Question

Following data gives age($X$ in years) of students in a particular school and their marks in G. K. test. Prepare a bivariate frequency distribution.
$X$: $11$ $11$ $10$ $13$ $12$ $10$ $12$ $12$ $13$ $10$ $11$ $13$ $12$ $12$ $11$ $13$ $10$ $10$ $11$
$Y$: $22$ $24$ $23$ $21$ $24$ $21$ $22$ $23$ $24$ $24$ $22$ $23$ $21$ $24$ $21$ $22$ $24$ $22$ $23$ $24$
Also obtain
(i) Marginal frequency distributions of age and marks in G.K.
(ii) Conditional frequency distribution of age when marks in G.K. are $23$.
(iii) Conditional frequency distribution of marks in G.K. when the age is $11$ years.

Answer 1

Write values of $X$ horizontally and $Y$ vertically.
$X$ takes values from $10$ to $13$ and $Y$ takes values from $21$ to $24$. Therefore the two way table will contain $16$ cells.
The first student is $11$ years old and has $22$ marks in G.K.
Therefore, we put tally mark in the cell corresponding to $X=11$ and $Y=22$ etc.
Bivariate frequency distribution table of $X$ and $Y$.

Y $\downarrow$ / X$\to$	10	11	12	13	$f_y$
21	$|$ (1)	_	$||$ (2)	$|$ (1)	4
22	$|$ (1)	$|||$ (3)	$|$ (1)	_	5
23	$||$ (2)	_	$|$ (1)	$|$ (1)	4
24	$|$ (1)	$||$ (2)	$||$ (2)	$||$ (2)	7
$f_x$	5	5	6	4	20

(i a.) Marginal frequency distribution of $X$:

X	10	11	12	13	Total
F	5	5	6	4	20

(i b.) Marginal frequency distribution of $Y$:

X	21	22	23	24	Total
F	4	5	4	7	20

(ii) Conditional frequency distribution of $X$ when $Y=23$

X	10	11	12	13	Total
F	2	2	1	1	4

(iii) Conditional frequency distribution of $Y$ when $X=11$

X	21	22	23	24	Total
F	0	3	0	2	5