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JEE Maths Formulas
Complete Formula Sheet

All Chapters — JEE Main & Advanced 2026 | 16 Topics | 400+ Formulas |

📏 Straight Lines⭕ Circles 🌀 Conics📐 Trigonometry ∫ Calculus→ Vectors 🎲 Probability🔢 Algebra
🔍

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Table of Contents

1
Straight Lines
2
Circles
3
Parabola
4
Ellipse
5
Hyperbola
6
Trigonometry
7
Inverse Trig
8
Limits
9
Derivatives
10
Integration
11
Diff. Equations
12
Vectors & 3D
13
Algebra
14
Sequences & Series
15
Probability
16
Statistics
📏

1. Straight Lines

14 Formulas
Important
Distance Formula
\[ d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \]
Key
Section Formula (Internal)
\[ x=\frac{mx_2+nx_1}{m+n},\quad y=\frac{my_2+ny_1}{m+n} \]
Section Formula (External)
\[ x=\frac{mx_2-nx_1}{m-n},\quad y=\frac{my_2-ny_1}{m-n} \]
Centroid of Triangle
\[ G=\!\left(\frac{x_1+x_2+x_3}{3},\,\frac{y_1+y_2+y_3}{3}\right) \]
Area of Triangle
\[ \Delta=\frac{1}{2}\bigl|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\bigr| \]
Important
Slope of Line
\[ m=\frac{y_2-y_1}{x_2-x_1}=\tan\theta \]
Key
Angle Between Two Lines
\[ \tan\theta=\left|\frac{m_1-m_2}{1+m_1m_2}\right| \]
Condition of Parallelism
\[ m_1=m_2 \]
For lines ax+by+c=0 and a'x+b'y+c'=0: a/a' = b/b'
Condition of Perpendicularity
\[ m_1\cdot m_2=-1 \]
aa' + bb' = 0
Distance: Point to Line
\[ d=\frac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}} \]
Distance Between Parallel Lines
\[ d=\frac{|c_1-c_2|}{\sqrt{a^2+b^2}} \]
Foot of Perpendicular from (x₁,y₁) to ax+by+c=0
\[ \frac{x-x_1}{a}=\frac{y-y_1}{b}=-\frac{ax_1+by_1+c}{a^2+b^2} \]
Condition: Three Collinear Points
\[ \begin{vmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{vmatrix}=0 \]
Point of Intersection of Lines
Solve simultaneously:
\(a_1x+b_1y+c_1=0\)
\(a_2x+b_2y+c_2=0\)
⭕

2. Circle

14 Formulas
Important
Standard Equation
\[ (x-h)^2+(y-k)^2=r^2 \]
Centre (h,k), radius r
Key
General Equation
\[ x^2+y^2+2gx+2fy+c=0 \]
Centre: (−g,−f); Radius: \(\sqrt{g^2+f^2-c}\)
x-axis Intercept
\[ 2\sqrt{g^2-c} \]
y-axis Intercept
\[ 2\sqrt{f^2-c} \]
Parametric Form
\[ x=h+r\cos\theta,\quad y=k+r\sin\theta \]
Important
Tangent at Point (x₁,y₁)
\[ xx_1+yy_1+g(x+x_1)+f(y+y_1)+c=0 \]
Tangent: Slope Form (circle x²+y²=a²)
\[ y=mx\pm a\sqrt{1+m^2} \]
Length of Tangent from External Point
\[ L=\sqrt{S_1}=\sqrt{x_1^2+y_1^2+2gx_1+2fy_1+c} \]
Chord of Contact
\[ T=0\Rightarrow xx_1+yy_1+g(x+x_1)+f(y+y_1)+c=0 \]
Orthogonal Circles Condition
\[ 2g_1g_2+2f_1f_2=c_1+c_2 \]
Radical Axis
\[ S_1-S_2=0 \]
Locus of points of equal tangent lengths to two circles
Power of a Point
\[ \text{Power}=x_1^2+y_1^2+2gx_1+2fy_1+c \]
🌀

3. Parabola

12 Formulas
Standard Parabolas
EquationVertexFocusDirectrixLR LengthAxis
\(y^2=4ax\)(0,0)(a,0)x=−a4ax-axis
\(y^2=-4ax\)(0,0)(−a,0)x=a4ax-axis
\(x^2=4ay\)(0,0)(0,a)y=−a4ay-axis
\(x^2=-4ay\)(0,0)(0,−a)y=a4ay-axis
Important
Parametric Form (y²=4ax)
\[ x=at^2,\quad y=2at \]
Tangent at Point (x₁,y₁)
\[ yy_1=2a(x+x_1) \]
Key
Tangent: Slope Form
\[ y=mx+\frac{a}{m} \]
Condition of tangency: c = a/m
Normal at Point (at², 2at)
\[ y=-tx+2at+at^3 \]
Important
Normal: Slope Form
\[ y=mx-2am-am^3 \]
Chord of Contact from (x₁,y₁)
\[ yy_1=2a(x+x_1) \]
Focal Chord: t₁·t₂ = −1
\[ t_1\cdot t_2=-1 \]
🥚

4. Ellipse

12 Formulas
Important
Standard Equation
\[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1,\quad a>b \]
Key
Eccentricity
\[ e=\sqrt{1-\frac{b^2}{a^2}},\quad 0
Also: b² = a²(1 − e²)
Foci & Directrices
\[ \text{Foci: }(\pm ae,0)\quad\text{Directrices: }x=\pm\frac{a}{e} \]
Latus Rectum Length
\[ LR=\frac{2b^2}{a} \]
Parametric Form
\[ x=a\cos\theta,\quad y=b\sin\theta \]
Tangent at (x₁,y₁)
\[ \frac{xx_1}{a^2}+\frac{yy_1}{b^2}=1 \]
Important
Tangent: Slope Form
\[ y=mx\pm\sqrt{a^2m^2+b^2} \]
Normal at (x₁,y₁)
\[ \frac{a^2x}{x_1}-\frac{b^2y}{y_1}=a^2-b^2 \]
Sum of Focal Radii
\[ SP+S'P=2a \]
Auxiliary Circle
\[ x^2+y^2=a^2 \]
📈

5. Hyperbola

12 Formulas
Important
Standard Equation
\[ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \]
Key
Eccentricity
\[ e=\sqrt{1+\frac{b^2}{a^2}},\quad e>1 \]
b² = a²(e² − 1)
Foci & Directrices
\[ \text{Foci: }(\pm ae,0)\quad\text{Directrices: }x=\pm\frac{a}{e} \]
Asymptotes
\[ y=\pm\frac{b}{a}x \]
Latus Rectum
\[ LR=\frac{2b^2}{a} \]
Parametric Form
\[ x=a\sec\theta,\quad y=b\tan\theta \]
Tangent: Slope Form
\[ y=mx\pm\sqrt{a^2m^2-b^2} \]
Rectangular Hyperbola
\[ xy=c^2,\quad e=\sqrt{2} \]
Parametric: (ct, c/t)
Difference of Focal Radii
\[ |SP-S'P|=2a \]
📐

6. Trigonometry

28 Formulas
Trigonometric Values Table
Angle0°30°45°60°90°
sin01/21/√2√3/21
cos1√3/21/√21/20
tan01/√31√3∞
Pythagorean Identities
\[\sin^2\theta+\cos^2\theta=1\] \[1+\tan^2\theta=\sec^2\theta\] \[1+\cot^2\theta=\csc^2\theta\]
Important
Sum / Difference Formulas
\[\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B\] \[\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B\] \[\tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}\]
Key
Double Angle Formulas
\[\sin 2\theta=2\sin\theta\cos\theta=\frac{2\tan\theta}{1+\tan^2\theta}\] \[\cos 2\theta=\cos^2\theta-\sin^2\theta=1-2\sin^2\theta=2\cos^2\theta-1\] \[\tan 2\theta=\frac{2\tan\theta}{1-\tan^2\theta}\]
Triple Angle Formulas
\[\sin 3\theta=3\sin\theta-4\sin^3\theta\] \[\cos 3\theta=4\cos^3\theta-3\cos\theta\] \[\tan 3\theta=\frac{3\tan\theta-\tan^3\theta}{1-3\tan^2\theta}\]
Sum-to-Product (Factor Formulae)
\[\sin C+\sin D=2\sin\frac{C+D}{2}\cos\frac{C-D}{2}\] \[\sin C-\sin D=2\cos\frac{C+D}{2}\sin\frac{C-D}{2}\] \[\cos C+\cos D=2\cos\frac{C+D}{2}\cos\frac{C-D}{2}\] \[\cos C-\cos D=-2\sin\frac{C+D}{2}\sin\frac{C-D}{2}\]
Product-to-Sum
\[2\sin A\cos B=\sin(A+B)+\sin(A-B)\] \[2\cos A\cos B=\cos(A-B)+\cos(A+B)\] \[2\sin A\sin B=\cos(A-B)-\cos(A+B)\]
Important
Sine Rule
\[ \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R \]
Important
Cosine Rule
\[\cos A=\frac{b^2+c^2-a^2}{2bc}\] \[\cos B=\frac{a^2+c^2-b^2}{2ac}\]
Area of Triangle
\[ \Delta=\frac{1}{2}ab\sin C=\frac{abc}{4R}=rs \]
r = inradius, s = semi-perimeter, R = circumradius
Half-Angle Formulas
\[\sin\frac{A}{2}=\sqrt{\frac{(s-b)(s-c)}{bc}}\] \[\cos\frac{A}{2}=\sqrt{\frac{s(s-a)}{bc}}\] \[\tan\frac{A}{2}=\sqrt{\frac{(s-b)(s-c)}{s(s-a)}}\]
🔄

7. Inverse Trigonometry

14 Formulas
Key
Complementary Relations
\[\sin^{-1}x+\cos^{-1}x=\frac{\pi}{2}\] \[\tan^{-1}x+\cot^{-1}x=\frac{\pi}{2}\] \[\sec^{-1}x+\csc^{-1}x=\frac{\pi}{2}\]
Important
tan⁻¹ Addition / Subtraction
\[\tan^{-1}x+\tan^{-1}y=\tan^{-1}\!\frac{x+y}{1-xy}\quad(xy<1)\] \[\tan^{-1}x-\tan^{-1}y=\tan^{-1}\!\frac{x-y}{1+xy}\]
sin⁻¹ Addition
\[ \sin^{-1}x+\sin^{-1}y=\sin^{-1}\!\left(x\sqrt{1-y^2}+y\sqrt{1-x^2}\right) \]
Negation Properties
\[\sin^{-1}(-x)=-\sin^{-1}x\] \[\cos^{-1}(-x)=\pi-\cos^{-1}x\] \[\tan^{-1}(-x)=-\tan^{-1}x\]
Double Angle (ITF)
\[2\sin^{-1}x=\sin^{-1}(2x\sqrt{1-x^2})\] \[2\cos^{-1}x=\cos^{-1}(2x^2-1)\] \[2\tan^{-1}x=\tan^{-1}\!\frac{2x}{1-x^2}=\sin^{-1}\!\frac{2x}{1+x^2}\]
Domain & Range
FunctionDomainRange
\(\sin^{-1}x\)[−1,1][−π/2, π/2]
\(\cos^{-1}x\)[−1,1][0, π]
\(\tan^{-1}x\)ℝ(−π/2, π/2)
🎯

8. Limits

14 Formulas
Important
Standard Trigonometric Limits
\[\lim_{x\to 0}\frac{\sin x}{x}=1\] \[\lim_{x\to 0}\frac{\tan x}{x}=1\] \[\lim_{x\to 0}\frac{1-\cos x}{x^2}=\frac{1}{2}\]
Key
Exponential & Log Limits
\[\lim_{x\to 0}\frac{e^x-1}{x}=1\] \[\lim_{x\to 0}\frac{\ln(1+x)}{x}=1\] \[\lim_{x\to 0}\frac{a^x-1}{x}=\ln a\]
Power Limit
\[ \lim_{x\to 0}\frac{x^n-a^n}{x-a}=na^{n-1} \]
e as a Limit
\[\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n=e\] \[\lim_{x\to 0}(1+x)^{1/x}=e\]
L'Hôpital's Rule (0/0 or ∞/∞)
\[ \lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)} \]
Squeeze Theorem
\[\text{If }g(x)\le f(x)\le h(x)\text{ and }\lim g=\lim h=L\Rightarrow\lim f=L\]
📉

9. Differentiation

24 Formulas
Standard Derivatives Table
f(x)f'(x)f(x)f'(x)
\(x^n\)\(nx^{n-1}\)\(e^x\)\(e^x\)
\(\sin x\)\(\cos x\)\(a^x\)\(a^x\ln a\)
\(\cos x\)\(-\sin x\)\(\ln x\)\(1/x\)
\(\tan x\)\(\sec^2 x\)\(\log_a x\)\(1/(x\ln a)\)
\(\cot x\)\(-\csc^2 x\)\(\sin^{-1}x\)\(1/\sqrt{1-x^2}\)
\(\sec x\)\(\sec x\tan x\)\(\cos^{-1}x\)\(-1/\sqrt{1-x^2}\)
\(\csc x\)\(-\csc x\cot x\)\(\tan^{-1}x\)\(1/(1+x^2)\)
Key
Product Rule
\[ (uv)'=u'v+uv' \]
Quotient Rule
\[ \left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^2} \]
Chain Rule
\[ \frac{d}{dx}f(g(x))=f'(g(x))\cdot g'(x) \]
Rolle's Theorem
\[\text{If }f(a)=f(b)\Rightarrow\exists\ c\in(a,b)\text{ s.t. }f'(c)=0\]
Mean Value Theorem (LMVT)
\[ f'(c)=\frac{f(b)-f(a)}{b-a} \]
Important
Tangent & Normal
Slope of tangent: \(m=\frac{dy}{dx}\bigg|_{(x_1,y_1)}\)
Tangent: \(y-y_1=m(x-x_1)\)
Normal: \(y-y_1=-\frac{1}{m}(x-x_1)\)
Tangent / Normal Lengths
Length of tangent: \(|y|\sqrt{1+\frac{1}{m^2}}\)
Length of normal: \(|y|\sqrt{1+m^2}\)
Maxima & Minima Conditions
\(f'(c)=0\) (critical point)
Max: \(f''(c)<0\);
Min: \(f''(c)>0\)
∫

10. Integration

28 Formulas
Standard Integrals Table
f(x)∫f(x)dxf(x)∫f(x)dx
\(x^n\)\(\frac{x^{n+1}}{n+1}+C\)\(e^x\)\(e^x+C\)
\(1/x\)\(\ln|x|+C\)\(a^x\)\(\frac{a^x}{\ln a}+C\)
\(\sin x\)\(-\cos x+C\)\(\cos x\)\(\sin x+C\)
\(\tan x\)\(\ln|\sec x|+C\)\(\cot x\)\(\ln|\sin x|+C\)
\(\sec^2 x\)\(\tan x+C\)\(\csc^2 x\)\(-\cot x+C\)
\(\sec x\tan x\)\(\sec x+C\)\(\csc x\cot x\)\(-\csc x+C\)
\(\frac{1}{\sqrt{1-x^2}}\)\(\sin^{-1}x+C\)\(\frac{1}{1+x^2}\)\(\tan^{-1}x+C\)
\(\frac{1}{\sqrt{a^2-x^2}}\)\(\sin^{-1}\frac{x}{a}+C\)\(\frac{1}{a^2+x^2}\)\(\frac{1}{a}\tan^{-1}\frac{x}{a}+C\)
\(\frac{1}{x^2-a^2}\)\(\frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right|+C\)\(\frac{1}{a^2-x^2}\)\(\frac{1}{2a}\ln\left|\frac{a+x}{a-x}\right|+C\)
\(\sqrt{a^2-x^2}\)\(\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\sin^{-1}\frac{x}{a}+C\)\(\sqrt{x^2+a^2}\)\(\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\ln\left|x+\sqrt{x^2+a^2}\right|+C\)
Key
Integration by Parts (ILATE)
\[ \int uv\,dx=u\int v\,dx-\int\!\left(u'\int v\,dx\right)dx \]
ILATE: Inverse trig, Log, Algebraic, Trig, Exponential
Important
Definite Integral Properties
\[\int_a^b f(x)\,dx=-\int_b^a f(x)\,dx\] \[\int_a^b f(x)\,dx=\int_a^b f(a+b-x)\,dx\] \[\int_0^a f(x)\,dx=\int_0^a f(a-x)\,dx\] \[\int_0^{2a} f(x)\,dx=2\int_0^a f(x)\,dx\text{ if }f(2a-x)=f(x)\]
Walli's Formula (Even n)
\[ \int_0^{\pi/2}\sin^n x\,dx=\frac{(n-1)!!}{n!!}\cdot\frac{\pi}{2} \]
Area Under Curve
\[ A=\int_a^b|f(x)|\,dx \]
Area Between Two Curves
\[ A=\int_a^b[f(x)-g(x)]\,dx \]
where f(x) ≥ g(x) in [a,b]
🔃

11. Differential Equations

10 Formulas
Important
Variable Separable
\[ f(y)\,dy=g(x)\,dx\Rightarrow\int f(y)\,dy=\int g(x)\,dx \]
Key
Linear First-Order DE
\[\frac{dy}{dx}+Py=Q\] IF \(=e^{\int P\,dx}\) \[y\cdot\text{IF}=\int Q\cdot\text{IF}\,dx+C\]
Homogeneous DE
\[\frac{dy}{dx}=f\!\left(\frac{y}{x}\right)\Rightarrow\text{Put }y=vx\]
Exact DE Condition
\[ M\,dx+N\,dy=0\text{ is exact if }\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x} \]
Order & Degree
Order = highest derivative; Degree = power of highest derivative (when rational)
Bernoulli's Equation
\[ \frac{dy}{dx}+Py=Qy^n;\quad\text{Put }v=y^{1-n} \]
→

12. Vectors & 3D Geometry

20 Formulas
Important
Dot Product
\[ \vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta=a_1b_1+a_2b_2+a_3b_3 \]
Important
Cross Product (Magnitude)
\[ |\vec{a}\times\vec{b}|=|\vec{a}||\vec{b}|\sin\theta \]
Direction: Right-hand rule; Area of parallelogram = |a×b|
Scalar Triple Product
\[ [\vec{a}\,\vec{b}\,\vec{c}]=\vec{a}\cdot(\vec{b}\times\vec{c}) \]
Volume of parallelepiped; = 0 if coplanar
Angle Between Vectors
\[ \cos\theta=\frac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|} \]
Projection of a on b
\[ \text{Proj}=\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|} \]
Direction Cosines
\[ l^2+m^2+n^2=1 \]
l=cosα, m=cosβ, n=cosγ
Key
Distance Between Points in 3D
\[ d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2} \]
Equation of Line (Symmetric Form)
\[ \frac{x-x_1}{l}=\frac{y-y_1}{m}=\frac{z-z_1}{n} \]
Equation of Plane
\[ ax+by+cz+d=0 \]
Normal vector: (a,b,c)
Important
Distance: Point to Plane
\[ d=\frac{|ax_1+by_1+cz_1+d|}{\sqrt{a^2+b^2+c^2}} \]
Angle Between Two Planes
\[ \cos\theta=\frac{|a_1a_2+b_1b_2+c_1c_2|}{\sqrt{\sum a_1^2}\sqrt{\sum a_2^2}} \]
Angle Between Line & Plane
\[ \sin\theta=\frac{|al+bm+cn|}{\sqrt{a^2+b^2+c^2}\sqrt{l^2+m^2+n^2}} \]
Skew Lines – Shortest Distance
\[ d=\frac{|(\vec{b}_1-\vec{b}_2)\cdot(\vec{d}_1\times\vec{d}_2)|}{|\vec{d}_1\times\vec{d}_2|} \]
🔢

13. Algebra (Quadratic, Complex, P&C, Binomial)

24 Formulas
Important
Quadratic Formula
\[ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \]
Discriminant D = b²−4ac
Roots: Sum & Product
\[ \alpha+\beta=-\frac{b}{a},\quad\alpha\beta=\frac{c}{a} \]
Nature of Roots
DNature
D > 0Real & distinct
D = 0Real & equal
D < 0Complex conjugate
Complex Number
\[ z=a+ib,\quad|z|=\sqrt{a^2+b^2},\quad\bar{z}=a-ib \]
Euler's Formula
\[ e^{i\theta}=\cos\theta+i\sin\theta \]
De Moivre's Theorem
\[ (\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta \]
Cube Roots of Unity
\[ 1+\omega+\omega^2=0,\quad\omega^3=1 \]
ω = e^(2πi/3) = (−1+i√3)/2
Key
Permutation & Combination
\[^nP_r=\frac{n!}{(n-r)!}\] \[^nC_r=\frac{n!}{r!(n-r)!}\] \[^nC_r+\,^nC_{r-1}=\,^{n+1}C_r\]
Important
Binomial Theorem
\[ (a+b)^n=\sum_{r=0}^n\binom{n}{r}a^{n-r}b^r \]
Total terms: n+1; Middle term: T_{(n/2)+1}
General Term of Binomial
\[ T_{r+1}=\binom{n}{r}a^{n-r}b^r \]
Matrices: Determinant (2×2)
\[ \det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc \]
Cramer's Rule
\[ x=\frac{D_x}{D},\quad y=\frac{D_y}{D},\quad z=\frac{D_z}{D} \]
De Morgan's Laws
\[(A\cup B)'=A'\cap B'\] \[(A\cap B)'=A'\cup B'\]
🔢

14. Sequences & Series

16 Formulas
Important
AP – General Term
\[ T_n=a+(n-1)d \]
Key
AP – Sum of n Terms
\[ S_n=\frac{n}{2}[2a+(n-1)d]=\frac{n}{2}(a+l) \]
AM of a and b
\[ AM=\frac{a+b}{2} \]
Important
GP – General Term
\[ T_n=ar^{n-1} \]
GP – Sum of n Terms
\[ S_n=\frac{a(1-r^n)}{1-r}\quad(r\ne1) \]
Key
GP – Sum to Infinity
\[ S_\infty=\frac{a}{1-r},\quad|r|<1 \]
GM of a and b
\[ GM=\sqrt{ab} \]
HP – General Term
\[ T_n=\frac{1}{a+(n-1)d} \]
HM of a and b = 2ab/(a+b)
Relation AM, GM, HM
\[ AM\ge GM\ge HM \]
Also: GM² = AM × HM
Sum of Natural Numbers
\[\sum_{k=1}^n k=\frac{n(n+1)}{2}\] \[\sum_{k=1}^n k^2=\frac{n(n+1)(2n+1)}{6}\] \[\sum_{k=1}^n k^3=\left(\frac{n(n+1)}{2}\right)^2\]
🎲

15. Probability

14 Formulas
Important
Addition Rule
\[ P(A\cup B)=P(A)+P(B)-P(A\cap B) \]
Mutually Exclusive Events
\[ P(A\cup B)=P(A)+P(B) \]
if A∩B = ∅
Key
Conditional Probability
\[ P(A|B)=\frac{P(A\cap B)}{P(B)} \]
Multiplication Rule
\[ P(A\cap B)=P(A)\cdot P(B|A) \]
Independent Events
\[ P(A\cap B)=P(A)\cdot P(B) \]
Important
Bayes' Theorem
\[ P(A_i|B)=\frac{P(A_i)\cdot P(B|A_i)}{\sum_j P(A_j)\cdot P(B|A_j)} \]
Binomial Distribution
\[ P(X=r)=\binom{n}{r}p^r q^{n-r} \]
Mean = np; Variance = npq; q = 1−p
Total Probability
\[ P(B)=\sum_{i=1}^n P(A_i)\cdot P(B|A_i) \]
Geometric Probability
\[ P=\frac{\text{Favourable Length/Area}}{\text{Total Length/Area}} \]
📊

16. Statistics

12 Formulas
Important
Mean (Ungrouped)
\[ \bar{x}=\frac{\sum x_i}{n} \]
Mean (Grouped)
\[ \bar{x}=\frac{\sum f_i x_i}{\sum f_i}=\frac{\sum f_i x_i}{N} \]
Key
Variance
\[ \sigma^2=\frac{\sum(x_i-\bar{x})^2}{n}=\frac{\sum x_i^2}{n}-\bar{x}^2 \]
Standard Deviation
\[ \sigma=\sqrt{\text{Variance}} \]
Coefficient of Variation
\[ CV=\frac{\sigma}{\bar{x}}\times100\% \]
Lower CV = more consistent data
Median
\[M=\frac{n+1}{2}\text{th term (odd n)}\] \[M=\frac{n/2\text{th}+(n/2+1)\text{th}}{2}\text{ (even n)}\]
Grouped Data Variance
\[ \sigma^2=\frac{\sum f_i x_i^2}{N}-\left(\frac{\sum f_i x_i}{N}\right)^2 \]
Range
\[ R=x_{max}-x_{min} \]