Mathematical Formulas

​||\dfrac{\text{numerator}}{\text{denominator}}\times100||

||\dfrac{\text{numerator}}{\text{denominator}}=\dfrac{\text{number sought}}{100}||

||a+b=b+a||

||a\times b=b\times a||

||(a+b)+c=a+(b+c)||

||(a\times b)\times c=a\times(b\times c)||

||a+0=0+a=a||

||a\times1=1\times a=a||

||a\times0=0\times a=0||

||a+-a=-a+a=0||

||a\times\dfrac{1}{a}=1||

||a\times(b\pm c)=a\times b\pm a\times c||

0 Degree

||y=b||

1st degree

||y=x||

Functional form

Symmetrical form

General form

||y=ax+b|||a|: rate of change (slope)

|b|: y-intercept||a=\dfrac{y_2-y_1}{x_2-x_1}||

||\dfrac{x}{a}+\dfrac{y}{b}=1|||a|: x-intercept

|b|: y-intercept

||Ax+By+C=0||

|\Rightarrow| Symmetrical||\begin{align}a_s&=\dfrac{-b_f}{a_f}\\b_s&=b_f\end{align}||

|\Rightarrow| Functional||\begin{align}a_f&=\dfrac{-b_s}{a_s}\\b_f&=b_s\end{align}||

|\Rightarrow| Functional||\begin{align}a_f&=\dfrac{-A}{B}\\b_f&=\dfrac{-C}{B}\end{align}||

|\Rightarrow| General

Find the common denominator and bring everything to the same side of the equation.

|\Rightarrow| General

Find the common denominator and bring everything to the same side of the equation.

|\Rightarrow| Symmetrical||\begin{align}a_s&=\dfrac{-C}{A}\\\\b_s&=\dfrac{-C}{B}\end{align}||

2nd degree

||y=x^2||

General form

Standard form

Factored form

||y=ax^2+bx+c||

||\begin{align}y&=\text{a}\big(b(x-h)\big)^2+k\\y&=\text{a }b^2(x-h)^2+k\\y&=a(x-h)^2+k\end{align}||

Two zeros||y=a(x-z_1)(x-z_2)||One unique zero||y=a(x-z_1)^2||

Number of zeros||\sqrt{b^2-4ac}||

Number of zeros||\sqrt{\dfrac{-k}{a}}||

Number of zeros

Directly accessible from the equation (see the box above).

Note: if there are no zeros, it's not possible to use this form.

Value of the zeros||\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}||

Value of the zeros||h\pm\sqrt{\dfrac{-k}{a}}||

Value of the zeros

|z_1| and |z_2|

Absolute value

||y=\vert x\vert||

Standard form

||\begin{align}y&=\text{a }\vert b(x-h)\vert+k\\y&=\text{a }\vert b\vert\times\vert x-h\vert+k\\y&=a\ \vert x-h\vert+k\end{align}||

Square root

||y=\sqrt{x}||

Standard form

||\begin{align}y&=\text{a}\sqrt{b(x-h)}+k\\[3pt]y&=\text{a}\sqrt b\sqrt{\pm(x-h)}+k\\[3pt]y&=a\sqrt{\pm(x-h)}+k\end{align}||

Exponential

||f(x)=c^x||

||f(x)=a(c)^{b(x-h)}+k||

||\begin{align}a^0&=1\\[3pt]a^1&=a\\[3pt]a^{-m}&=\dfrac{1}{a^m}\\[3pt]a^{^{\frac{\large{m}}{\large{n}}}}&=\sqrt[\large{n}]{a^m}\\[3pt]a^m=a^n&\!\!\ \Leftrightarrow\ m=n\\[3pt]a^ma^n&=a^{m+n}\\[3pt]\dfrac{a^m}{a^n}&=a^{m-n}\\[3pt](ab)^m&=a^mb^m\\[3pt](a^m)^{^{\Large{n}}}&=a^{mn}\\[3pt]\left(\dfrac{a}{b}\right)^m&=\dfrac{a^m}{b^m}\\[3pt]\sqrt[\large{n}]{ab}&=\sqrt[\large{n}]{a}\ \sqrt[\large{n}]{b}\\[3pt]\sqrt[\large{n}]{\dfrac{a}{b}}&=\dfrac{\sqrt[\large{n}]{a}}{\sqrt[\large{n}]{b}}\end{align}||

Logarithmic

||f(x)=\log_cx||

||f(x)=a\log_c(b(x-h))+k||

||\begin{align}\log_c1&=0\\[3pt]\log_cc&=1\\[3pt]c^{\log_{\large{c}}m}&=m\\[3pt]\log_cc^m&=m\\[3pt]\log_cm=\log_cn\ &\Leftrightarrow\ m=n\\[3pt]\log_c(mn)&=\log_cm+\log_cn\\[3pt]\log_c\left(\dfrac{m}{n}\right)&=\log_cm-\log_cn\\[3pt]\log_c(m^n)&=n\log_cm\\[3pt]\log_cm&=\dfrac{\log_sm}{\log_sc}\end{align}||

One is the inverse of the other||x=c^y\ \Longleftrightarrow\ y=\log_cx||

Sine

||f(x)=\sin x||

||f(x)=a\sin\big(b(x-h)\big)+k||

||\begin{align}\vert a\vert&=\dfrac{\max-\min}{2}\\[3pt]\vert b \vert&=\dfrac{2\pi}{\text{period}}\\[3pt]\text{Range}f&=[k-a,k+a]\end{align}||Zeros: An infinite number of the form |(x_1+nP)| and |(x_2+nP)| where |x_1| and |x_2| are consecutive zeros, |n\in\mathbb{Z}| and |P| is the period.

Cosine

||f(x)=\cos x||

||f(x)=a\cos\big(b(x-h)\big)+k||

Tangent

||f(x)=\tan x||

||f(x)=a\tan\big(b(x-h)\big)+k||

||\vert b\vert=\dfrac{\pi}{\text{period}}\\[3pt]\text{Dom}\ f=\mathbb{R}\backslash\left\{\left(h+\dfrac{P}{2}\right)+nP\right\}||where |n\in\mathbb{Z}| and |P| is the period.

Zeros: An infinite number of the form |x_1+nP| where |x_1| is a zero, |n\in\mathbb{Z}| and |P| is the period.

Arcsine

||f(x)=\arcsin(x)||or||f(x)=\sin^{-1}(x)||

||f(x)=a\arcsin\big(b(x-h)\big)+k||

Arccosine

||f(x)=\arccos(x)||or||f(x)=\cos^{-1}(x)||

||f(x)=a\arccos\big(b(x-h)\big)+k||

Arctangent

||f(x)=\arctan(x)||or||f(x)=\tan^{-1}(x)||

||f(x)=a\arctan\big(b(x-h)\big)+k||

||\sin^2\theta+\cos^2\theta=1||

||1+\tan^2\theta=sec^2\theta||

||1+\text{cotan}^2\theta=\text{cosec}^2\theta||

||\begin{align}\sin(a+b)&=\sin a\cos b+\cos a\sin b\\[3pt]\sin(a-b)&=\sin a\cos b-\cos a\sin b\\[3pt]\cos(a+b)&=\cos a\cos b-\sin a\sin b\\[3pt]\cos(a-b)&=\cos a\cos b+\sin a\sin b\\[3pt]\tan(a+b)&=\dfrac{\tan a+\tan b}{1-\tan a\tan b}\\[3pt]\tan(a-b)&=\dfrac{\tan a-\tan b}{1+\tan a\tan b}\end{align}||

||\begin{align}\sin2x&=2\sin x\cos x\\[3pt]\cos2x&=1-2\sin^2x\\[3pt]\tan2x&=\dfrac{2}{\text{cotan}x-\tan x}\\[3pt]\sin(-\theta)&=-\sin\theta\\[3pt]\cos(-\theta)&=\cos\theta\\[3pt]\sin\left(\theta+\dfrac{\pi}{2}\right)&=\cos\theta\\[3pt]\cos\left(\theta+\dfrac{\pi}{2}\right)&=-\sin\theta\end{align}||

Triangle

The sum of all sides

|A =\dfrac{b\times h}{2}|

|A = \sqrt{p(p-a)(p-b)(p-c)}|
where
|p=\dfrac{a+b+c}{2}=| half-perimeter

|A=\dfrac{ab\sin C}{2}|
where |C=| measure of the angle located between sides |a| and |b|

Square

|P=4 \times s|

|\begin{align} A &= s \times s\\
A &= s^2
\end{align}|

Rectangle

|\begin{align} P &= b+h+b+h\\
P &= 2(b+h)
\end{align}|

|A=bh|

Rhombus

P=|4 \times s|

|A=\dfrac{D\times d}{2}|

Parallelogram

The sum of all sides

|A=bh|

Trapezoid

The sum of all sides

|A=\dfrac{(B+b)\times h}{2}|

Regular polygon

|P=n \times s|

|A=\dfrac{san}{2}|

Any polygon

The sum of all sides

The sum of the areas of all the triangles that make up the polygon

Circle

|\begin{align} d &= 2r\\\\
r &= \frac{d}{2}
\end{align}|

||\begin{align} C &= \pi d\\\\
C &= 2 \pi r
\end{align}||

|A=\pi r^2|

Circular arc and sector of a circle

|\displaystyle \frac{\text{Central angle}}{360^o}=\frac{\text{Arc length}}{2\pi r}|

|\displaystyle \frac{\text{Central angle}}{360^o}=\frac{\text{Area of sector}}{\pi r^2}|

Theorems related to radii, diameters, chords and arcs:

• The radii of a circle are congruent.

• The diameter is the longest chord in a circle.

• In the same circle or in two isometric circles, two isometric chords are located at the same distance from the centre and vice versa.

• Any diameter perpendicular to a chord divides that chord and each of the arcs it subtends into two isometric parts.

• In a circle, two arcs are congruent if and only if they are subtended by congruent chords.

Theorems related to angles:

• Connecting any point on a circle to the endpoints of a diameter forms a right angle.

• The measure of an inscribed angle is half that of the arc formed between its sides.

• An angle whose vertex lies between the circle and its centre measures half the sum of the lengths of the arcs between its extended sides.

• An angle whose vertex lies outside a circle measures half the difference between the lengths of the arcs between its sides.

Theorems relating to the secants and tangents of the circle:

• Any line perpendicular to the endpoint of a ray is tangent to the circle and vice versa.

• Two parallel lines, secant or tangent to a circle, intercept two isometric arcs on the circle.

• If two tangents are drawn from point |P| outside a circle with centre |O,| to points |A| and |B| on the circle, then line |OP| is the angle bisector of angle |APB| and |\mathrm{m}\overline{PA}=\mathrm{m}\overline{PB}.|

• If the extension of a chord |\overline{AB}| intersects the extension of a chord |\overline{CD}| at a point |P| outside the circle, then the product of |\mathrm{m}\overline{PA}| and |\mathrm{m}\overline{PB}| is equal to the product of |\mathrm{m}\overline{PC}| and |\mathrm{m}\overline{PD}.|

• If from point |P| outside a circle a line tangent to the circle is drawn at |C| and another line intersects the circle at |A| and |B|, then the product of |\mathrm{m}\overline{PA}| and |\mathrm{m}\overline{PB}| is equal to the square of |\mathrm{m}\overline{PC}.|

• When two chords intersect inside a circle, the product of the measures of the segments of one equals the product of the measures of the segments of the other.

|\dfrac{n(n-3)}{2}|

|n-3|

|180(n-2)|

|\dfrac{180(n-2)}{n}|

​Prism and cylinder

Sum of the areas of the lateral faces of the solid

|A_L=P_b\times h|

​Sum of the areas of all faces of the solid

|A_T = A_L+2A_b|

​|V=A_b\times h|

​Pyramid and cone

​Sum of the areas of the lateral faces of the solid

|A_L=\displaystyle \frac{P_b\times a}{2}|

Sum of the areas of all faces of the solid

|A_T = A_L+A_b|

​|V=\displaystyle \frac{A_b\times h}{3}|

​Sphere

|A=4\pi r^2|

​|V=\displaystyle \frac{4\pi r^3}{3}|

• Pythagorean theorem
  ​In any right triangle, the sum of the square of the legs |(a| and |b)| is equal to the square of the hypotenuse |(c).|||a^2+b^2 = c^2||

• In any triangle, the measure of any one side is smaller than the sum of the measures of the other two sides.

• In any isosceles triangle, the angles opposite the congruent sides are congruent.

• In any right triangle, the acute angles are complementary |(90^\circ).|

• Any right triangle with a |30^\circ| angle has a side opposite the |30^\circ| angle that is equal to half the hypotenuse.

Altitude to Hypotenuse theorem

In a right triangle, the height |(h)| drawn from the right angle is the proportional mean between the 2 segments it creates on the hypotenuse |(m| and |n).|||\dfrac{m}{h}=\dfrac{h}{n}\quad\text{or}\quad h^2=mn||

Product of the Sides theorem

In a right triangle, the product of the hypotenuse |(c)| and the corresponding height |(h)| is equal to the product of legs |(a| and |b).|||ch=ab\quad\text{or}\quad h=\dfrac{ab}{c}||

Proportional Mean theorem

In a right triangle, each leg |(a| and |b)| is the proportional mean between its projection onto hypotenuse |(m| or |n)| and the entire hypotenuse |(c).|||\dfrac{m}{a}=\dfrac{a}{c}\quad\text{or}\quad a^2=mc\\\dfrac{n}{c}=\dfrac{b}{c}\quad\text{or}\quad b^2=nc||

||k=\dfrac{\text{Length of image figure}}{\text{Length of initial figure}}||

||k^2=\dfrac{\text{Area of image figure}}{\text{Area of initial figure}}||

​||k^3=\dfrac{\text{Volume of image solid}}{\text{Volume of initial solid}}||

||a=\Vert \overrightarrow{u}\Vert \cos \theta|| ||b=\Vert \overrightarrow{u}\Vert \sin \theta||

Consider the vector |\overrightarrow{AB}| where |A(x_1, y_1)| and |B(x_2, y_2)|

The components are: ||a=x_2-x_1\\b=y_2-y_1||

Consider the vector |\overrightarrow{u}=(a,b)|

The magnitude is: ||\Vert\overrightarrow{v}\Vert=\sqrt{a^2+b^2}||

Consider the vector  |\overrightarrow{AB}| where |A(x_1, y_1)| and |B(x_2, y_2)|

The magnitude is:  ||\Vert\overrightarrow{AB}\Vert=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}||

|\theta=\tan^{-1}\left(\displaystyle\frac{b}{a}\right)|

• If​ |a>0,\ b>0\ \Rightarrow\ \theta| is correct.

• If​ |a<0,\ b>0\ \Rightarrow\ \theta+180^o.|

• If |a<0,\ b<0\ \Rightarrow\ \theta+180^o.|

• If |a>0,\ b<0\ \Rightarrow\ \theta+360^o.|

Consider |\overrightarrow{u}=(a,b)| and |\overrightarrow{v}=(c,d)|

Therefore, |\overrightarrow{u}+\overrightarrow{v}=(a+c,b+d)|

|\Vert \overrightarrow{u}+\overrightarrow{v}\Vert=\Vert \overrightarrow{u}\Vert+\Vert \overrightarrow{v}\Vert-2\Vert \overrightarrow{u}\Vert\ \Vert \overrightarrow{v}\Vert\ \cos\theta|

where |\theta =\ \Large{\mid} \normalsize 180^o - \mid \theta_\overrightarrow{u}-\theta_\overrightarrow{v}\mid \Large{\mid}|

Consider |\overrightarrow{u}=(a,b)| and |\overrightarrow{v}=(c,d)|

Therefore, |\overrightarrow{u}-\overrightarrow{v}=(a-c,b-d)|​

|\Vert \overrightarrow{u}+\overrightarrow{v}\Vert=\Vert \overrightarrow{u}\Vert+\Vert \overrightarrow{v}\Vert-2\Vert \overrightarrow{u}\Vert\ \Vert \overrightarrow{v}\Vert\ \cos\theta|

where |\theta=\mid \theta_\overrightarrow{u}-\theta_\overrightarrow{v}\mid| if |\mid \theta_\overrightarrow{u}-\theta_\overrightarrow{v}\mid<180^o|
and |\theta = 180^o - \mid \theta_\overrightarrow{u}-\theta_\overrightarrow{v}\mid| otherwise

Consider scalar |k| and vector |\overrightarrow{u}=(a,b)|

Therefore, |k\overrightarrow{u}=(ka,kb)|
||\begin{align}\Vert k \overrightarrow{u} \Vert &= k \times \Vert\overrightarrow{u}\Vert \\ \theta_{k \overrightarrow{u}} &= \theta_{\overrightarrow{u}} \end{align}||

If the scalar product equals |0,| the vectors are perpendicular.

Using components

Consider |\overrightarrow{u}=(a,b)| and |\overrightarrow{v}=(c,d)|

Then, |\overrightarrow{u}\cdot \overrightarrow{v}=ac+bd|

Using the magnitude and direction

|\overrightarrow{u}\cdot \overrightarrow{v}=\Vert\overrightarrow{u}\Vert\times \Vert\overrightarrow{v}\Vert\times \cos\theta|

1) The sum of two vectors is a vector.

2) Commutativity

|\overrightarrow{u}+\overrightarrow{v}=\overrightarrow{v}+\overrightarrow{u}|

3) Associativity

|(\overrightarrow{u} + \overrightarrow{v}) + \overrightarrow{w} = \overrightarrow{u} + (\overrightarrow{v} + \overrightarrow{w})|

4) Existence of a neutral (identity) element

|\overrightarrow{u}+\overrightarrow{0}=\overrightarrow{0}+\overrightarrow{u}=\overrightarrow{u}|

​5) Existence of opposites

|\overrightarrow{u}+(-\overrightarrow{u})=-\overrightarrow{u}+\overrightarrow{u}=\overrightarrow{0}|

1) The product of a vector and a scalar is always a vector.

2) Associativity

|k_1(k_2\overrightarrow{u})=(k_1k_2)\overrightarrow{u}|

​3) Existence of a neutral (identity) element

|1\times \overrightarrow{u}=\overrightarrow{u}\times 1=\overrightarrow{u}|

​4) Distributivity over vector addition

|k(\overrightarrow{u}+\overrightarrow{v})=k\overrightarrow{u}+k\overrightarrow{v}|

5) Distributivity over scalar addition

|(k_1+k_2)\overrightarrow{u}=k_1\overrightarrow{u}+k_2\overrightarrow{v}|

1) Commutativity

|\overrightarrow{u}\cdot \overrightarrow{v}=\overrightarrow{v}\cdot \overrightarrow{u}|

​2) Scalar associativity

|k_1\overrightarrow{u}\cdot k_2\overrightarrow{v}=k_1k_2(\overrightarrow{u}\cdot\overrightarrow{v})|

​3) Distributivity over a vector sum

|\overrightarrow{u}\cdot(\overrightarrow{v}+\overrightarrow{w})=(\overrightarrow{u}\cdot\overrightarrow{v})+(\overrightarrow{u}\cdot\overrightarrow{w})|

Displacements

||\begin{align}\Delta x&=x_2-x_1\\[3pt]\Delta y&=y_2-y_1\end{align}||

Distance between two points

||d(A,B)=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}||

Division point formula

Part-to-whole ratio

Part-to-part ratio

||\begin{align}x_p&=x_1+\dfrac{r}{s}(x_2-x_1)\\[3pt]y_p&=y_1+\dfrac{r}{s}(y_2-y_1)\end{align}||

||\begin{align}x_p&=x_1+\dfrac{r}{r+s}(x_2-x_1)\\[3pt]y_p&=y_1+\dfrac{r}{r+s}(y_2-y_1)\end{align}||

Midpoint formula

||(x_m,y_m)=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)||

Slope (rate of change) of a line

||a=\dfrac{\Delta y}{\Delta x}=\dfrac{y_2-y_1}{x_2-x_1}||

Relative position of two lines with equations of the form |y=ax+b|

Coinciding parallel lines

Disjoint parallel lines

Perpendicular lines

||\begin{align}a_1&=a_2\\[3pt]b_1&=b_2\end{align}||

||\begin{align}​a_1&=a_2\\[3pt]b_1&\neq b_2\end{align}||

||a_1=-\dfrac{1}{a_2}||

​Translation

||t_{(a,b)}:(x,y)\stackrel{t}{\mapsto}(x+a,y+b)||

​||t^{-1}_{(a,b)}=t_{(-a,-b)}:(x,y)\stackrel{t}{\mapsto}(x-a,y-b)||

​Rotation

||\begin{align}r_{(O,90^\circ)}&:(x,y)\stackrel{r}{\mapsto}(-y,x)\\[3pt]r_{(O,-270^\circ)}&:(x,y)\stackrel{r}{\mapsto}(-y,x)\\[3pt]r_{(O,180^\circ)}&:(x,y)\stackrel{r}{\mapsto}(-x,-y)\\[3pt]r_{(O,-90^\circ)}&:(x,y)\stackrel{r}{\mapsto}(y,-x)\\[3pt]r_{(O,270^\circ)}&:(x,y)\stackrel{r}{\mapsto}(y,-x)\end{align}||

||\begin{align}​r^{-1}_{(O,90^\circ)}&=r_{(O,-90^\circ)}\\[3pt]r^{-1}_{(O,-270^\circ)}&=r_{(O,270^\circ)}\\[3pt]r^{-1}_{(O,180^\circ)}&=r_{(O,180^\circ)}\\[3pt]r^{-1}_{(O,-90^\circ)}&=r_{(O,90^\circ)}\\[3pt]r^{-1}_{(O,270^\circ)}&=r_{(O,-270^\circ)}\end{align}||

​Reflection

(Symmetry)

||\begin{align}​s_x&:(x,y)\stackrel{s}{\mapsto}(x,-y)\\[3pt]s_y&:(x,y)\stackrel{s}{\mapsto}(-x,y)\\[3pt]s_{\small/}&:(x,y)\stackrel{s}{\mapsto}(y,x)\\[3pt]s_{\tiny\backslash}&:(x,y)\stackrel{s}{\mapsto}(-y,-x)\end{align}||

||\begin{align}​s^{-1}_x&=s_x\\[3pt]s^{-1}_y&=s_y\\[3pt]s^{-1}_{\small/}&=s_{\small/}\\[3pt]s^{-1}_{\tiny\backslash}&=s_{\tiny\backslash}\end{align}||

​​Dilation

||h_{(O,k)}:(x,y)\stackrel{h}{\mapsto}(kx,ky)||

​||h^{-1}_{(O,k)}=h_{\left(\frac{1}{k},\frac{1}{k}\right)}:(x,y)\stackrel{h}{\mapsto}\left(\dfrac{x}{k},\dfrac{y}{k}\right)||

Circle

Geometric locus of all points located at an equal distance from the centre.

||x^2+y^2=r^2|| ||(x-h)^2+(y-k)^2=r^2||

|r:| radius

|(h,k):| Centre of circle

Ellipse

Geometric locus of all points for which the sum of the distances to the two foci is constant.

||\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1|| ||\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1||

||\begin{align}a&=\dfrac{\text{Horizontal axis}}{2}\\b&=\dfrac{\text{Vertical Axis}}{2}\end{align}|| |(h,k):| Centre of the ellipse

Hyperbola

Geometric locus of all points for which the absolute value of the difference in distance to the two foci is constant.

||\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=\pm1|| ||\dfrac{(x-h)^2}{a^2}-\dfrac{(y-k)^2}{b^2}=\pm1||

Asymptotes: ||\begin{align}y&=\dfrac{b}{a}(x-h)+k\\y&=-\dfrac{b}{a}(x-h)+k\end{align}|| |(h,k):| Centre of the hyperbola

Parabola

Geometric locus of all points located at an equal distance from the directrix and the focal point.

​||(x-h)^2=4c(y-k)|| ||(y-k)^2=4c(x-h)||

||\vert c\vert :\dfrac{\text{Distance focus-directrix}}{2}|| |(h,k):| Vertex of the parabola

​Probability

||\text{Probability}=\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}}||

​​Complementary probability

||\mathbb{P}(A')=1-P(A)||

Probability of mutually exclusive events

||\mathbb{P}(A\cup B)=\mathbb{P}(A)+\mathbb{P}(B)||

​Probability of non-mutually exclusive events

||\mathbb{P}(A\cup B)=\mathbb{P}(A)+\mathbb{P}(B)-\mathbb{P}(A\cap B)||

Conditional probability

||\mathbb{P}(B\mid A)=\mathbb{P}_A(B)=\dfrac{\mathbb{P}(B\cap A)}{\mathbb{P}(A)}||

Expected gain

||\mathbb{E}[\text{Gain}]=\text{Probability of winning}\times\text{Net gain}+\text{Probability of losing}\times\text{Net loss}||

​Mathematical expectation

||\mathbb{E}[X]=x_1\mathbb{P}(x_1)+x_2\mathbb{P}(x_2)+\ldots+x_n\mathbb{P}(x_n)||where the possible outcomes of |X| are the values |x_1, \ldots, x_n.|

​​Mean

||\overline{x}=\dfrac{\sum x_i}{n}||

||\overline{x}=\dfrac{\sum x_i n_i}{n}||

Approximate mean: ||\overline{x}=\dfrac{\sum m_i n_i}{n}||

​​Median

||\text{Rank}_\text{median}=\left(\dfrac{n+1}{2}\right)|| If |n| is odd, the median is obtained directly.

If |n| is even, the median is obtained by calculating the mean of the two central data values.

||\text{Rank}_\text{median}=\left(\dfrac{n+1}{2}\right)|| If |n| is odd, the median is obtained directly.

If |n| is even, the median is obtained by calculating the mean of the two central data values.

Medial class:

The class that contains the median.

The median of a grouped-data distribution is often estimated by calculating the middle of the medial class.

​Mode

The most frequent data value

The most frequent data value

Modal class:

The class with the largest frequency

​Range

||R=x_\text{max}-x_\text{min}||

||R=\text{Value}_\text{Max}-\text{Value}_\text{Min}||

||R=\text{Boundary}_\text{upper}-\text{Boundary}_\text{lower}||

​Interquartile range

||IR=Q_3-Q_1||

||IR=Q_3-Q_1||

||IR=Q_3-Q_1||

​Quarter range

||Q=\dfrac{EI}{2}=\dfrac{Q_3-Q_1}{2}||

||Q=\dfrac{EI}{2}=\dfrac{Q_3-Q_1}{2}||

||Q=\dfrac{EI}{2}=\dfrac{Q_3-Q_1}{2}||

Mean deviation

||MD=\dfrac{\sum\mid x_i-\overline{x}\mid}{n}||

||MD=\dfrac{\sum n_i\mid X_i-\overline{x}\mid}{n}||

||MD=\dfrac{\sum n_i \mid m_i-\overline{x}\mid}{n}||

Standard deviation

||\sigma=\sqrt{\dfrac{\sum (x_i-\overline{x})^2}{n}}||

||\sigma=\sqrt{\dfrac{\sum n_i(X_i-\overline{x})^2}{n}}||

||\sigma=\sqrt{\dfrac{\sum n_i (m_i-\overline{x})^2}{n}}||

Quintile rank

||R_5(x)\approx\left(\dfrac{\text{No. of data values greater than } x+\dfrac{\text{No. of data values equal to }x}{2}}{\text{Total number of data values}}\right) \times 5|| If the result is not a whole number, round up.

Percentile rank

||R_{100}(x)\approx\left(\dfrac{\text{No. of data values less than } x+\dfrac{\text{No. of data values equal to }x}{2}}{\text{Total number of data values}}\right) \times 100|| If the result is not a whole number, round up to the next whole number, unless the result is |99.|

||r\approx\pm\left(1-\dfrac{l}{L}\right)|| where |L| represents the length and |l,| the width of the rectangle that encompasses the scatter plot.

The sign of |r| depends on the direction of the scatter plot.