Question

Find the mean and median of the following data:
 $
  i)\,\,13,17,16,14,11,13,10,16,11,18,12,17 \\
  ii)\,36,72,46,42,60,45,53,46,51,49 \\
  $

Answer 1

Hint: To find the mean of given data we first find the sum of observation and then divide it by the number of observations to get the mean of the data.
To find median we first arrange given terms in either ascending or descending order and then to see that the number of observations are odd or even. If odd then median will be $ \dfrac{{N + 1}}{2} $ th term and if even then median will be $ \dfrac{N}{2}\,\,and\,\,\dfrac{N}{2} + 1 $ terms of the series.

Complete step-by-step answer:
Given observation or terms are
 $ \,13,17,16,14,11,13,10,16,11,18,12,17 $
We first find the meaning of a given observation.
As, given observations are single observations. So we use direct methods to find their meaning.
In this method we first find the sum of given observations.
Sum of observations given as:
 $\Rightarrow \dfrac{{\,13 + 17 + 16 + 14 + 11 + 13 + 10 + 16 + 11 + 18 + 12 + 17}}{{12}} $
 $
   = \dfrac{{168}}{{10}} \\
   = 16.8 \;
  $
Therefore, from above we see that mean of the data is $ 16.8 $
Now, we will calculate the median of the given observation.
As, data is in single observation. So, to find the median of given data we first arrange the given observation in either ascending order or descending order.
Arranging given observation in ascending order. WE have,
 $ 10,11,11,12,13,13,14,16,16,17,17,18 $
Now, we will see observations are in even numbers or in odd numbers.
We see that there are $ 12 $ observation.
So, there are even numbers of observations.
For even number of observation median is given as:
Median = $ \dfrac{N}{2}\,\,and\,\,\,\dfrac{N}{2} + 1 $
Therefore, median of given data is given as:
$\Rightarrow \dfrac{{12}}{2} = 6th\,\,and\,7th\,\,terms $ of the above series.
Hence, medians are $ 13\,\,and\,\,14 $ .
Therefore, from above we see that the mean is $ 16.8\, $ and the median are $ 13\,\,and\,\,14 $ of given data.
We take average of the two terms then median becomes $13.5$

ii)
Given observation or terms are
 $ \,36,72,46,42,60,45,53,46,51,49 $
We first find the meaning of given observation.
As, given observations are single observations. So we use direct methods to find their meaning.
In this method we first find the sum of given observations.
Sum of observations given as:
\[
\Rightarrow \dfrac{{36 + 72 + 46 + 42 + 60 + 45 + 53 + 46 + 51 + 49}}{{10}} \\
   = \dfrac{{500}}{{10}} \\
   = 50 \;
 \]
Therefore, from above we see that mean of the data is $ 50 $
Now, we will calculate the median of the given observation.
As, data is in single observation. So, to find the median of given data we first arrange the given observation in either ascending order or descending order.
Arranging given observation in ascending order. We have,
 $ \,36,42,45,46,46,49,51,53,60,72 $
Now, we will see observations are in even numbers or in odd numbers.
We see that there are $ 10 $ observation.
So, there are even numbers of observations.
For even number of observation median is given as:
Median = $ \dfrac{N}{2}\,\,and\,\,\,\dfrac{N}{2} + 1 $
Therefore, median of given data is given as:
$\Rightarrow \dfrac{{10}}{2} = 5th\,\,and\,\,6th\,\,terms $ of the above series.
Hence, medians are $ 46\,\,and\,\,49 $ .
Therefore, from above we see that the mean is $ 50 $ and median are $ 46\,\,and\,\,49 $ of given data.
We take average of the two terms then median becomes $47.5$

Note: There are different methods to find mean and median of data. Therefore, we first see which type of data is given as data can be as in form of single observations, discrete data form or in continuous form. Then we will apply the corresponding formula or method to find the mean and median of the data and hence the solution of the given problem.