Question

Find the mean of following data 30, 32, 24, 34, 26, 28, 30, 35, 33, 25
(i) Show that the sum of deviations of all the given observations from the mean is zero.
(ii) Find the median of the given data.

Answer 1

Hint: For solving this problem we use simple formulas of mean, mean deviation from mean and median. The mean \['\bar{x}'\] of a data is given as \[\bar{x}=\dfrac{\text{sum of observations}}{\text{number of observations}}\]. From the mean we calculate the sum of mean deviations as \[\sum\limits_{i}{\left( {{x}_{i}}-\bar{x} \right)}\] where \[{{x}_{i}}\] represents \[{{i}^{th}}\] observation in the data. Finally for finding the median of data first we arrange the given observations in ascending order, then, the middle term gives the median of the data. If the number of observations is even then the average of the middle two terms gives the median.

Complete step-by-step solution:
First, let us write down the given observation that is
30, 32, 24, 34, 26, 28, 30, 35, 33, 25
Here, we can see that there are 10 observations.
So, the number of observations is 10.
Let us assume that \[\bar{x}\] is the mean of given data.
We know that the formula of mean is
\[\bar{x}=\dfrac{\text{sum of observations}}{\text{number of observations}}\]
By applying the mean formulae to given observations we get
\[\begin{align}
  & \Rightarrow \bar{x}=\dfrac{30+32+24+34+26+28+30+35+33+25}{10} \\
 & \Rightarrow \bar{x}=\dfrac{297}{10}=29.7 \\
\end{align}\]
Therefore, the mean of given data is 29.7.
Now, let us solve the first part.
(i) For finding the sum of deviations let us create a table having the given observations in one column and deviation about mean that is \[\left( {{x}_{i}}-\bar{x} \right)\] on other column.

Observations (\[{{x}_{i}}\])	Deviations about mean \[\left( {{x}_{i}}-\bar{x} \right)\]
30	0.3
32	2.3
24	-5.7
34	4.3
26	-3.7
28	-1.7
30	0.3
35	5.3
33	3.3
25	-4.7

\[\begin{align}
&\Rightarrow \sum\limits_{i}{\left( {{x}_{i}}-\bar{x} \right)}=0.3+2.3+\left( -5.7 \right)+4.3+\left( -3.7 \right)+\left( -1.7 \right)+0.3+5.3+3.3+\left( -4.7 \right) \\
 & \Rightarrow \sum\limits_{i}{\left( {{x}_{i}}-\bar{x} \right)}=0 \\
\end{align}\]
Therefore, the sum of deviations of given data from mean is zero.
Now, let us solve the second part.
(ii) We know that the median of the observations is the middle value of observations after arranging them in ascending or descending order.
By arranging the given data in ascending order we get
24, 25, 26, 28, 30, 30, 32, 33, 34, 35
Here, we can see that there are even a number of observations. So, the median is the average of the middle two terms.
We know that the middle two terms are 30, 30
So, we know that the average of two equal numbers is the same as that number. So, the average of 30, 30 is also 30. So, we can say that the median is 30.

Note: Students may make mistakes in calculating the sum of deviations. We know that mean deviation is given by formula \[\sigma =\dfrac{\sum{\left| {{x}_{i}}-\bar{x} \right|}}{N}\]. But students consider the sum of deviations as \[\left| {{x}_{i}}-\bar{x} \right|\] which is wrong. There is no need for modulus for the sum of deviations. But while calculating the mean deviation then we need to consider the modulus. Also, while calculating the median we have to take repeated numbers also in ascending order. Students may miss repeated numbers and proceed for the solution which gives the wrong answer.