Question

Consider the given data with frequency distribution

xi	3	8	11	10	5	4
fi	5	2	3	2	4	4

Match each entry in List-I to the correct entries in List-II. 

	List-I		List-II
(P)	The mean of the above data is	(1)	2.5
(Q)	The median of the above data is	(2)	5
(R)	The mean deviation from the mean of the above data is	(3)	6
(S)	The mean deviation from the median of the above data is	(4)	2.7
		(5)	2.4

The correct option is: 

• A. (P)\(\rightarrow\)(3),(Q)\(\rightarrow\)(2),(R)\(\rightarrow\)(4),(S)\(\rightarrow\)(5)
• B. (P)\(\rightarrow\)(3) (Q)\(\rightarrow\)(2) (R)\(\rightarrow\)(1) (S)\(\rightarrow\)(5)
• C. (P)\(\rightarrow\)(2) (Q)\(\rightarrow\)(3) (R)\(\rightarrow\)(4) (S)\(\rightarrow\)(1)
• D. (P)\(\rightarrow\)(3) (Q)\(\rightarrow\)(3) (R)\(\rightarrow\)(5) (S)\(\rightarrow\)(5)

Answer 1

The Correct Option is A

Mean, Median, and Mean Deviation Calculations 

The correct answer is (A).

Mean Calculation:

Mean is given by:

\[ \frac{3 \times 5 + 8 \times 2 + 11 \times 3 + 10 \times 2 + 5 \times 4 + 4 \times 4}{5 + 2 + 3 + 2 + 4 + 4} \]

Simplifying the equation:

\[ = \frac{15 + 16 + 33 + 20 + 20 + 16}{20} = \frac{120}{20} = 6 \]

Median Calculation:

Median is the average of the 10^th and 11^th observations:

\[ = \frac{1}{2}(5 + 5) = 5 \]

Mean Deviation about the Mean:

Mean deviation about the mean is:

\[ \frac{54}{20} = 2.7 \]

Mean Deviation about the Median:

Mean deviation about the median is:

\[ \frac{48}{20} = 2.4 \]

Answer 2

First, arrange data in ascending order

xi	3	4	5	8	10	11
fi	5	4	4	2	2	3

 

 

\(\begin{array}{|c|c|c|c|c|} \hline x_i & f_i & f_i x_i & f_i\left|x_i-\bar{x}\right| & f_i\left|x_i-N\right| \\ \hline 3 & 5 & 15 & 15 & 10 \\ \hline 4 & 4 & 16 & 8 & 4 \\ \hline 5 & 4 & 20 & 4 & 0 \\ \hline 8 & 2 & 16 & 4 & 6 \\ \hline 10 & 2 & 20 & 8 & 10 \\ \hline 11 & 3 & 33 & 15 & 18 \\ \hline  & \Sigma f_i=20 & \Sigma f_i x_i=120 & \text { sum }=54 & \text { sum }=48 \\ \hline \end{array}\)

Mean = \(\frac{\sum f_ix_i}{\sum f_i}\)
\(=\frac{120}{20}=6\)

Median = \(\frac{n^{th}+(n^{th}+1)\ \text{observation}}{2}\)
\(n=\frac{\sum f_i}{2}=\frac{20}{2}=10\)
\(Median\ =\frac{10^{th} +11^{th}}{2}=\frac{5+5}{2}=5\)

Mean Deviation about mean
 \(\frac{\sum f_i|x_i-\bar{x}|}{\sum f_i}=\frac{54}{20}=2.7\)

Mean Deviation about Median
\(\frac{\sum f_i|x_i-M|}{\sum f_i}=\frac{48}{20}=2.4\)

So, the correct option is (A):(P)\(\rightarrow\)(3),(Q)\(\rightarrow\)(2),(R)\(\rightarrow\)(4),(S)\(\rightarrow\)(5)