Mathematics · JEE

Statistics and Probability Formula Sheet for JEE

62+ JEE formulas in this unit

Quick answer

The Statistics and Probability JEE formula sheet lists 62+ important formulas for JEE Main and Advanced, including essential identities from Measures of dispersion; calculation of mean, median, mode of grouped and ungrouped data, Calculation of standard deviation, variance and mean deviation for grouped and ungrouped data, Probability: Probability of an event, addition and multiplication theorems of probability, Baye's theorem, probability distribution of a random variable. Revise essential formulas first, then practise MCQs on Goodmarks.

Download-free JEE mathematics formula revision for Statistics and Probability. This unit-wise formula list covers 62+ exam-relevant results across Measures of dispersion; calculation of mean, median, mode of grouped and ungrouped data, Calculation of standard deviation, variance and mean deviation for grouped and ungrouped data, Probability: Probability of an event, addition and multiplication theorems of probability, Baye's theorem, probability distribution of a random variable, organised by subtopic for quick last-minute revision.

JEE Formula Sheet

62 formulas across 4 subtopics — organised for JEE Main & Advanced revision

Practise MCQs for this unit

Essential: 30Important: 24Supplementary: 8

Measures of dispersion; calculation of mean, median, mode of grouped and ungrouped data

Arithmetic mean of ungrouped dataEssential

\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i

Variables: $x_i$ : observations, $n$ : number of observations.
Conditions: Applies to numerical data.
Where used in JEE: Finding average of raw data; further computation of variance, standard deviation, mean deviation.

Arithmetic mean using frequenciesEssential

\bar{x}=\frac{\sum f_i x_i}{\sum f_i}

Variables: $x_i$ : observation/value, $f_i$ : corresponding frequency, $N=\sum f_i$ : total frequency.
Conditions: For discrete frequency distribution.
Where used in JEE: Mean of frequency tables.

Mean of grouped continuous dataEssential

\bar{x}=\frac{\sum f_i m_i}{\sum f_i}

Variables: $m_i$ : class mark of the $i$ -th class, $f_i$ : frequency, $N=\sum f_i$ .
Conditions: For grouped continuous data; class mark $m_i=\frac{l_i+u_i}{2}$ .
Where used in JEE: Approximate mean of grouped data.

Assumed mean method for meanImportant

\bar{x}=a+\frac{\sum f_i d_i}{\sum f_i}

d_i=x_i-a

Variables: $a$ : assumed mean, $x_i$ : value or class mark, $f_i$ : frequency.
Conditions: Used for discrete or grouped data.
Where used in JEE: Quick calculation of mean when values are large.

Step-deviation method for meanImportant

\bar{x}=a+h\frac{\sum f_i u_i}{\sum f_i}

u_i=\frac{x_i-a}{h}

Variables: $a$ : assumed mean, $h$ : common factor/class width, $x_i$ : value or class mark, $f_i$ : frequency.
Conditions: Useful when deviations are multiples of a common number $h$ .
Where used in JEE: Grouped data with equal class widths.

Combined mean of two groupsImportant

\bar{x}=\frac{n_1\bar{x}_1+n_2\bar{x}_2}{n_1+n_2}

Variables: $n_1,n_2$ : sizes of groups, $\bar{x}_1,\bar{x}_2$ : their means.
Conditions: For combining two data sets.
Where used in JEE: Mixture/combined data problems.

Median of ordered ungrouped dataEssential

\text{Median}=x_{\frac{n+1}{2}}

for odd

n

;

\text{Median}=\frac{x_{n/2}+x_{n/2+1}}{2}

for even

n

Variables: $x_1\le x_2\le \cdots \le x_n$ : ordered observations.
Conditions: Data must be arranged in ascending or descending order.
Where used in JEE: Central tendency of raw data.

Median for discrete frequency distributionImportant

\text{Median}=\text{value whose cumulative frequency first exceeds }\frac{N}{2}

Variables: $N=\sum f_i$ : total frequency.
Conditions: Values arranged in increasing order and cumulative frequencies computed.
Where used in JEE: Median from frequency table.

Median of grouped continuous dataEssential

\text{Median}=l+\left(\frac{\frac{N}{2}-c_f}{f}\right)h

Variables: $l$ : lower boundary of median class, $N$ : total frequency, $c_f$ : cumulative frequency before median class, $f$ : frequency of median class, $h$ : class width.
Conditions: Median class is the class whose cumulative frequency first exceeds $N/2$ .
Where used in JEE: Median of grouped data.

Mode of ungrouped/discrete dataImportant

\text{Mode}=\text{observation with highest frequency}

Variables: Frequency indicates repetition count of each observation.
Conditions: May be non-unique if multiple values share highest frequency.
Where used in JEE: Most frequent value in raw/discrete data.

Mode of grouped continuous dataEssential

\text{Mode}=l+\left(\frac{f_1-f_0}{2f_1-f_0-f_2}\right)h

Variables: $l$ : lower boundary of modal class, $h$ : class width, $f_1$ : frequency of modal class, $f_0$ : frequency of class preceding modal class, $f_2$ : frequency of class succeeding modal class.
Conditions: Modal class is class with maximum frequency.
Where used in JEE: Mode from grouped frequency distribution.

Empirical relation among mean, median and modeImportant

\text{Mode}=3\,\text{Median}-2\bar{x}

Variables: $\bar{x}$ : mean.
Conditions: Approximate relation for moderately skewed distributions.
Where used in JEE: Estimating one central measure from the other two.

RangeSupplementary

\text{Range}=L-S

Variables: $L$ : largest observation, $S$ : smallest observation.
Conditions: Simple measure of dispersion.
Where used in JEE: Quick spread estimation.

Coefficient of rangeSupplementary

\frac{L-S}{L+S}

Variables: $L$ : largest observation, $S$ : smallest observation.
Conditions: Defined when $L+S\ne 0$ .
Where used in JEE: Comparing dispersion of two series.

Quartiles for ungrouped dataSupplementary

Q_1=\text{value of }\left(\frac{n+1}{4}\right)\text{th item},\quad Q_3=\text{value of }\left(\frac{3(n+1)}{4}\right)\text{th item}

Variables: $n$ : number of ordered observations.
Conditions: Data arranged in order; interpolation may be used if needed.
Where used in JEE: Quartile deviation and box-type summaries.

Quartiles for grouped dataSupplementary

Q_k=l+\left(\frac{\frac{kN}{4}-c_f}{f}\right)h,\ k=1,3

Variables: $l$ : lower boundary of quartile class, $N$ : total frequency, $c_f$ : cumulative frequency before quartile class, $f$ : frequency of quartile class, $h$ : class width.
Conditions: Quartile class determined using $N/4$ or $3N/4$ .
Where used in JEE: Finding quartiles in grouped data.

Interquartile range and quartile deviationSupplementary

\text{IQR}=Q_3-Q_1,\quad \text{QD}=\frac{Q_3-Q_1}{2}

Variables: $Q_1,Q_3$ : first and third quartiles.
Where used in JEE: Measures of dispersion resistant to extremes.

Coefficient of quartile deviationSupplementary

\frac{Q_3-Q_1}{Q_3+Q_1}

Variables: $Q_1,Q_3$ : quartiles.
Conditions: Defined when $Q_3+Q_1\ne 0$ .
Where used in JEE: Comparing relative spread using quartiles.

Calculation of standard deviation, variance and mean deviation for grouped and ungrouped data

Mean deviation about a value for ungrouped dataImportant

\text{MD about }a=\frac{1}{n}\sum_{i=1}^{n}|x_i-a|

Variables: $x_i$ : observations, $a$ : chosen central value, $n$ : number of observations.
Conditions: Usually $a$ is mean or median.
Where used in JEE: Average absolute deviation in raw data.

Mean deviation about a value for frequency distributionEssential

\text{MD about }a=\frac{1}{N}\sum f_i|x_i-a|

Variables: $x_i$ : values or class marks, $f_i$ : frequencies, $N=\sum f_i$ , $a$ : chosen central value.
Conditions: For grouped data use class marks.
Where used in JEE: Mean deviation from mean/median for frequency tables.

Mean deviation about meanImportant

\text{MD}_{\bar{x}}=\frac{1}{N}\sum f_i|x_i-\bar{x}|

Variables: $\bar{x}$ : arithmetic mean, $x_i$ : values/class marks, $f_i$ : frequencies, $N=\sum f_i$ .
Conditions: For ungrouped data take all $f_i=1$ .
Where used in JEE: Standard mean deviation computation.

Mean deviation about medianImportant

\text{MD}_{M}=\frac{1}{N}\sum f_i|x_i-M|

Variables: $M$ : median, $x_i$ : values/class marks, $f_i$ : frequencies.
Conditions: Mean deviation is minimum when taken about median.
Where used in JEE: Problems involving least absolute deviation.

Variance of ungrouped dataEssential

\sigma^2=\frac{1}{n}\sum_{i=1}^{n}(x_i-\bar{x})^2

Variables: $x_i$ : observations, $\bar{x}$ : mean, $n$ : number of observations.
Conditions: Population variance form used in school/JEE statistics.
Where used in JEE: Dispersion of raw data.

Standard deviation of ungrouped dataEssential

\sigma=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_i-\bar{x})^2}

Variables: $x_i$ : observations, $\bar{x}$ : mean.
Where used in JEE: Spread of raw data; comparison using coefficient of variation.

Variance for frequency distributionEssential

\sigma^2=\frac{1}{N}\sum f_i(x_i-\bar{x})^2

Variables: $x_i$ : values or class marks, $f_i$ : frequencies, $\bar{x}$ : mean, $N=\sum f_i$ .
Conditions: For grouped data use class marks.
Where used in JEE: Variance of discrete/grouped data.

Standard deviation for frequency distributionEssential

\sigma=\sqrt{\frac{1}{N}\sum f_i(x_i-\bar{x})^2}

Variables: $x_i$ : values or class marks, $f_i$ : frequencies, $N=\sum f_i$ .
Where used in JEE: Standard deviation of discrete/grouped data.

Computational formula for varianceEssential

\sigma^2=\frac{\sum f_i x_i^2}{N}-\left(\frac{\sum f_i x_i}{N}\right)^2

Variables: $x_i$ : values or class marks, $f_i$ : frequencies, $N=\sum f_i$ .
Conditions: For ungrouped data, take all $f_i=1$ .
Where used in JEE: Fast variance calculation.

Assumed mean method for varianceImportant

\sigma^2=\frac{\sum f_i d_i^2}{N}-\left(\frac{\sum f_i d_i}{N}\right)^2

d_i=x_i-a

Variables: $a$ : assumed mean, $x_i$ : value/class mark, $f_i$ : frequency.
Conditions: For ungrouped/discrete/grouped data.
Where used in JEE: Simplified variance computation.

Step-deviation method for varianceImportant

\sigma=h\sqrt{\frac{\sum f_i u_i^2}{N}-\left(\frac{\sum f_i u_i}{N}\right)^2}

u_i=\frac{x_i-a}{h}

Variables: $a$ : assumed mean, $h$ : common factor/class width, $x_i$ : value/class mark, $f_i$ : frequency.
Conditions: Useful when deviations are integral after scaling by $h$ .
Where used in JEE: Grouped data variance with equal class intervals.

Combined variance of two groupsImportant

\sigma^2=\frac{n_1\left(\sigma_1^2+(\bar{x}_1-\bar{x})^2\right)+n_2\left(\sigma_2^2+(\bar{x}_2-\bar{x})^2\right)}{n_1+n_2}

Variables: $n_1,n_2$ : sizes, $\bar{x}_1,\bar{x}_2$ : means, $\sigma_1,\sigma_2$ : standard deviations, $\bar{x}$ : combined mean.
Conditions: For combining two data sets.
Where used in JEE: Mixture/merged data problems.

Effect of origin and scale on meanImportant

y=\frac{x-a}{h}

, then

\bar{y}=\frac{\bar{x}-a}{h}

; equivalently, if

y=ax+b

, then

\bar{y}=a\bar{x}+b

Variables: $a,b,h$ : constants with $h\ne 0$ .
Where used in JEE: Data transformation and coding method.

Effect of origin and scale on variance and standard deviationEssential

y=\frac{x-a}{h}

, then

\sigma_y^2=\frac{\sigma_x^2}{h^2}

\sigma_y=\frac{\sigma_x}{|h|}

; equivalently, if

y=ax+b

, then

\sigma_y^2=a^2\sigma_x^2

\sigma_y=|a|\sigma_x

Variables: $a,b,h$ : constants, $\sigma_x,\sigma_y$ : standard deviations.
Conditions: Change of origin does not affect variance or standard deviation; change of scale does.
Where used in JEE: Coding transformations and comparison problems.

Coefficient of variationEssential

\mathrm{CV}=\frac{\sigma}{\bar{x}}\times 100\%

Variables: $\sigma$ : standard deviation, $\bar{x}$ : mean.
Conditions: Usually used when $\bar{x}>0$ . Smaller CV implies greater consistency.
Where used in JEE: Comparing variability/consistency of two series.

Probability: Probability of an event, addition and multiplication theorems of probability

Equally likely probabilityEssential

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

Variables: $n(E)$ : number of favorable outcomes, $n(S)$ : total number of outcomes in sample space $S$ .
Conditions: Finite sample space with equally likely outcomes.
Where used in JEE: Basic probability counting problems.

Bounds of probabilityEssential

0\le P(A)\le 1

Variables: $A$ : event.
Where used in JEE: Checking validity of probability values.

Probability of sample space and impossible eventEssential

P(S)=1,\quad P(\varnothing)=0

Variables: $S$ : sample space, $\varnothing$ : impossible event.
Where used in JEE: Basic axioms and simplification.

Complement ruleEssential

P(A')=1-P(A)

Variables: $A'$ : complement of event $A$ .
Where used in JEE: At least one / not occurring type problems.

Probability of difference of eventsImportant

P(A-B)=P(A\cap B')=P(A)-P(A\cap B)

Variables: $A-B$ : event that $A$ occurs and $B$ does not.
Where used in JEE: Exactly one / only one event problems.

Addition theorem for two eventsEssential

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

Variables: $A,B$ : events.
Where used in JEE: Union of two events; at least one event occurs.

Addition theorem for mutually exclusive eventsEssential

A\cap B=\varnothing

, then

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

Variables: $A,B$ : mutually exclusive events.
Conditions: Events cannot occur together.
Where used in JEE: Either-or without overlap.

Addition theorem for three eventsImportant

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

Variables: $A,B,C$ : events.
Where used in JEE: At least one of three events occurs.

Conditional probabilityEssential

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

Variables: $A,B$ : events.
Conditions: Defined for $P(B)>0$ .
Where used in JEE: Sequential and dependent event problems.

Multiplication theorem for two eventsEssential

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

Variables: $A,B$ : events.
Conditions: Conditional probabilities must be defined.
Where used in JEE: Joint probability and successive draws.

Multiplication theorem for three eventsImportant

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

Variables: $A,B,C$ : events.
Conditions: Required conditional probabilities exist.
Where used in JEE: Sequential dependent trials.

Independent events criterionEssential

A\text{ and }B\text{ independent }\iff P(A\cap B)=P(A)P(B)

Variables: $A,B$ : events.
Conditions: Equivalent to $P(A\mid B)=P(A)$ when $P(B)>0$ .
Where used in JEE: Testing independence.

Probability of intersection for independent eventsImportant

A_1,A_2,\dots,A_n

are independent, then

P\left(\bigcap_{i=1}^{n}A_i\right)=\prod_{i=1}^{n}P(A_i)

Variables: $A_i$ : independent events.
Conditions: Mutual independence assumed.
Where used in JEE: All events occur.

Probability of complements of independent eventsImportant

A

and

B

are independent, then

A'

and

B'

are independent, and

P(A'\cap B')=(1-P(A))(1-P(B))

Variables: $A',B'$ : complements.
Where used in JEE: Neither event occurs.

At least one event occurs for independent eventsImportant

A_1,A_2,\dots,A_n

are independent, then

P\left(\bigcup_{i=1}^{n}A_i\right)=1-\prod_{i=1}^{n}(1-P(A_i))

Variables: $A_i$ : independent events.
Conditions: Uses complement of none occurring.
Where used in JEE: At least one success problems.

Baye's theorem, probability distribution of a random variable

Total probability theoremEssential

B_1,B_2,\dots,B_n

form a partition of

S

, then

P(A)=\sum_{i=1}^{n}P(B_i)P(A\mid B_i)

Variables: $B_i$ : mutually exclusive and exhaustive events, $A$ : any event.
Conditions: Each $P(B_i)>0$ .
Where used in JEE: Multi-case probability computation.

Bayes' theoremEssential

P(B_j\mid A)=\frac{P(B_j)P(A\mid B_j)}{\sum_{i=1}^{n}P(B_i)P(A\mid B_i)}

Variables: $B_1,B_2,\dots,B_n$ : partition events, $A$ : observed event.
Conditions: $P(A)>0$ , $P(B_i)>0$ .
Where used in JEE: Posterior probability; source/urn/defect problems.

Bayes' theorem for two eventsImportant

P(B\mid A)=\frac{P(B)P(A\mid B)}{P(B)P(A\mid B)+P(B')P(A\mid B')}

Variables: $A,B$ : events, $B'$ : complement of $B$ .
Conditions: $P(A)>0$ .
Where used in JEE: Binary-cause problems.

Probability mass function conditionsEssential

p(x)=P(X=x),\quad p(x)\ge 0,\quad \sum_x p(x)=1

Variables: $X$ : discrete random variable, $p(x)$ : probability mass function.
Conditions: Sum over all possible values of $X$ .
Where used in JEE: Checking/constructing probability distributions.

Distribution function of a random variableImportant

F(x)=P(X\le x)

Variables: $X$ : random variable, $F(x)$ : cumulative distribution function.
Conditions: For discrete random variable in JEE, $F$ is a step function.
Where used in JEE: Finding cumulative probabilities.

Expectation of a discrete random variableEssential

E(X)=\mu=\sum_x x\,p(x)

Variables: $X$ : discrete random variable, $p(x)=P(X=x)$ .
Conditions: Sum over all possible values of $X$ .
Where used in JEE: Mean of probability distribution.

Expectation of a function of random variableImportant

E[g(X)]=\sum_x g(x)\,p(x)

Variables: $g(X)$ : function of random variable $X$ , $p(x)=P(X=x)$ .
Conditions: For discrete distributions.
Where used in JEE: Finding moments and transformed expectations.

Linearity of expectationImportant

E(aX+b)=aE(X)+b

and more generally

E(X\pm Y)=E(X)\pm E(Y)

Variables: $a,b$ : constants, $X,Y$ : random variables.
Conditions: No independence required for linearity.
Where used in JEE: Expectation of transformed or combined variables.

Variance of a discrete random variableEssential

\operatorname{Var}(X)=\sigma^2=E\big[(X-\mu)^2\big]=\sum_x (x-\mu)^2p(x)

Variables: $\mu=E(X)$ , $p(x)=P(X=x)$ .
Where used in JEE: Dispersion of probability distribution.

Variance formula using second momentEssential

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

Variables: $E(X^2)=\sum x^2p(x)$ .
Where used in JEE: Fast variance computation for random variables.

Standard deviation of a random variableEssential

\sigma=\sqrt{\operatorname{Var}(X)}

Variables: $\sigma$ : standard deviation of $X$ .
Where used in JEE: Spread of a probability distribution.

Effect of linear transformation on expectation and varianceImportant

E(aX+b)=aE(X)+b,\quad \operatorname{Var}(aX+b)=a^2\operatorname{Var}(X)

Variables: $a,b$ : constants, $X$ : random variable.
Where used in JEE: Transforming random variables.

Bernoulli distributionSupplementary

For

X\in\{0,1\}

P(X=x)=p^x(1-p)^{1-x}

E(X)=p

\operatorname{Var}(X)=p(1-p)

Variables: $p=P(X=1)$ , $0\le p\le 1$ .
Conditions: Single success-failure trial.
Where used in JEE: Indicator random variable; one-trial probability models.

Binomial distributionSupplementary

X\sim B(n,p)

, then

P(X=r)=\binom{n}{r}p^r q^{n-r}

r=0,1,2,\dots,n

q=1-p

;

E(X)=np

\operatorname{Var}(X)=npq

Variables: $n$ : number of independent trials, $p$ : success probability, $q=1-p$ .
Conditions: Independent identical Bernoulli trials.
Where used in JEE: Number of successes in repeated trials.

Frequently asked questions

What are the important Statistics and Probability formulas for JEE?

This page lists 62+ JEE-relevant Statistics and Probability formulas organised by subtopic. Start with essential formulas, then important identities before supplementary shortcuts.

Is this Statistics and Probability formula sheet aligned with JEE Main?

Yes. Every formula is mapped to the JEE Main Mathematics syllabus for Statistics and Probability, covering Measures of dispersion; calculation of mean, median, mode of grouped and ungrouped data, Calculation of standard deviation, variance and mean deviation for grouped and ungrouped data, Probability: Probability of an event, addition and multiplication theorems of probability, Baye's theorem, probability distribution of a random variable.

How should I revise the Statistics and Probability formula sheet before JEE?

Revise essential formulas daily, important ones every 2–3 days, and supplementary results weekly. After each pass, solve 10–15 MCQs to test recall under exam conditions.

Where can I practise Statistics and Probability MCQs after revising formulas?

Use the Online Practice or MCQs pages for the same unit on Goodmarks to convert formula recall into problem-solving speed.

Does this replace NCERT for Statistics and Probability?

No — use this formula sheet for quick revision alongside NCERT and your coaching notes. Formulas here are a condensed reference, not a substitute for concept building.

JEE Formula Sheet

Measures of dispersion; calculation of mean, median, mode of grouped and ungrouped data

Calculation of standard deviation, variance and mean deviation for grouped and ungrouped data

Probability: Probability of an event, addition and multiplication theorems of probability

Baye's theorem, probability distribution of a random variable

Popular questions in Statistics and Probability

Frequently asked questions

What are the important Statistics and Probability formulas for JEE?

Is this Statistics and Probability formula sheet aligned with JEE Main?

How should I revise the Statistics and Probability formula sheet before JEE?

Where can I practise Statistics and Probability MCQs after revising formulas?

Does this replace NCERT for Statistics and Probability?

Related topics