Chapter 13 – Statistics and Probability (JEE)
PART A – STATISTICS
1. Meaning of Statistics
Statistics is the branch of mathematics that deals with the collection,
analysis, interpretation, and presentation of numerical data.
2. Measures of Central Tendency
These measures describe the **central or representative value** of a data set.
Mean (Arithmetic Mean)
For ungrouped data:
$$\bar{x} = \frac{\sum x}{n}$$
For grouped data:
$$\bar{x} = \frac{\sum f x}{\sum f}$$
Assumed Mean Method
$$\bar{x} = a + \frac{\sum f d}{\sum f}$$
Step Deviation Method
$$\bar{x} = a + h\frac{\sum f u}{\sum f}$$
3. Weighted Mean
$$\bar{x}_w = \frac{\sum w x}{\sum w}$$
Used when observations have different importance.
4. Mean Deviation
Mean deviation about mean:
$$\text{MD} = \frac{\sum |x-\bar{x}|}{n}$$
5. Variance
$$\sigma^2 = \frac{\sum (x-\bar{x})^2}{n}$$
6. Standard Deviation
$$\sigma = \sqrt{\frac{\sum (x-\bar{x})^2}{n}}$$
Standard deviation measures the **spread of data**.
7. Short-Cut Method for Standard Deviation
$$\sigma = \sqrt{\frac{\sum f d^2}{\sum f} - \left(\frac{\sum f d}{\sum f}\right)^2}$$
8. Coefficient of Variation
$$\text{CV} = \frac{\sigma}{\bar{x}} \times 100$$
Lower CV → more consistent data.
9. Comparison of Data Sets
To compare two distributions:
- Compare their means
- Compare their standard deviations
- Use coefficient of variation
PART B – PROBABILITY
10. Concept of Probability
Probability measures the likelihood of an event occurring.
11. Classical Definition of Probability
$$P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$$
12. Sample Space
Sample space is the set of all possible outcomes of an experiment.
13. Event
An event is any subset of the sample space.
14. Types of Events
- Impossible event
- Sure event
- Simple event
- Compound event
- Complementary event
15. Probability Axioms
- $0 \le P(E) \le 1$
- $P(S) = 1$
- $P(A') = 1 - P(A)$
16. Addition Theorem of Probability
$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
17. Mutually Exclusive Events
If $A \cap B = \varnothing$, then:
$$P(A \cup B) = P(A) + P(B)$$
18. Conditional Probability
$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$
19. Multiplication Theorem
$$P(A \cap B) = P(A)P(B|A)$$
20. Independent Events
Two events are independent if:
$$P(A|B) = P(A)$$
21. Bayes’ Theorem
$$P(A_i|B) = \frac{P(A_i)P(B|A_i)}{\sum P(A_j)P(B|A_j)}$$
Extremely important for JEE probability questions.
22. Common JEE Traps
- Forgetting conditional probability
- Wrong sample space
- Confusing independent and mutually exclusive events
- Ignoring complementary events
23. Final Revision Checklist
You are exam-ready if you can:
- Compute mean and standard deviation quickly
- Compare data sets using CV
- Form correct sample spaces
- Apply Bayes’ theorem confidently
- Solve multi-step probability problems