Statistics and Probability formulas for JEE Main & Advanced

23 formula entries for Statistics and Probability, in the order the chapter builds them. Each shows the formula, when to use it and the first trap to avoid.

Mean (Arithmetic Mean)

\overset{ˉ}{x} = \frac{\sum x_{i}}{n}

(ungrouped),

\overset{ˉ}{x} = \frac{\sum f_{i} x_{i}}{\sum f_{i}}

(grouped)

When to use. To find the central tendency of a dataset; baseline for variance and standard deviation calculations.

Trap. Forgetting to use class marks (midpoints) instead of class boundaries in grouped data.

Median

Median = l + (\frac{\frac{n}{2} - c f}{f}) \times h

(grouped)

When to use. When data has outliers or is skewed; preferred over mean for ordinal data.

Trap. Forgetting to sort data before finding median in ungrouped case.

Mode

Mode = l + (\frac{f_{1} - f_{0}}{2 f_{1} - f_{0} - f_{2}}) \times h

(grouped)

When to use. To find the most common value; useful for categorical or discrete data.

Trap. Assuming mode is unique; data can be multimodal.

Range

Range = \max - \min

When to use. Quick estimate of spread; used in quality control or small datasets.

Trap. Ignoring outliers that drastically change range.

Mean Deviation

MD = \frac{\sum ∣ x_{i} - \overset{ˉ}{x} ∣}{n}

(ungrouped),

MD = \frac{\sum f_{i} ∣ x_{i} - \overset{ˉ}{x} ∣}{\sum f_{i}}

(grouped)

When to use. When outliers are present but you need a simple dispersion measure.

Trap. Forgetting absolute values; using squared deviations instead.

Variance and Standard Deviation

σ^{2} = \frac{\sum (x_{i} - \overset{ˉ}{x})^{2}}{n}

(population),

s^{2} = \frac{\sum (x_{i} - \overset{ˉ}{x})^{2}}{n - 1}

(sample),

σ = \sqrt{σ^{2}}

When to use. To quantify spread; basis for many statistical tests and probability distributions.

Trap. Dividing by n-1 when JEE says 'variance': unless 'sample variance' is explicitly stated, use the population formula (divide by n).

Combined Mean and Variance

{\overset{ˉ}{x}}_{12} = \frac{n_{1} {\overset{ˉ}{x}}_{1} + n_{2} {\overset{ˉ}{x}}_{2}}{n_{1} + n_{2}}

σ_{12}^{2} = \frac{n_{1} (σ_{1}^{2} + d_{1}^{2}) + n_{2} (σ_{2}^{2} + d_{2}^{2})}{n_{1} + n_{2}}

where

d_{i} = {\overset{ˉ}{x}}_{i} - {\overset{ˉ}{x}}_{12}

When to use. When given summary statistics of subgroups and need overall statistics.

Trap. Simply averaging variances without adjusting for mean differences.

Shortcut Method for Variance

σ^{2} = \frac{\sum x_{i}^{2}}{n} - {(\frac{\sum x_{i}}{n})}^{2}

When to use. When raw data is available; saves time in calculations.

Trap. Forgetting to square the mean; miscomputing sum of squares.

Probability - Basic Terms

P (E) = \frac{n (E)}{n (S)}

(equally likely outcomes)

When to use. Foundation for all probability problems; define sample space first.

Trap. Assuming outcomes are equally likely when they are not.

Addition Theorem of Probability

P (A \cup B) = P (A) + P (B) - P (A \cap B)

When to use. When finding probability of at least one of two events occurring.

Trap. Forgetting to subtract intersection when events are not mutually exclusive.

Mutually Exclusive and Exhaustive Events

P (A \cap B) = 0

(mutually exclusive),

P (A \cup B) = 1

(exhaustive)

When to use. To simplify probability calculations; check if events partition sample space.

Trap. Assuming mutually exclusive events are also exhaustive (not necessarily).

Conditional Probability

P (A ∣ B) = \frac{P (A \cap B)}{P (B)}

P (B) \neq 0

When to use. When additional information is given; basis for Bayes' theorem.

Trap. Forgetting to check $P (B) > 0$ ; dividing by zero.

Multiplication Theorem and Independent Events

P (A \cap B) = P (A) \cdot P (B)

(independent),

P (A \cap B) = P (A) \cdot P (B ∣ A)

(general)

When to use. When events are independent (e.g., repeated trials); otherwise use conditional.

Trap. Assuming independence without checking; using product for dependent events.

Bayes' Theorem

P (A_{i} ∣ B) = \frac{P (A_{i}) \cdot P (B ∣ A_{i})}{\sum_{j} P (A_{j}) \cdot P (B ∣ A_{j})}

When to use. When you have prior probabilities and new evidence; reverse conditional probability.

Trap. Forgetting to normalize by total probability of B.

Random Variable and Probability Distribution

P (X = x_{i}) = p_{i}

, with

\sum p_{i} = 1

p_{i} \geq 0

When to use. To model random experiments numerically; basis for expectation and variance.

Trap. Forgetting that sum of probabilities must be 1.

Expectation (Mean) of a Random Variable

E (X) = \sum x_{i} p_{i}

When to use. To find average outcome in the long run; used in games, insurance, etc.

Trap. Forgetting to multiply each value by its probability.

Variance of a Random Variable

V a r (X) = E (X^{2}) - [E (X)]^{2} = \sum (x_{i} - μ)^{2} p_{i}

When to use. To quantify risk or variability; needed for binomial and other distributions.

Trap. Forgetting to compute $E (X^{2})$ correctly; using $E (X)^{2}$ instead of $[E (X)]^{2}$ .

Binomial Distribution

P (X = k) = (\binom{n}{k}) p^{k} (1 - p)^{n - k}

k = 0, 1, \dots, n

When to use. When experiment has exactly two outcomes, repeated n times independently.

Trap. Forgetting that trials must be independent and probability constant.

Mean and Variance of Binomial Distribution

μ = n p

σ^{2} = n p (1 - p)

When to use. Quickly compute expected value and spread for binomial random variable.

Trap. Using $n p$ for variance; forgetting $1 - p$ factor.

Using Complement and Symmetry

P (at least one) = 1 - P (no success)

When to use. When direct calculation of many probabilities is tedious; common in binomial problems.

Trap. Forgetting to consider all cases for 'none'.

Probability Tree Diagrams

No formula; use tree to compute

P (A \cap B) = P (A) \cdot P (B ∣ A)

When to use. When experiment has sequential steps (e.g., drawing without replacement).

Trap. Forgetting to update probabilities after each stage (conditional).

Conditional Probability and Total Probability Theorem

P (B) = \sum P (A_{i}) \cdot P (B ∣ A_{i})

When to use. When B can occur via multiple mutually exclusive routes.

Trap. Forgetting that $A_{i}$ must be mutually exclusive and exhaustive.

Solving for Missing Frequencies in Statistics

Set

\overset{ˉ}{x} = \frac{\sum f_{i} x_{i}}{\sum f_{i}}

or median formula equal to given value.

When to use. When frequency distribution has missing values and a measure of central tendency is given.

Trap. Forgetting to include unknown frequencies in total n.

Read the full chapter: Statistics and Probability for JEE, with what JEE asks and the common traps.