Question

Construct a bivariate frequency distribution table of the marks obtained by students in English (X) and statistics (Y).

Marks in statistics(X)	37	20	46	28	35	26	41	48	32	23	20	39	47	33	27	26
Marks in English(Y)	30	32	41	33	29	43	30	21	44	38	47	24	32	21	20	21

Construct a bivariate frequency distribution table for the given data by taking class interval 20 – 30, 30 – 40 etc. for both X and Y. Also find the marginal distribution and conditional frequency distribution of Y where X lies between 30 – 40.

Answer 1

Hint: For solving this question you should know about the Bivariate frequency distribution table. In this question first we will make the Bivariate frequency distribution table and then find marginal frequency distribution of English and of Statistics. And then we find the conditional frequency distribution of Y when X lies between 30 – 40.

Complete step-by-step solution:
According to our question we will find the Bivariate frequency distribution table, and marginal distribution table for both subjects and find the conditional frequency distribution table.
So, if we see our marks obtained by students in subjects, then:

Marks in statistics(X)	37	20	46	28	35	26	41	48	32	23	20	39	47	33	27	26
Marks in English(Y)	30	32	41	33	29	43	30	21	44	38	47	24	32	21	20	21

As we can see that the minimum marks obtained in any of subjects from English and Statistics is 20 and the maximum marks obtained in any of subjects from English and Statistics is 50.
So, we will start by making a class interval from 20 with the class size of 10. So, we will get the class intervals as 20-30, 30-40 and 40-50.
It is known that the frequency distribution of a single variable is called univariate distribution. When a data set consists of a large mass of observations, they may be summarized by using a two-way table.
A two-way table is associated with two variables, say X and Y. For each variable, a number of classes can be defined keeping in view the same considerations as in the univariate case.
When there are m classes for X and n classes for Y, there will be m × n cells in the two-way table. The classes of one variable may be arranged horizontally, and the classes of another variable may be arranged vertically in the two way table. By going through the pairs of values of X and Y, we can find the frequency for each cell.
The whole set of cell frequencies will then define a bivariate frequency distribution.
So, the Bivariate frequency distribution table:

$Y\backslash X$	20 – 30	30 – 40	40 – 50	\[{{f}_{y}}\]
20 – 30	11 (2)	11 (2)	1 (1)	5
30 – 40	111 (3)	11 (2)	11 (2)	7
40 – 50	11 (2)	1 (1)	1 (1)	4
\[{{f}_{x}}\]	7	5	4	16

Marginal frequency distribution of English (X): -

X	20 – 30	30 – 40	40 – 50	Total
F	7	5	4	16

Marginal frequency distribution of Statistics (Y): -

X	20 – 30	30 – 40	40 – 50	Total
F	5	7	4	16

A conditional distribution is a probability distribution for a sub-population. In other words, it shows the probability that a randomly selected item in a sub-population has a characteristic you’re interested in.
Conditional frequency distribution of y when x lies between 30 – 40: -

X	20 – 30	30 – 40	40 – 50	Total
F	2	2	1	5

So, all tables are formed.

Note: During calculating the bivariate table for the given data always mind that all the data should be calculated and counted on. If any data is left then all the questions will be wrong and the difference will be different for every different number. It can be the same if that difference comes in that difference.