Mathematics and Statistics · Class 11 Commerce
Ch 13Bivariate Frequency Distribution and Chi-Square Statistic — Class 11 Mathematics and Statistics, concept-first.
Every measure met so far in this course — mean, median, quartiles, standard deviation, skewness — describes just one variable at a time: the marks of a class in a single subject, or the daily wages of one factory's workers.
Key concepts
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Bivariate Frequency Distribution
Bivariate data records a pair for every unit of observation. When there are many pairs, they are summarised in a two-way (bivariate) frequency table: the values or class-intervals of label the rows, those of label the co…
Most relevant Q&A
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Bivariate Data and the Bivariate Frequency Distribution
Every measure met so far in this course — mean, median, quartiles, standard deviation, skewness — describes just one variable at a time: the marks of a class in a single subject, or the daily wages of…
Marginal and Conditional Frequency Distributions
A two-way table actually carries three kinds of distribution inside it. The cell frequencies describe the two variables jointly; but by looking only at the totals, or only at a single row or column, t…
Attributes, Contingency Tables and Independence
Not every characteristic can be measured as a number. Sex (male / female), literacy (literate / illiterate), or whether a customer liked a product (yes / no) are qualitative characteristics, called at…
The Chi-Square Statistic and Its Interpretation
Once every cell has an observed frequency and an expected-under-independence frequency , a single number is needed to summarise how far apart the whole observed table is from what independence would p…
Exercises
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- Q8The paired values $(x, y)$ of 10 units are given below, where $x \in \{10, 20, 30\}$ and $y \in \{5, 10, 15\}$: $(10,5),\ (10,10),\ (20,10),…Free
- Q9From the bivariate frequency table constructed in Exercise 1, find the marginal distributions of $x$ and $y$, and hence compute the means $\…Free
- Q10100 candidates are classified by whether they were coached and whether they passed an examination, as shown. Compute the expected frequencie…Preview
- Q11Using the bivariate frequency table constructed in Exercise 1, find the conditional distribution of $y$ given $x = 20$, and express it in re…Preview
- Q12200 people are classified by whether they were vaccinated and whether they contracted a disease, as shown. Compute the chi-square statistic…Preview
More questions
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- Example 1The following are the paired scores $(x, y)$ of 12 students in two short tests, where $x$ is the score in Test A and $y$ the score in Test B…Free
- Example 2From the bivariate frequency table constructed in Worked Example 1, write down (i) the marginal distribution of $x$ and (ii) the marginal di…Free
- Example 3Using the bivariate frequency table of Worked Example 1, find the conditional distribution of $y$ given $x = 2$, and express it in relative…Free
- Example 4For the following contingency table of 100 customers classified by sex and whether they liked a product, compute the expected frequency of e…Preview
- Example 5Using the observed table and the expected frequencies from Worked Example 4, compute the chi-square statistic and comment on whether sex and…Preview
- Example 6200 buyers are classified by region (Urban / Rural) and by which of three brands X, Y, Z they bought, as shown. Compute the chi-square stati…Preview
- Example 7In a survey of 100 people classified by literacy and employment status, the observed frequencies are shown. Compute the chi-square statistic…Preview