The marginal distribution of X is obtained from the:

Row totals. Totals across each class of X.

Which of the following is a categorical variable?

Blood group. Blood group is an attribute, not a measurement.

A contingency table shows frequencies of:

Two categorical variables. It cross-classifies two attributes.

The expected frequency of a cell is:

(Rᵢ × Cⱼ)/N. Eᵢⱼ = RᵢCⱼ/N.

Maharashtra State Board · HSC Commerce · Class 11 (Std XI)

Class 11 Maths and Statistics Part 2 Chapter 4: Bivariate Frequency Distribution

Short notes, key concepts, quick revision and practice MCQs with answers

When two variables are recorded for each unit, the data are bivariate. The chapter classifies and tabulates bivariate data, finds marginal and conditional distributions, handles categorical variables in contingency tables and computes the chi square statistic comparing observed and expected frequencies.

Key concepts: Bivariate Frequency Distribution

Bivariate data: Observations on two variables for each individual, e.g. height and weight of students, written as pairs (x, y).
Bivariate frequency distribution: A two-way table: classes of X in rows, classes of Y in columns and tally marks/frequencies in the cells. With m classes of X and n classes of Y there are mn cells.
Marginal distributions: Row totals give the frequency distribution of X; column totals give the distribution of Y. The grand total is N.
Conditional distributions: The frequency distribution of X for a fixed class of Y (a column), or of Y for a fixed class of X (a row).
Categorical variables: Qualitative characteristics (attributes) such as gender, blood group or colour that are classified into categories rather than measured.
Contingency table: A two-way table of observed frequencies of two categorical variables, with row totals Rᵢ, column totals Cⱼ and grand total N.
Expected frequencies: Eᵢⱼ = (Rᵢ × Cⱼ)/N. The expected frequencies have the same row, column and grand totals as the observed ones.
Chi square statistic: χ² = Σ (Oᵢⱼ − Eᵢⱼ)²/Eᵢⱼ. It is never negative; χ² = 0 when observed and expected frequencies agree exactly. A larger value indicates a greater difference.

Quick revision: important points

Sum of all cell frequencies = N.
Σ O = Σ E = N.
An r × c contingency table has rc cells.
Height, weight and marks are measurable (numerical) variables, not categorical ones.
Compute (O − E)²/E cell by cell and add.

Practice MCQs with answers

Data collected on two variables for each individual are called:
1. Univariate data
2. Bivariate data
3. Categorical data only
4. Time series
Show answer
Answer: B. Bivariate data
Bi means two.
The marginal distribution of X is obtained from the:
1. Cell frequencies of one column
2. Row totals
3. Grand total
4. Tally marks
Show answer
Answer: B. Row totals
Totals across each class of X.
Which of the following is a categorical variable?
1. Height
2. Weight
3. Blood group
4. Income
Show answer
Answer: C. Blood group
Blood group is an attribute, not a measurement.
A contingency table shows frequencies of:
1. One numerical variable
2. Two categorical variables
3. Time and value
4. Only percentages
Show answer
Answer: B. Two categorical variables
It cross-classifies two attributes.
The expected frequency of a cell is:
1. Rᵢ + Cⱼ
2. Rᵢ × Cⱼ × N
3. (Rᵢ × Cⱼ)/N
4. N/(Rᵢ × Cⱼ)
Show answer
Answer: C. (Rᵢ × Cⱼ)/N
Eᵢⱼ = RᵢCⱼ/N.

All Class 11 Maths and Statistics chapters

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