Statistics and Probability Formula Sheet — JEE Main Mathematics

Every key Statistics and Probability formula, definition and theorem for JEE Main Mathematics in one place — with common examiner traps and worked examples. Free to read; blurt from memory, then check your gaps.

Syllabus — topics coveredNTA · 7 sub-topics

Measures of dispersion
Mean, median, mode of grouped and ungrouped data
Standard deviation, variance and mean deviation
Probability of an event
Addition and multiplication theorems of probability
Baye's theorem
Probability distribution of a random variable

Measures of Central Tendency

Central tendency: A single value that represents the 'centre' of a data set. The three standard averages are the

mean

(arithmetic average), the

median

(the middle value when ordered) and the

mode

(the most frequent value).

★ Remember · Grouped-data formulas

Mean

\overset{x}{ˉ} = \frac{\sum f _{i} x _{i}}{\sum f _{i}}

; median

= l + \frac{N /2 - C}{f} \times h

(median class); mode

= l + \frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}} \times h

(modal class), where l,h are the lower limit and width, C the cumulative frequency before the class, f its frequency.

Other averages & facts

▸Combined mean of two groups: $\frac{n _{1} x ˉ _{1} + n _{2} x ˉ _{2}}{n _{1} + n _{2}}$ .
▸Weighted mean $\frac{\sum w _{i} x _{i}}{\sum w _{i}}$ ; geometric mean $(x_{1} \dots x_{n})^{1/ n}$ ; harmonic mean $\frac{n}{\sum ( 1/ x _{i} )}$ .
▸Median is unaffected by extreme values; the mean is the most affected.
▸For two positive numbers, $A.M. \geq G.M. \geq H.M.$ and $G.M.^{2} = A.M. \cdot H.M.$

🎯 Exam · Empirical relation

For a moderately skewed distribution,

Mode = 3 Median - 2 Mean

. Knowing any two of the three lets you estimate the third — a frequent shortcut in objective questions.

★ Remember · Shift & scale the mean

If every observation changes as

x_{i} \to a + b x_{i}

, the mean changes the same way:

\overset{x}{ˉ} \to a + b \overset{x}{ˉ}

. This 'step-deviation' idea (

d_{i} = (x_{i} - a) / h

) turns ugly numbers into easy ones before computing

\overset{x}{ˉ}

⚠️ Watch out · Mean of means is not the mean

The average of two group means is

not

the combined mean unless the groups are equal in size — you must weight by the counts

n_{1}, n_{2}

. Forgetting the weights is the classic combined-mean error.

🚫 Examiner Trap · Examiner traps

(1) Mean is pulled toward outliers; MEDIAN is the robust centre — pick the right one for skewed data. (2) For RIGHT-skew: mode

<

median

<

mean (the empirical rule Mode

= 3 M e d ian - 2

Mean). (3) Mode can be missing or multiple; median needs the data SORTED first. (4) Combined mean is the WEIGHTED average

\frac{n _{1} x ˉ _{1} + n _{2} x ˉ _{2}}{n _{1} + n _{2}}

, not

\frac{x ˉ _{1} + x ˉ _{2}}{2}

Measures of Dispersion

Dispersion: How spread out the data are about a central value. The main measures are the

range

, the

mean deviation

, the

variance

σ^{2}

and the

standard deviation

σ

— the larger they are, the more scattered the data.

Spread formulas (range, mean deviation, variance, standard deviation), two distributions with small and large standard deviation, and the coefficient of variation — The spread formulas, small vs large sigma, and the coefficient of variation.

★ Remember · Variance & standard deviation

σ^{2} = \frac{1}{N} \sum f_{i} (x_{i} - \overset{x}{ˉ})^{2} = \frac{\sum f _{i} x _{i}^{2}}{N} - \overset{x}{ˉ}^{2}

and

σ = σ^{2}

. The second (shortcut) form avoids computing each deviation and is the one to use in practice.

Key properties

▸Range $= x_{ma x} - x_{min}$ ; mean deviation $= \frac{1}{N} \sum f_{i} ∣ x_{i} - a ∣$ (about mean or median).
▸Variance and SD are always $\geq 0$ ; SD shares the data's units, variance does not.
▸SD is unchanged by a shift of origin but scales: $x \to a + b x \Rightarrow σ \to ∣ b ∣ σ$ .
▸For the first n natural numbers, $σ^{2} = \frac{n ^{2} - 1}{12}$ .

🎯 Exam · Coefficient of variation

CV = \frac{σ}{x ˉ} \times 100

is unit-free, so it compares the consistency of two different data sets. The series with the

smaller CV

is more consistent / more stable.

★ Remember · Mean deviation is least about the median

The mean deviation is

minimum

when taken about the median, and the sum of squared deviations is minimum about the mean. These extremal facts are tested directly.

⚠️ Watch out · Don't drop the square / the modulus

Variance uses

(x_{i} - \overset{x}{ˉ})^{2}

(squares) while mean deviation uses

∣ x_{i} - a ∣

(absolute values). Mixing them — or forgetting to subtract

\overset{x}{ˉ}^{2}

in the shortcut

\sum f_{i} x_{i}^{2} / N - \overset{x}{ˉ}^{2}

— is the usual slip.

🚫 Examiner Trap · Examiner traps

(1) Variance and SD are NEVER negative; SD shares the data's units, variance does not. (2) Use the shortcut

σ^{2} = \frac{\sum f _{i} x _{i}^{2}}{N} - \overset{x}{ˉ}^{2}

— and the

\overset{x}{ˉ}^{2}

is subtracted at the END. (3) SD is unchanged by a shift of ORIGIN but SCALES with the scale:

x \to a + b x \Rightarrow σ \to ∣ b ∣ σ

. (4) Coefficient of variation

= \frac{σ}{x ˉ} \times 100

compares consistency (smaller

=

more consistent); for first n naturals

σ^{2} = \frac{n ^{2} - 1}{12}

Probability — Basics & Addition Theorem

Probability of an event: For equally likely outcomes,

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

, where S is the

sample space

(all outcomes) and the event

E \subseteq S

. Axiomatically,

0 \leq P (E) \leq 1

P (S) = 1

and

P (ϕ) = 0

A Venn diagram for the union of two events with the addition theorem, the classical definition and axioms, and a table of event relations — The union of two events, the axioms, and common event relations.

★ Remember · Addition theorem

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

— subtract the overlap counted twice. For

mutually exclusive

events (

A \cap B = ϕ

) this is just

P (A) + P (B)

. The complement rule:

P (A^{'}) = 1 - P (A)

Useful event translations

▸'At least one of A,B' $= P (A \cup B)$ ; 'none' $= P (A^{'} \cap B^{'}) = 1 - P (A \cup B)$ .
▸'Exactly one of A,B' $= P (A) + P (B) - 2 P (A \cap B)$ .
▸Mutually exclusive $=$ cannot happen together; exhaustive $=$ together cover $S$ .
▸Odds in favour of $A$ are $P (A) : P (A^{'})$ .

🎯 Exam · Count carefully, then divide

Most basic problems are really

counting

problems: find n(S) and n(E) using permutations/combinations, then

P (E) = n (E) / n (S)

. Make sure 'favourable' and 'total' are counted the same way (ordered vs unordered).

⚠️ Watch out · Mutually exclusive is about overlap, not chance

A \cap B = ϕ

means the events

share no outcome

— it does not mean they are equally likely or independent. Only for disjoint events may you add probabilities without subtracting

P (A \cap B)

🚫 Examiner Trap · Examiner traps

(1) Addition:

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

— only drop the last term when the events are MUTUALLY EXCLUSIVE. (2) Mutually exclusive (

A \cap B = \emptyset

) is NOT the same as independent — don't confuse them. (3)

0 \leq P (E) \leq 1

; complement

P (A^{'}) = 1 - P (A)

. (4) 'At least one'

= 1 - P (none)

; 'neither A nor B'

= P (A^{'} \cap B^{'}) = 1 - P (A \cup B)

Conditional Probability & Independence

Conditional probability: The probability of E given that F has occurred:

P (E ∣ F) = \frac{P ( E \cap F )}{P ( F )}

, provided

P (F) \neq = 0

. Knowing F shrinks the sample space to F, so we measure E only within F.

★ Remember · Multiplication & independence

Multiplication theorem:

P (E \cap F) = P (F) P (E ∣ F) = P (E) P (F ∣ E)

. Events are

independent

when

P (E \cap F) = P (E) P (F)

, equivalently

P (E ∣ F) = P (E)

— knowing F tells you nothing about E.

Properties

▸ $P (S ∣ F) = 1$ and $P (A^{'} ∣ F) = 1 - P (A ∣ F)$ .
▸If $E, F$ are independent, so are $(E^{'}, F)$ , $(E, F^{'})$ and $(E^{'}, F^{'})$ .
▸Three events independent: all three pairs AND $P (A \cap B \cap C) = P (A) P (B) P (C)$ .
▸Use a $probability tree$ : multiply along a branch, add across branches.

🎯 Exam · 'And' vs 'given'

P (E \cap F)

is 'both happen';

P (E ∣ F)

is 'E happens assuming F did'. They are linked by

P (E \cap F) = P (F) P (E ∣ F)

— pick the multiplication rule for 'and', the ratio for 'given'.

⚠️ Watch out · Independent is not mutually exclusive

For events with non-zero probabilities, mutually exclusive (

P (E \cap F) = 0

) and independent (

P (E \cap F) = P (E) P (F) > 0

)

cannot both hold

. Treat the two ideas as opposites, not synonyms.

🚫 Examiner Trap · Examiner traps

(1)

P (E ∣ F) = \frac{P ( E \cap F )}{P ( F )}

needs

P (F) \neq = 0

— and

P (E ∣ F) \neq = P (F ∣ E)

in general. (2) Multiplication:

P (E \cap F) = P (F) P (E ∣ F)

; if INDEPENDENT this is just P(E)P(F). (3) Independent

\neq =

mutually exclusive — for nonzero probabilities ME events are actually DEPENDENT. (4) Three events independent needs ALL pairs AND the triple

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

Total Probability & Bayes' Theorem

Partition of the sample space: Events

E_{1}, E_{2}, \dots, E_{n}

form a

partition

of S if they are pairwise disjoint (

E_{i} \cap E_{j} = ϕ

), exhaustive (

⋃ E_{i} = S

) and each has

P (E_{i}) > 0

. These are the mutually exclusive 'causes' behind an observed event A.

★ Remember · The two formulas

Total probability:

P (A) = \sum_{j} P (E_{j}) P (A ∣ E_{j})

. Bayes' theorem:

P (E_{i} ∣ A) = \frac{P ( E _{i} ) P ( A ∣ E _{i} )}{\sum _{j} P ( E _{j} ) P ( A ∣ E _{j} )}

— the numerator is exactly one term of the (shared) denominator.

How to apply

▸Identify the partition ${E_{i}}$ (the 'cause' that occurred first, e.g. which bag/machine).
▸ $P (E_{i})$ = prior (before seeing A); $P (E_{i} ∣ A)$ = posterior (after seeing A).
▸Total probability gives P(A); Bayes $reverses$ the condition to give $P (E_{i} ∣ A)$ .
▸All the posteriors $P (E_{i} ∣ A)$ for fixed A add up to $1$ .

🎯 Exam · Tree first, then read off

Draw a tree: first branch into the causes

E_{i}

(weights

P (E_{i})

), then into A/A' (weights

P (A ∣ E_{i})

). P(A) is the sum of the A-paths;

P (E_{i} ∣ A)

is that cause's A-path divided by P(A).

⚠️ Watch out · Don't swap the conditional

P (A ∣ E_{i})

and

P (E_{i} ∣ A)

are different — Bayes exists precisely to convert one into the other. The data usually give

P (A ∣ E_{i})

(reliability); the question asks the reversed

P (E_{i} ∣ A)

🚫 Examiner Trap · Examiner traps

(1) The

E_{i}

must PARTITION S (disjoint, exhaustive, each

P (E_{i}) > 0

) for total probability / Bayes to apply. (2) Total probability

P (A) = \sum_{i} P (E_{i}) P (A ∣ E_{i})

gives the prior-to-evidence link. (3) Bayes REVERSES the condition:

P (E_{i} ∣ A) = \frac{P ( E _{i} ) P ( A ∣ E _{i} )}{\sum _{j} P ( E _{j} ) P ( A ∣ E _{j} )}

— the numerator is ONE term of the denominator sum. (4) Don't swap prior

P (E_{i})

and posterior

P (E_{i} ∣ A)

Random Variable & Probability Distribution

Random variable: A real-valued function X on the sample space. Its

probability distribution

lists the values

x_{i}

with probabilities

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

, where every

p_{i} \geq 0

and

\sum p_{i} = 1

A probability distribution with mean and variance formulas, a bar chart of P(X=x) with the mean marked, and key facts — A distribution, its mean and variance, and how they behave.

★ Remember · Mean & variance

Mean (expectation)

μ = E (X) = \sum x_{i} p_{i}

. Variance

σ^{2} = Var (X) = \sum x_{i}^{2} p_{i} - μ^{2} = E (X^{2}) - [E (X)]^{2}

, and SD

= Var (X)

Key facts

▸ $E (a X + b) = a E (X) + b$ (linear) and $Var (a X + b) = a^{2} Var (X)$ .
▸ $E (X)$ is a weighted average — the long-run mean if the experiment were repeated.
▸ $Var (X) \geq 0$ ; it is $0$ only if X is constant.
▸Two distributions can share a mean but differ in variance (spread).

🎯 Exam · Build the table, then sum

Tabulate

x_{i}

p_{i}

x_{i} p_{i}

and

x_{i}^{2} p_{i}

. The column sums give

\sum p_{i} = 1

(a check), E(X) and

E (X^{2})

directly, then

Var (X) = E (X^{2}) - [E (X)]^{2}

⚠️ Watch out · Variance is E(X²) − μ², not E(X²) − μ

Subtract the

square

of the mean:

Var (X) = E (X^{2}) - μ^{2}

. Also confirm

\sum p_{i} = 1

before trusting any mean — an unnormalised table silently corrupts every later value.

🚫 Examiner Trap · Examiner traps

(1) A valid distribution needs

\sum p_{i} = 1

and every

p_{i} \geq 0

. (2) Mean

μ = E (X) = \sum x_{i} p_{i}

(a weighted average, the long-run mean). (3) Variance

= E (X^{2}) - [E (X)]^{2}

— compute

E (X^{2}) = \sum x_{i}^{2} p_{i}

, NOT

[E (X)]^{2}

. (4)

E (a X + b) = a E (X) + b

but

Var (a X + b) = a^{2} Var (X)

(the b vanishes, the a is squared).

Bernoulli Trials & Binomial Distribution

Binomial distribution: For n independent

Bernoulli trials

(two outcomes, constant success probability p), the number of successes X has

P (X = x) =^{n} C_{x} p^{x} q^{n - x}

x = 0, 1, \dots, n

, where

q = 1 - p

. Written

X \sim B (n, p)

★ Remember · Mean & variance

Mean

= n p

, variance

= npq

, SD

= n pq

. Since

q < 1

, the mean always exceeds the variance — a quick way to recover n and p when given both (np and npq).

Bernoulli conditions & uses

▸Bernoulli trials: fixed $n$ , independent, exactly two outcomes, constant $p$ .
▸Sampling WITH replacement $\to$ Bernoulli; WITHOUT replacement $\to$ not (trials dependent).
▸ $P (at least one) = 1 - q^{n}$ ; $P (at most r) = \sum_{x = 0}^{r}^{n} C_{x} p^{x} q^{n - x}$ .
▸The distribution is symmetric when $p = \frac{1}{2}$ and skewed otherwise; the peak is near np.

🎯 Exam · Read off n, p, x

Pin down n (number of trials), p (success probability) and the required x, then plug into

^{n} C_{x} p^{x} q^{n - x}

. 'At least'/'at most' just sum the relevant x values (use the complement

1 - q^{n}

for 'at least one').

⚠️ Watch out · Without replacement breaks independence

If items are drawn

without replacement

, the success probability changes each draw, so the trials are not Bernoulli and the binomial formula does not apply. Check 'with/without replacement' before using B(n,p).

🚫 Examiner Trap · Examiner traps

(1) Binomial needs FIXED n, INDEPENDENT trials, TWO outcomes and CONSTANT p — sampling WITHOUT replacement breaks independence (use hypergeometric). (2)

P (X = x) =^{n} C_{x} p^{x} q^{n - x}

with

q = 1 - p

. (3) Mean

= n p

, Variance

= npq

, so Mean

>

Variance ALWAYS (since

q < 1

) — a quick check. (4) 'At least one'

= 1 - q^{n}

; the most probable count is near np.

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Frequently Asked Questions

What are the most important Statistics and Probability formulas for JEE Main?

This Statistics and Probability formula sheet covers all the high-yield Mathematics formulas, definitions and theorems you need for JEE Main, across Measures of dispersion, Mean, median, mode of grouped and ungrouped data, Standard deviation, variance and mean deviation, Probability of an event, Addition and multiplication theorems of probability — each shown with the key result and, where useful, a worked example.

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