{2} \right)}^{th}}={{\left( \dfrac{9+1}{2} \right)}^{th}}={{5}^{th}}$ point in the ordered series would be median. The value is 16. So, the median is 16.

Note: We need to be careful about the data being grouped data or discrete data. As for different types of data, the formula will be different. We also find the odd and even notion of the given observations.