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Edexcel IAL Mathematics Formula Book

Reproduced from the official Pearson Edexcel IAL Mathematics & Further Mathematics Formula Books (2018 specification). Both booklets are provided in every examination — this page mirrors their combined contents.
This page reproduces the formulas from the official Pearson Edexcel International A Level Mathematics and Further Mathematics Formula Books. Cross-reference against the printed booklet before your exam. Pearson & Edexcel are trademarks of Pearson Education Limited; WhateverGo is an independent study tool with no affiliation.
Pure Mathematics (P1 – P4)

Pure Mathematics P1

Mensuration

Surface area of sphere$$\text{Surface area} = 4\pi r^2$$
Area of curved surface of cone$$A = \pi r \times \text{slant height}$$

Cosine rule

$$a^2 = b^2 + c^2 - 2bc\cos A$$

Pure Mathematics P2

Arithmetic series

$$u_n = a + (n-1)d$$
$$S_n = \tfrac{1}{2}n(a+l) = \tfrac{1}{2}n\bigl[2a+(n-1)d\bigr]$$

Geometric series

$$u_n = ar^{n-1}$$
$$S_n = \dfrac{a(r^n - 1)}{r - 1} \qquad S_\infty = \dfrac{a}{1-r} \quad (|r|<1)$$

Logarithms and exponentials

$$\log_a x = \dfrac{\log_b x}{\log_b a}$$

Binomial series

$$(a+b)^n = a^n + \tbinom{n}{1}a^{n-1}b + \tbinom{n}{2}a^{n-2}b^2 + \cdots + b^n \quad (n \in \mathbb{Z}^+)$$
where$$\tbinom{n}{r} = \dfrac{n!}{r!\,(n-r)!}$$
$$(1+x)^n = 1+nx+\dfrac{n(n-1)}{2!}x^2+\cdots+\tbinom{n}{r}x^r+\cdots \quad (|x|<1,\; n \in \mathbb{Q})$$

Numerical integration

The trapezium rule$$\int_a^b y\,dx \approx \tfrac{1}{2}h\bigl\{(y_0 + y_{n-1}) + 2(y_1 + y_2 + \cdots + y_{n-1})\bigr\}$$
where$$h = \dfrac{b-a}{n}$$

Pure Mathematics P3

Candidates sitting Pure Mathematics P3 may also require those formulae listed under Pure Mathematics P1 and P2.

Logarithms and exponentials

$$e^{\ln a} = a$$

Trigonometric identities

$$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$$
$$\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$$
$$\tan(A \pm B) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A\tan B} \quad \bigl(A \pm B \neq (k+\tfrac{1}{2})\pi\bigr)$$
$$\sin A + \sin B = 2\sin\!\tfrac{A+B}{2}\cos\!\tfrac{A-B}{2}$$
$$\sin A - \sin B = 2\cos\!\tfrac{A+B}{2}\sin\!\tfrac{A-B}{2}$$
$$\cos A + \cos B = 2\cos\!\tfrac{A+B}{2}\cos\!\tfrac{A-B}{2}$$
$$\cos A - \cos B = -2\sin\!\tfrac{A+B}{2}\sin\!\tfrac{A-B}{2}$$

Differentiation

$f(x)$$f'(x)$$f(x)$$f'(x)$
$\tan kx$$k\sec^2 kx$$\sec x$$\sec x\tan x$
$\cot x$$-\csc^2 x$$\csc x$$-\csc x\cot x$
Product rule$$\dfrac{d}{dx}(uv) = u\dfrac{dv}{dx}+v\dfrac{du}{dx}$$
Quotient rule$$\dfrac{d}{dx}\!\left(\dfrac{u}{v}\right) = \dfrac{v\,\frac{du}{dx}-u\,\frac{dv}{dx}}{v^2}$$
Chain rule$$\dfrac{dy}{dx} = \dfrac{dy}{du}\cdot\dfrac{du}{dx}$$

Integration

$f(x)$$\displaystyle\int f(x)\,dx$$f(x)$$\displaystyle\int f(x)\,dx$
$\sec^2 kx$$\dfrac{1}{k}\tan kx$$\tan x$$\ln|\sec x|$
$\cot x$$\ln|\sin x|$

Pure Mathematics P4

Candidates sitting Pure Mathematics P4 may also require those formulae listed under Pure Mathematics P1, P2 and P3.

Binomial series

$$(1+x)^n = 1+nx+\dfrac{n(n-1)}{2!}x^2+\cdots+\tbinom{n}{r}x^r+\cdots \quad (|x|<1,\; n \in \mathbb{Q})$$

Integration

$f(x)$$\displaystyle\int f(x)\,dx$
$\csc x$$-\ln|\csc x + \cot x|$ , $\ln|\tan\tfrac{1}{2}x|$
$\sec x$$\ln|\sec x + \tan x|$ , $\\ln|\\tan(\\tfrac{1}{2}x + \\tfrac{1}{4}\\pi)|$
Integration by parts$$\int u\,\dfrac{dv}{dx}\,dx = uv - \int v\,\dfrac{du}{dx}\,dx$$
Further Pure Mathematics (FP1 – FP3)

Summations (FP1)

Standard sums$$\sum_{r=1}^{n} r^2 = \tfrac{1}{6}\,n(n+1)(2n+1)$$$$\sum_{r=1}^{n} r^3 = \tfrac{1}{4}\,n^2(n+1)^2$$

Numerical Solution of Equations (FP1)

Newton-Raphson for $f(x)=0$$$x_{n+1} = x_n - \dfrac{f(x_n)}{f'(x_n)}$$

Conics (FP1)

Parabola and Rectangular Hyperbola
ParabolaRectangular Hyperbola
Standard Form$y^2=4ax$$xy=c^2$
Parametric$(at^2,\,2at)$$\left(ct,\,\dfrac{c}{t}\right)$
Foci$(a,0)$Not required
Directrices$x=-a$Not required

Matrix Transformations (FP1)

Anticlockwise rotation through $\theta$ about $O$$$\begin{pmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{pmatrix}$$
Reflection in the line $y=(\tan\theta)\,x$$$\begin{pmatrix}\cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta\end{pmatrix}$$
In FP1, $\theta$ will be a multiple of $45^\circ$.

Area of a Sector (FP2)

Polar coordinates$$A = \dfrac{1}{2}\int r^2\,d\theta$$

Complex Numbers (FP2)

Euler's formula$$e^{i\theta} = \cos\theta+i\sin\theta$$
De Moivre's theorem$$\{r(\cos\theta+i\sin\theta)\}^n = r^n(\cos n\theta+i\sin n\theta)$$
Roots of $z^n=1$$$z = e^{\frac{2\pi k i}{n}}, \quad k=0,1,2,\ldots,n-1$$

Maclaurin's and Taylor's Series (FP2)

Maclaurin ($a=0$)$$f(x) = f(0) + xf'(0) + \dfrac{x^2}{2!}f''(0) + \cdots + \dfrac{x^r}{r!}f^{(r)}(0) + \cdots$$
Taylor (about $x=a$)$$f(x) = f(a) + (x-a)f'(a) + \dfrac{(x-a)^2}{2!}f''(a) + \cdots + \dfrac{(x-a)^r}{r!}f^{(r)}(a) + \cdots$$
Standard expansions$$e^x = 1+x+\dfrac{x^2}{2!}+\cdots+\dfrac{x^r}{r!}+\cdots \quad\text{for all }x$$$$\ln(1+x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \cdots + (-1)^{r+1}\dfrac{x^r}{r}+\cdots \quad (-1 < x \le 1)$$$$\sin x = x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \cdots + (-1)^r\dfrac{x^{2r+1}}{(2r+1)!}+\cdots \quad\text{for all }x$$$$\cos x = 1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} - \cdots + (-1)^r\dfrac{x^{2r}}{(2r)!}+\cdots \quad\text{for all }x$$$$\arctan x = x - \dfrac{x^3}{3} + \dfrac{x^5}{5} - \cdots + (-1)^r\dfrac{x^{2r+1}}{2r+1}+\cdots \quad (-1 \le x \le 1)$$

Vectors (FP3)

Resolved part of $\mathbf{a}$ in the direction of $\mathbf{b}$$$\dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|}$$
Point dividing $AB$ in ratio $\lambda:\mu$$$\dfrac{\mu\mathbf{a}+\lambda\mathbf{b}}{\lambda+\mu}$$
Vector product$$\mathbf{a}\times\mathbf{b} = |\mathbf{a}||\mathbf{b}|\sin\theta\,\hat{\mathbf{n}} = \begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix} = \begin{pmatrix}a_2 b_3-a_3 b_2\\a_3 b_1-a_1 b_3\\a_1 b_2-a_2 b_1\end{pmatrix}$$
Scalar triple product$$\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c}) = \begin{vmatrix}a_1&a_2&a_3\\b_1&b_2&b_3\\c_1&c_2&c_3\end{vmatrix} = \mathbf{b}\cdot(\mathbf{c}\times\mathbf{a}) = \mathbf{c}\cdot(\mathbf{a}\times\mathbf{b})$$
Line through $A$ (pos. vector $\mathbf{a}$) with direction $\mathbf{b}$$$\text{Cartesian: }\dfrac{x-a_1}{b_1}=\dfrac{y-a_2}{b_2}=\dfrac{z-a_3}{b_3}(=\lambda)$$
Plane through $A$ with normal $\mathbf{n}=n_1\mathbf{i}+n_2\mathbf{j}+n_3\mathbf{k}$$$n_1 x+n_2 y+n_3 z+d=0 \quad\text{where }d=-\mathbf{a}\cdot\mathbf{n}$$
Plane through non-collinear points $A,B,C$$$\mathbf{r}=\mathbf{a}+\lambda(\mathbf{b}-\mathbf{a})+\mu(\mathbf{c}-\mathbf{a}) = (1-\lambda-\mu)\mathbf{a}+\lambda\mathbf{b}+\mu\mathbf{c}$$
Plane through point $\mathbf{a}$, parallel to $\mathbf{b}$ and $\mathbf{c}$$$\mathbf{r}=\mathbf{a}+s\mathbf{b}+t\mathbf{c}$$
Perp. distance from $(\alpha,\beta,\gamma)$ to $n_1 x+n_2 y+n_3 z+d=0$$$\dfrac{|n_1\alpha+n_2\beta+n_3\gamma+d|}{\sqrt{n_1^2+n_2^2+n_3^2}}$$

Hyperbolic Functions (FP3)

Fundamental identity$$\cosh^2 x - \sinh^2 x = 1$$
Double argument$$\sinh 2x = 2\sinh x\cosh x \qquad \cosh 2x = \cosh^2 x + \sinh^2 x$$
Logarithmic forms$$\operatorname{arccosh} x = \ln\!\left\{x+\sqrt{x^2-1}\right\} \quad (x\ge 1)$$$$\operatorname{arcsinh} x = \ln\!\left\{x+\sqrt{x^2+1}\right\}$$$$\operatorname{artanh} x = \dfrac{1}{2}\ln\!\left(\dfrac{1+x}{1-x}\right) \quad (|x|<1)$$

Conics (FP3)

EllipseParabolaHyperbolaRect. Hyperbola
Standard$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$$y^2=4ax$$\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$$xy=c^2$
Parametric$(a\cos\theta,b\sin\theta)$$(at^2,2at)$$(a\sec\theta,b\tan\theta)$$(ct,\frac{c}{t})$
Eccentricity$e<1$; $b^2=a^2(1-e^2)$$e=1$$e>1$; $b^2=a^2(e^2-1)$$e=\sqrt{2}$
Foci$(\pm ae,0)$$(a,0)$$(\pm ae,0)$$(\pm\sqrt{2}c,\pm\sqrt{2}c)$
Directrices$x=\pm\dfrac{a}{e}$$x=-a$$x=\pm\dfrac{a}{e}$$x+y=\pm\sqrt{2}c$
Asymptotesnonenone$\dfrac{x}{a}=\pm\dfrac{y}{b}$$x=0,\,y=0$

Differentiation (FP3)

$f(x)$$f'(x)$
$\arcsin x$$\dfrac{1}{\sqrt{1-x^2}}$
$\arccos x$$-\dfrac{1}{\sqrt{1-x^2}}$
$\arctan x$$\dfrac{1}{1+x^2}$
$\sinh x$$\cosh x$
$\cosh x$$\sinh x$
$\tanh x$$\operatorname{sech}^2 x$
$\operatorname{arcsinh} x$$\dfrac{1}{\sqrt{1+x^2}}$
$\operatorname{arccosh} x$$\dfrac{1}{\sqrt{x^2-1}}$
$\operatorname{artanh} x$$\dfrac{1}{1-x^2}$

Integration (FP3)

$+\text{constant};\; a > 0$ where relevant
$f(x)$$\displaystyle\int f(x)\,dx$
$\sinh x$$\cosh x$
$\cosh x$$\sinh x$
$\tanh x$$\ln\cosh x$
$\dfrac{1}{\sqrt{a^2-x^2}}$$\arcsin\!\left(\dfrac{x}{a}\right) \quad (|x|
$\dfrac{1}{a^2+x^2}$$\dfrac{1}{a}\arctan\!\left(\dfrac{x}{a}\right)$
$\dfrac{1}{\sqrt{x^2-a^2}}$$\operatorname{arccosh}\!\left(\dfrac{x}{a}\right),\;\ln\!\left\{x+\sqrt{x^2-a^2}\right\} \;\;(x>a)$
$\dfrac{1}{\sqrt{a^2+x^2}}$$\operatorname{arcsinh}\!\left(\dfrac{x}{a}\right),\;\ln\!\left\{x+\sqrt{a^2+x^2}\right\}$
$\dfrac{1}{a^2-x^2}$$\dfrac{1}{2a}\ln\!\left|\dfrac{a+x}{a-x}\right|,\;\dfrac{1}{a}\operatorname{artanh}\!\left(\dfrac{x}{a}\right) \;\;(|x|<a)$
$\dfrac{1}{x^2-a^2}$$\dfrac{1}{2a}\ln\!\left|\dfrac{x-a}{x+a}\right|$

Arc Length and Surface Area (FP3)

Arc length (Cartesian)$$s = \int\sqrt{1+\left(\dfrac{dy}{dx}\right)^2}\,dx$$
Arc length (parametric)$$s = \int\sqrt{\left(\dfrac{dx}{dt}\right)^2+\left(\dfrac{dy}{dt}\right)^2}\,dt$$
Surface area of revolution (about $x$-axis)$$S_x = 2\pi\int y\,ds = 2\pi\int y\sqrt{1+\left(\dfrac{dy}{dx}\right)^2}\,dx = 2\pi\int y\sqrt{\left(\dfrac{dx}{dt}\right)^2+\left(\dfrac{dy}{dt}\right)^2}\,dt$$
Mechanics (M1 – M3)

Mechanics M1

There are no formulae given for M1 in addition to those candidates are expected to know. Candidates sitting M1 may also require those formulae listed under Pure Mathematics P1.

Centres of Mass (M2)

Candidates sitting M2 may also require those formulae listed under Pure Mathematics P1, P2, P3 and P4.

Uniform bodies$$\text{Triangular lamina: }\dfrac{2}{3}\text{ along median from vertex}$$$$\text{Circular arc, radius }r\text{, angle }2\alpha:\;\dfrac{r\sin\alpha}{\alpha}\text{ from centre}$$$$\text{Sector of circle, radius }r\text{, angle }2\alpha:\;\dfrac{2r\sin\alpha}{3\alpha}\text{ from centre}$$

Motion in a Circle (M3)

Candidates sitting M3 may also require those formulae listed under Mechanics M2, and Pure Mathematics P1, P2, P3 and P4.

Transverse velocity$$v = r\dot{\theta}$$
Transverse acceleration$$\dot{v} = r\ddot{\theta}$$
Radial acceleration$$-r\dot{\theta}^2 = -\dfrac{v^2}{r}$$

Centres of Mass (M3)

Uniform bodies$$\text{Solid hemisphere, radius }r:\;\dfrac{3}{8}r\text{ from centre}$$$$\text{Hemispherical shell, radius }r:\;\dfrac{1}{2}r\text{ from centre}$$$$\text{Solid cone or pyramid of height }h:\;\dfrac{1}{4}h\text{ above base}$$$$\text{Conical shell of height }h:\;\dfrac{1}{3}h\text{ above base}$$

Universal Law of Gravitation (M3)

Gravitational force$$\text{Force} = \dfrac{Gm_1 m_2}{d^2}$$
Statistics (S1 – S3)

Probability

Addition law$$P(A\cup B) = P(A)+P(B)-P(A\cap B)$$
Conditional probability$$P(A\cap B) = P(A)\,P(B\mid A)$$
Bayes' theorem$$P(A\mid B) = \dfrac{P(B\mid A)\,P(A)}{P(B\mid A)\,P(A)+P(B\mid A')\,P(A')}$$

Discrete distributions

For a discrete random variable $X$ taking values $x_i$ with probabilities $P(X=x_i)$:
Expectation (mean)$$E(X) = \mu = \sum x_i\,P(X=x_i)$$
Variance$$\text{Var}(X) = \sigma^2 = \sum (x_i-\mu)^2\,P(X=x_i) = \sum x_i^2\,P(X=x_i)-\mu^2$$
For a function $g(X)$$$E\bigl(g(X)\bigr) = \sum g(x_i)\,P(X=x_i)$$

Continuous distributions

Standard continuous distribution:
Distribution of $X$P.D.F.MeanVariance
Normal $N(\mu,\sigma^2)$$\dfrac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$$\mu$$\sigma^2$

Correlation and regression

For a set of $n$ pairs of values $(x_i,y_i)$:
Summary statistics$$S_{xx} = \sum x_i^2 - \dfrac{(\sum x_i)^2}{n} \qquad S_{yy} = \sum y_i^2 - \dfrac{(\sum y_i)^2}{n} \qquad S_{xy} = \sum x_i y_i - \dfrac{(\sum x_i)(\sum y_i)}{n}$$
Product moment correlation coefficient$$r = \dfrac{S_{xy}}{\sqrt{S_{xx}\,S_{yy}}}$$
Regression coefficient of $y$ on $x$$$b = \dfrac{S_{xy}}{S_{xx}}$$
Least squares regression line of $y$ on $x$$$y = a+bx, \qquad a = \bar{y}-b\bar{x}$$

The Normal Distribution Function

$\Phi(z)$, defined as $\Phi(z)=\dfrac{1}{\sqrt{2\pi}}\displaystyle\int_{-\infty}^{z}e^{-\frac{1}{2}t^2}\,dt$

$z$$\Phi(z)$$z$$\Phi(z)$$z$$\Phi(z)$$z$$\Phi(z)$$z$$\Phi(z)$
0.000.50000.500.69151.000.84131.500.93322.000.9772
0.010.50400.510.69501.010.84381.510.93452.020.9783
0.020.50800.520.69851.020.84611.520.93572.040.9793
0.030.51200.530.70191.030.84851.530.93702.060.9803
0.040.51600.540.70541.040.85081.540.93822.080.9812
0.050.51990.550.70881.050.85311.550.93942.100.9821
0.060.52390.560.71231.060.85541.560.94062.120.9830
0.070.52790.570.71571.070.85771.570.94182.140.9838
0.080.53190.580.71901.080.85991.580.94292.160.9846
0.090.53590.590.72241.090.86211.590.94412.180.9854
0.100.53980.600.72571.100.86431.600.94522.200.9861
0.110.54380.610.72911.110.86651.610.94632.220.9868
0.120.54780.620.73241.120.86861.620.94742.240.9875
0.130.55170.630.73571.130.87081.630.94842.260.9881
0.140.55570.640.73891.140.87291.640.94952.280.9887
0.150.55960.650.74221.150.87491.650.95052.300.9893
0.160.56360.660.74541.160.87701.660.95152.320.9898
0.170.56750.670.74861.170.87901.670.95252.340.9904
0.180.57140.680.75171.180.88101.680.95352.360.9909
0.190.57530.690.75491.190.88301.690.95452.380.9913
0.200.57930.700.75801.200.88491.700.95542.400.9918
0.210.58320.710.76111.210.88691.710.95642.420.9922
0.220.58710.720.76421.220.88881.720.95732.440.9927
0.230.59100.730.76731.230.89071.730.95822.460.9931
0.240.59480.740.77041.240.89251.740.95912.480.9934
0.250.59870.750.77341.250.89441.750.95992.500.9938
0.260.60260.760.77641.260.89621.760.96082.550.9946
0.270.60640.770.77941.270.89801.770.96162.600.9953
0.280.61030.780.78231.280.89971.780.96252.650.9960
0.290.61410.790.78521.290.90151.790.96332.700.9965
0.300.61790.800.78811.300.90321.800.96412.750.9970
0.310.62170.810.79101.310.90491.810.96492.800.9974
0.320.62550.820.79391.320.90661.820.96562.850.9978
0.330.62930.830.79671.330.90821.830.96642.900.9981
0.340.63310.840.79951.340.90991.840.96712.950.9984
0.350.63680.850.80231.350.91151.850.96783.000.9987
0.360.64060.860.80511.360.91311.860.96863.050.9989
0.370.64430.870.80781.370.91471.870.96933.100.9990
0.380.64800.880.81061.380.91621.880.96993.150.9992
0.390.65170.890.81331.390.91771.890.97063.200.9993
0.400.65540.900.81591.400.91921.900.97133.250.9994
0.410.65910.910.81861.410.92071.910.97193.300.9995
0.420.66280.920.82121.420.92221.920.97263.350.9996
0.430.66640.930.82381.430.92361.930.97323.400.9997
0.440.67000.940.82641.440.92511.940.97383.450.9998
0.450.67360.950.82891.450.92651.950.97443.500.9998
0.460.67720.960.83151.460.92791.960.97503.600.9998
0.470.68080.970.83401.470.92921.970.97563.700.9999
0.480.68440.980.83651.480.93061.980.97613.800.9999
0.490.68790.990.83891.490.93191.990.97673.901.0000
1.000.84131.500.93322.000.97724.001.0000

Percentage Points of the Normal Distribution

Values $z$ such that $P(Z>z)=1-\Phi(z)=p$ where $Z\sim N(0,1)$.

$p$$z$$p$$z$
0.50000.00000.05001.6449
0.40000.25330.02501.9600
0.30000.52440.01002.3263
0.20000.84160.00502.5758
0.15001.03640.00103.0902
0.10001.28160.00053.2905

S2 Note

Candidates sitting S2 may also require those formulae listed under Statistics S1, and also those listed under Pure Mathematics P1, P2, P3 and P4.

Standard Discrete Distributions (S2)

Distribution$P(X=x)$MeanVariance
Binomial $B(n,p)$$\dbinom{n}{x}p^x(1-p)^{n-x}$$np$$np(1-p)$
Poisson $\mathrm{Po}(\lambda)$$e^{-\lambda}\dfrac{\lambda^x}{x!}$$\lambda$$\lambda$

Continuous Random Variables (S2)

Expectation (mean)$$E(X) = \mu = \int x\,f(x)\,dx$$
Variance$$\text{Var}(X) = \sigma^2 = \int (x-\mu)^2\,f(x)\,dx = \int x^2\,f(x)\,dx - \mu^2$$
For a function $g(X)$$$E\bigl(g(X)\bigr) = \int g(x)\,f(x)\,dx$$
Cumulative distribution function$$F(x_0) = P(X\le x_0) = \int_{-\infty}^{x_0} f(t)\,dt$$

Standard Continuous Distributions (S2)

DistributionP.D.F.MeanVariance
Uniform on $[a,b]$$\dfrac{1}{b-a}$$\dfrac{1}{2}(a+b)$$\dfrac{1}{12}(b-a)^2$

Binomial Cumulative Distribution Function

Tabulated values are $P(X\le x)$ where $X$ has a binomial distribution with index $n$ and parameter $p$.

$n=5$

$x$$p=0.05$$0.10$$0.15$$0.20$$0.25$$0.30$$0.35$$0.40$$0.45$$0.50$
00.77380.59050.44370.32770.23730.16810.11600.07780.05030.0312
10.97740.91850.83520.73730.63280.52820.42840.33700.25620.1875
20.99880.99140.97340.94210.89650.83690.76480.68260.59310.5000
31.00000.99950.99780.99330.98440.96920.94600.91300.86880.8125
41.00001.00000.99990.99970.99900.99760.99470.98980.98150.9688

$n=6$

$x$$p=0.05$$0.10$$0.15$$0.20$$0.25$$0.30$$0.35$$0.40$$0.45$$0.50$
00.73510.53140.37710.26210.17800.11760.07540.04670.02770.0156
10.96720.88570.77650.65540.53390.42020.31910.23330.16360.1094
20.99780.98420.95270.90110.83060.74430.64710.54430.44150.3438
30.99990.99870.99410.98300.96240.92950.88260.82080.74470.6563
41.00000.99990.99960.99840.99540.98910.97770.95900.93080.8906
51.00001.00001.00000.99990.99980.99930.99820.99590.99170.9844

$n=7$

$x$$p=0.05$$0.10$$0.15$$0.20$$0.25$$0.30$$0.35$$0.40$$0.45$$0.50$
00.69830.47830.32060.20970.13350.08240.04900.02800.01520.0078
10.95560.85030.71660.57670.44490.32940.23380.15860.10240.0625
20.99620.97430.92620.85200.75640.64710.53230.41990.31640.2266
30.99980.99730.98790.96670.92940.87400.80020.71020.60830.5000
41.00000.99980.99880.99530.98710.97120.94440.90370.84710.7734
51.00001.00000.99990.99960.99870.99620.99100.98120.96430.9375
61.00001.00001.00001.00000.99990.99980.99940.99840.99630.9922

$n=8$

$x$$p=0.05$$0.10$$0.15$$0.20$$0.25$$0.30$$0.35$$0.40$$0.45$$0.50$
00.66340.43050.27250.16780.10010.05760.03190.01680.00840.0039
10.94280.81310.65720.50330.36710.25530.16910.10640.06320.0352
20.99420.96190.89480.79690.67850.55180.42780.31540.22010.1445
30.99960.99500.97860.94370.88620.80590.70640.59410.47700.3633
41.00000.99960.99710.98960.97270.94200.89390.82630.73960.6367
51.00001.00000.99980.99880.99580.98870.97470.95020.91150.8555
61.00001.00001.00000.99990.99960.99870.99640.99150.98190.9648
71.00001.00001.00001.00001.00000.99990.99980.99930.99830.9961

$n=9$

$x$$p=0.05$$0.10$$0.15$$0.20$$0.25$$0.30$$0.35$$0.40$$0.45$$0.50$
00.63020.38740.23160.13420.07510.04040.02070.01010.00460.0020
10.92880.77480.59950.43620.30030.19600.12110.07050.03850.0195
20.99160.94700.85910.73820.60070.46280.33730.23180.14950.0898
30.99940.99170.96610.91440.83430.72970.60890.48260.36140.2539
41.00000.99910.99440.98040.95110.90120.82830.73340.62140.5000
51.00000.99990.99940.99690.99000.97470.94640.90060.83420.7461
61.00001.00001.00000.99970.99870.99570.98880.97500.95020.9102
71.00001.00001.00001.00000.99990.99960.99860.99620.99090.9805
81.00001.00001.00001.00001.00001.00000.99990.99970.99920.9980

$n=10$

$x$$p=0.05$$0.10$$0.15$$0.20$$0.25$$0.30$$0.35$$0.40$$0.45$$0.50$
00.59870.34870.19690.10740.05630.02820.01350.00600.00250.0010
10.91390.73610.54430.37580.24400.14930.08600.04640.02330.0107
20.98850.92980.82020.67780.52560.38280.26160.16730.09960.0547
30.99900.98720.95000.87910.77590.64960.51380.38230.26600.1719
40.99990.99840.99010.96720.92190.84970.75150.63310.50440.3770
51.00000.99990.99860.99360.98030.95270.90510.83380.73840.6230
61.00001.00000.99990.99910.99650.98940.97400.94520.89800.8281
71.00001.00001.00000.99990.99960.99840.99520.98770.97260.9453
81.00001.00001.00001.00001.00000.99990.99950.99830.99550.9893
91.00001.00001.00001.00001.00001.00001.00000.99990.99970.9990

$n=12$

$x$$p=0.05$$0.10$$0.15$$0.20$$0.25$$0.30$$0.35$$0.40$$0.45$$0.50$
00.54040.28240.14220.06870.03170.01380.00570.00220.00080.0002
10.88160.65900.44350.27490.15840.08500.04240.01960.00830.0032
20.98040.88910.73580.55830.39070.25280.15130.08340.04210.0193
30.99780.97440.90780.79460.64880.49250.34670.22530.13450.0730
40.99980.99570.97610.92740.84240.72370.58330.43820.30440.1938
51.00000.99950.99540.98060.94560.88220.78730.66520.52690.3872
61.00000.99990.99930.99610.98570.96140.91540.84180.73930.6128
71.00001.00000.99990.99940.99720.99050.97450.94270.88830.8062
81.00001.00001.00000.99990.99960.99830.99440.98470.96440.9270
91.00001.00001.00001.00001.00000.99980.99920.99720.99210.9807
101.00001.00001.00001.00001.00001.00000.99990.99970.99890.9968
111.00001.00001.00001.00001.00001.00001.00001.00000.99990.9998

$n=15$

$x$$p=0.05$$0.10$$0.15$$0.20$$0.25$$0.30$$0.35$$0.40$$0.45$$0.50$
00.46330.20590.08740.03520.01340.00470.00160.00050.00010.0000
10.82900.54900.31860.16710.08020.03530.01420.00520.00170.0005
20.96380.81590.60420.39800.23610.12680.06170.02710.01070.0037
30.99450.94440.82270.64820.46130.29690.17270.09050.04240.0176
40.99940.98730.93830.83580.68650.51550.35190.21730.12040.0592
50.99990.99780.98320.93890.85160.72160.56430.40320.26080.1509
61.00000.99970.99640.98190.94340.86890.75480.60980.45220.3036
71.00001.00000.99940.99580.98270.95000.88680.78690.65350.5000
81.00001.00000.99990.99920.99580.98480.95780.90500.81820.6964
91.00001.00001.00000.99990.99920.99630.98760.96620.92310.8491
101.00001.00001.00001.00000.99990.99930.99720.99070.97450.9408
111.00001.00001.00001.00001.00000.99990.99950.99810.99370.9824
121.00001.00001.00001.00001.00001.00000.99990.99970.99890.9963
131.00001.00001.00001.00001.00001.00001.00001.00000.99990.9995
141.00001.00001.00001.00001.00001.00001.00001.00001.00001.0000

$n=20$

$x$$p=0.05$$0.10$$0.15$$0.20$$0.25$$0.30$$0.35$$0.40$$0.45$$0.50$
00.35850.12160.03880.01150.00320.00080.00020.00000.00000.0000
10.73580.39170.17560.06920.02430.00760.00210.00050.00010.0000
20.92450.67690.40490.20610.09130.03550.01210.00360.00090.0002
30.98410.86700.64770.41140.22520.10710.04440.01600.00490.0013
40.99740.95680.82980.62960.41480.23750.11820.05100.01890.0059
50.99970.98870.93270.80420.61720.41640.24540.12560.05530.0207
61.00000.99760.97810.91330.78580.60800.41660.25000.12990.0577
71.00000.99960.99410.96790.89820.77230.60100.41590.25200.1316
81.00000.99990.99870.99000.95910.88670.76240.59560.41430.2517
91.00001.00000.99980.99740.98610.95200.87820.75530.59140.4119
101.00001.00001.00000.99940.99610.98290.94680.87250.75070.5881
111.00001.00001.00000.99990.99910.99490.98040.94350.86920.7483
121.00001.00001.00001.00000.99980.99870.99400.97900.94200.8684
131.00001.00001.00001.00001.00000.99970.99850.99350.97860.9423
141.00001.00001.00001.00001.00001.00000.99970.99840.99360.9793
151.00001.00001.00001.00001.00001.00001.00000.99970.99850.9941
161.00001.00001.00001.00001.00001.00001.00001.00000.99970.9987
171.00001.00001.00001.00001.00001.00001.00001.00001.00000.9998
181.00001.00001.00001.00001.00001.00001.00001.00001.00001.0000

$n=25$

$x$$p=0.05$$0.10$$0.15$$0.20$$0.25$$0.30$$0.35$$0.40$$0.45$$0.50$
00.27740.07180.01720.00380.00080.00010.00000.00000.00000.0000
10.64240.27120.09310.02740.00700.00160.00030.00010.00000.0000
20.87290.53710.25370.09820.03210.00900.00210.00040.00010.0000
30.96590.76360.47110.23400.09620.03320.00970.00240.00050.0001
40.99280.90200.68210.42070.21370.09050.03200.00950.00230.0005
50.99880.96660.83850.61670.37830.19350.08260.02940.00860.0020
60.99980.99050.93050.78000.56110.34070.17340.07360.02580.0073
71.00000.99770.97450.89090.72650.51180.30610.15360.06390.0216
81.00000.99950.99200.95320.85060.67690.46680.27350.13400.0539
91.00000.99990.99790.98270.92870.81060.63030.42460.24240.1148
101.00001.00000.99950.99440.97030.90220.77120.58580.38430.2122
111.00001.00000.99990.99850.98930.95580.87460.73230.54260.3450
121.00001.00001.00000.99960.99660.98250.93960.84620.69370.5000
131.00001.00001.00000.99990.99910.99400.97450.92220.81730.6550
141.00001.00001.00001.00000.99980.99820.99070.96560.90400.7878
151.00001.00001.00001.00001.00000.99950.99710.98680.95600.8852
161.00001.00001.00001.00001.00000.99990.99920.99570.98260.9461
171.00001.00001.00001.00001.00001.00000.99980.99880.99420.9784
181.00001.00001.00001.00001.00001.00001.00000.99970.99840.9927
191.00001.00001.00001.00001.00001.00001.00000.99990.99960.9980
201.00001.00001.00001.00001.00001.00001.00001.00000.99990.9995

$n=30$

$x$$p=0.05$$0.10$$0.15$$0.20$$0.25$$0.30$$0.35$$0.40$$0.45$$0.50$
00.21460.04240.00760.00120.00020.00000.00000.00000.00000.0000
10.55350.18370.04800.01050.00200.00030.00000.00000.00000.0000
20.81220.41140.15140.04420.01060.00210.00030.00000.00000.0000
30.93920.64740.32170.12270.03740.00930.00190.00030.00000.0000
40.98440.82450.52450.25520.09790.03020.00750.00150.00020.0000
50.99670.92680.71060.42750.20260.07660.02330.00570.00110.0002
60.99940.97420.84740.60700.34810.15950.05860.01720.00400.0007
70.99990.99220.93020.76080.51430.28140.12380.04350.01210.0026
81.00000.99800.97220.87130.67360.43150.22470.09400.03120.0081
91.00000.99950.99030.93890.80340.58880.35750.17630.06940.0214
101.00000.99990.99710.97440.89430.73040.50780.29150.13500.0494
111.00001.00000.99920.99050.94930.84070.65480.43110.23270.1002
121.00001.00000.99980.99690.97840.91550.78020.57850.35920.1808
131.00001.00001.00000.99910.99180.95990.87370.71450.50250.2923
141.00001.00001.00000.99980.99730.98310.93480.82460.64480.4278
151.00001.00001.00000.99990.99920.99360.96990.90290.76910.5722
161.00001.00001.00001.00000.99980.99790.98760.95190.86440.7077
171.00001.00001.00001.00000.99990.99940.99550.97880.92860.8192
181.00001.00001.00001.00001.00000.99980.99860.99170.96660.8998
191.00001.00001.00001.00001.00001.00000.99960.99710.98620.9506
201.00001.00001.00001.00001.00001.00000.99990.99910.99550.9786
211.00001.00001.00001.00001.00001.00001.00000.99970.99850.9922
221.00001.00001.00001.00001.00001.00001.00000.99990.99960.9976
231.00001.00001.00001.00001.00001.00001.00001.00000.99990.9994
241.00001.00001.00001.00001.00001.00001.00001.00001.00000.9999
251.00001.00001.00001.00001.00001.00001.00001.00001.00001.0000

$n=40$

$x$$p=0.05$$0.10$$0.15$$0.20$$0.25$$0.30$$0.35$$0.40$$0.45$$0.50$
00.12850.01480.00150.00010.00000.00000.00000.00000.00000.0000
10.39910.08050.01210.00150.00010.00000.00000.00000.00000.0000
20.67670.22280.04860.00790.00100.00010.00000.00000.00000.0000
30.86190.42310.13020.02850.00470.00060.00010.00000.00000.0000
40.95200.62900.26330.07590.01600.00260.00030.00000.00000.0000
50.98610.79370.43250.16130.04330.00860.00130.00010.00000.0000
60.99660.90050.60670.28590.09620.02380.00440.00060.00010.0000
70.99930.95810.75590.43710.18200.05530.01240.00210.00020.0000
80.99990.98450.86460.59310.29980.11100.03030.00610.00090.0001
91.00000.99490.93280.73180.43950.19590.06440.01560.00270.0003
101.00000.99850.97010.83920.58390.30870.12150.03520.00740.0011
111.00000.99960.98800.91250.71510.44060.20530.07090.01790.0032
121.00000.99990.99570.95680.82090.57720.31430.12850.03860.0083
131.00001.00000.99860.98060.89860.70320.44080.21120.07510.0192
141.00001.00000.99960.99210.94560.80740.57210.31740.13260.0403
151.00001.00000.99990.99710.97380.88490.69460.44020.21420.0769
161.00001.00001.00000.99900.98880.93670.79780.56810.31850.1341
171.00001.00001.00000.99970.99530.96800.87610.68850.43910.2148
181.00001.00001.00000.99990.99830.98520.93010.79110.56510.3179
191.00001.00001.00001.00000.99940.99370.96370.87200.68440.4373
201.00001.00001.00001.00000.99980.99760.98270.92560.78700.5627
211.00001.00001.00001.00001.00000.99910.99250.96080.86690.6821
221.00001.00001.00001.00001.00000.99970.99700.98110.92330.7852
231.00001.00001.00001.00001.00000.99990.99890.99170.95950.8659
241.00001.00001.00001.00001.00001.00000.99960.99660.98040.9231
251.00001.00001.00001.00001.00001.00000.99990.99880.99140.9597
261.00001.00001.00001.00001.00001.00001.00000.99960.99660.9808
271.00001.00001.00001.00001.00001.00001.00000.99990.99880.9917
281.00001.00001.00001.00001.00001.00001.00001.00000.99960.9968
291.00001.00001.00001.00001.00001.00001.00001.00000.99990.9989
301.00001.00001.00001.00001.00001.00001.00001.00001.00000.9997
311.00001.00001.00001.00001.00001.00001.00001.00001.00000.9999
321.00001.00001.00001.00001.00001.00001.00001.00001.00001.0000

$n=50$

$x$$p=0.05$$0.10$$0.15$$0.20$$0.25$$0.30$$0.35$$0.40$$0.45$$0.50$
00.07690.00520.00030.00000.00000.00000.00000.00000.00000.0000
10.27940.03380.00290.00020.00000.00000.00000.00000.00000.0000
20.54050.11170.01420.00130.00010.00000.00000.00000.00000.0000
30.76040.25030.04600.00570.00050.00000.00000.00000.00000.0000
40.89640.43120.11210.01850.00210.00020.00000.00000.00000.0000
50.96220.61610.21940.04800.00700.00070.00010.00000.00000.0000
60.98820.77020.36130.10340.01940.00250.00020.00000.00000.0000
70.99680.87790.51880.19040.04530.00730.00080.00010.00000.0000
80.99920.94210.66810.30730.09160.01830.00250.00020.00000.0000
90.99980.97550.79110.44370.16370.04020.00670.00080.00010.0000
101.00000.99060.88010.58360.26220.07890.01600.00220.00020.0000
111.00000.99680.93720.71070.38160.13900.03420.00570.00060.0000
121.00000.99900.96990.81390.51100.22290.06610.01330.00180.0002
131.00000.99970.98680.88940.63700.32790.11630.02800.00450.0005
141.00000.99990.99470.93930.74810.44680.18780.05400.01040.0013
151.00001.00000.99810.96920.83690.56920.28010.09550.02200.0033
161.00001.00000.99930.98560.90170.68390.38890.15610.04270.0077
171.00001.00000.99980.99370.94490.78220.50600.23690.07650.0164
181.00001.00000.99990.99750.97130.85940.62160.33560.12730.0325
191.00001.00001.00000.99910.98610.91520.72640.44650.19740.0595
201.00001.00001.00000.99970.99370.95220.81390.56100.28620.1013
211.00001.00001.00000.99990.99740.97490.88130.67010.39000.1611
221.00001.00001.00001.00000.99900.98770.92900.76600.50190.2399
231.00001.00001.00001.00000.99960.99440.96040.84380.61340.3359
241.00001.00001.00001.00000.99990.99760.97930.90220.71600.4439
251.00001.00001.00001.00001.00000.99910.99000.94270.80340.5561
261.00001.00001.00001.00001.00000.99970.99550.96860.87210.6641
271.00001.00001.00001.00001.00000.99990.99810.98400.92200.7601
281.00001.00001.00001.00001.00001.00000.99930.99240.95560.8389
291.00001.00001.00001.00001.00001.00000.99970.99660.97650.8987
301.00001.00001.00001.00001.00001.00000.99990.99860.98840.9405
311.00001.00001.00001.00001.00001.00001.00000.99950.99470.9675
321.00001.00001.00001.00001.00001.00001.00000.99980.99780.9836
331.00001.00001.00001.00001.00001.00001.00000.99990.99910.9923
341.00001.00001.00001.00001.00001.00001.00001.00000.99970.9967
351.00001.00001.00001.00001.00001.00001.00001.00000.99990.9987
361.00001.00001.00001.00001.00001.00001.00001.00001.00000.9995
371.00001.00001.00001.00001.00001.00001.00001.00001.00000.9998
381.00001.00001.00001.00001.00001.00001.00001.00001.00001.0000

Poisson Cumulative Distribution Function

$P(X\le x)$ where $X$ has a Poisson distribution with parameter $\lambda$.

$\lambda = 0.5$ to $5.0$

$x$$\lambda=0.5$$1.0$$1.5$$2.0$$2.5$$3.0$$3.5$$4.0$$4.5$$5.0$
00.60650.36790.22310.13530.08210.04980.03020.01830.01110.0067
10.90980.73580.55780.40600.28730.19910.13590.09160.06110.0404
20.98560.91970.80880.67670.54380.42320.32080.23810.17360.1247
30.99820.98100.93440.85710.75760.64720.53660.43350.34230.2650
40.99980.99630.98140.94730.89120.81530.72540.62880.53210.4405
51.00000.99940.99550.98340.95800.91610.85760.78510.70290.6160
61.00000.99990.99910.99550.98580.96650.93470.88930.83110.7622
71.00001.00000.99980.99890.99580.98810.97330.94890.91340.8666
81.00001.00001.00000.99980.99890.99620.99010.97860.95970.9319
91.00001.00001.00001.00000.99970.99890.99670.99190.98290.9682
101.00001.00001.00001.00000.99990.99970.99900.99720.99330.9863
111.00001.00001.00001.00001.00000.99990.99970.99910.99760.9945
121.00001.00001.00001.00001.00001.00000.99990.99970.99920.9980
131.00001.00001.00001.00001.00001.00001.00000.99990.99970.9993
141.00001.00001.00001.00001.00001.00001.00001.00000.99990.9998
151.00001.00001.00001.00001.00001.00001.00001.00001.00000.9999

$\lambda = 5.5$ to $10.0$

$x$$\lambda=5.5$$6.0$$6.5$$7.0$$7.5$$8.0$$8.5$$9.0$$9.5$$10.0$
00.00410.00250.00150.00090.00060.00030.00020.00010.00010.0000
10.02660.01740.01130.00730.00470.00300.00190.00120.00080.0005
20.08840.06200.04300.02960.02030.01380.00930.00620.00420.0028
30.20170.15120.11180.08180.05910.04240.03010.02120.01490.0103
40.35750.28510.22370.17300.13210.09960.07440.05500.04030.0293
50.52890.44570.36900.30070.24140.19120.14960.11570.08850.0671
60.68600.60630.52650.44970.37820.31340.25620.20680.16490.1301
70.80950.74400.67280.59870.52460.45300.38560.32390.26870.2202
80.89440.84720.79160.72910.66200.59250.52310.45570.39180.3328
90.94620.91610.87740.83050.77640.71660.65300.58740.52180.4579
100.97470.95740.93320.90150.86220.81590.76340.70600.64530.5830
110.98900.97990.96610.94670.92080.88810.84870.80300.75200.6968
120.99550.99120.98400.97300.95730.93620.90910.87580.83640.7916
130.99830.99640.99290.98720.97840.96580.94860.92610.89810.8645
140.99940.99860.99700.99430.98970.98270.97260.95850.94000.9165
150.99980.99950.99880.99760.99540.99180.98620.97800.96650.9513
160.99990.99980.99960.99900.99800.99630.99340.98890.98230.9730
171.00000.99990.99980.99960.99920.99840.99700.99470.99110.9857
181.00001.00000.99990.99990.99970.99930.99870.99760.99570.9928
191.00001.00001.00001.00000.99990.99970.99950.99890.99800.9965
201.00001.00001.00001.00001.00000.99990.99980.99960.99910.9984
211.00001.00001.00001.00001.00001.00000.99990.99980.99960.9993
221.00001.00001.00001.00001.00001.00001.00000.99990.99990.9997

S3 Note

Candidates sitting S3 may also require those formulae listed under Statistics S1 and S2.

Expectation Algebra (S3)

For independent $X$ and $Y$$$E(XY) = E(X)\,E(Y) \qquad \text{Var}(aX\pm bY) = a^2\,\text{Var}(X)+b^2\,\text{Var}(Y)$$

Sampling Distributions (S3)

Unbiased estimators$$\bar{X}\text{ is unbiased for }\mu,\quad \text{Var}(\bar{X})=\dfrac{\sigma^2}{n}$$$$S^2\text{ is unbiased for }\sigma^2,\quad S^2=\dfrac{\sum(X_i-\bar{X})^2}{n-1}$$
For sample from $N(\mu,\sigma^2)$$$\dfrac{\bar{X}-\mu}{\sigma/\sqrt{n}}\sim N(0,1)$$
Two independent samples from $N(\mu_x,\sigma_x^2)$ and $N(\mu_y,\sigma_y^2)$$$\dfrac{(\bar{X}-\bar{Y})-(\mu_x-\mu_y)}{\sqrt{\dfrac{\sigma_x^2}{n_x}+\dfrac{\sigma_y^2}{n_y}}}$$

Spearman's Rank (S3)

Spearman's rank correlation coefficient$$r_s = 1 - \dfrac{6\sum d^2}{n(n^2-1)}$$

Non-parametric Tests (S3)

Goodness-of-fit and contingency tables$$\sum\dfrac{(O_i-E_i)^2}{E_i}\sim\chi^2_\nu$$

Percentage Points of the $\chi^2$ Distribution Function

Values which a random variable with the $\chi^2$ distribution on $\nu$ degrees of freedom exceeds with the probability shown.

$\nu$0.9950.9900.9750.9500.9000.1000.0500.0250.0100.005
10.0000.0000.0010.0040.0162.7053.8415.0246.6357.879
20.0100.0200.0510.1030.2114.6055.9917.3789.21010.597
30.0720.1150.2160.3520.5846.2517.8159.34811.34512.838
40.2070.2970.4840.7111.0647.7799.48811.14313.27714.860
50.4120.5540.8311.1451.6109.23611.07012.83215.08616.750
60.6760.8721.2371.6352.20410.64512.59214.44916.81218.548
70.9891.2391.6902.1672.83312.01714.06716.01318.47520.278
81.3441.6462.1802.7333.49013.36215.50717.53520.09021.955
91.7352.0882.7003.3254.16814.68416.91919.02321.66623.589
102.1562.5583.2473.9404.86515.98718.30720.48323.20925.188
112.6033.0533.8164.5755.58017.27519.67521.92024.72526.757
123.0743.5714.4045.2266.30418.54921.02623.33726.21728.300
133.5654.1075.0095.8927.04219.81222.36224.73627.68829.819
144.0754.6605.6296.5717.79021.06423.68526.11929.14131.319
154.6015.2296.2627.2618.54722.30724.99627.48830.57832.801
165.1425.8126.9087.9629.31223.54226.29628.84532.00034.267
175.6976.4087.5648.67210.08524.76927.58730.19133.40935.718
186.2657.0158.2319.39010.86525.98928.86931.52634.80537.156
196.8447.6338.90710.11711.65127.20430.14432.85236.19138.582
207.4348.2609.59110.85112.44328.41231.41034.17037.56639.997
218.0348.89710.28311.59113.24029.61532.67135.47938.93241.401
228.6439.54210.98212.33814.04230.81333.92436.78140.28942.796
239.26010.19611.68913.09114.84832.00735.17238.07641.63844.181
249.88610.85612.40113.84815.65933.19636.41539.36442.98045.558
2510.52011.52413.12014.61116.47334.38237.65240.64644.31446.928
2611.16012.19813.84415.37917.29235.56338.88541.92345.64248.290
2711.80812.87914.57316.15118.11436.74140.11343.19446.96349.645
2812.46113.56515.30816.92818.93937.91641.33744.46148.27850.993
2913.12114.25616.04717.70819.76839.08842.55745.72249.58852.336
3013.78714.95316.79118.49320.59940.25643.77346.97950.89253.672

Critical Values for Correlation Coefficients

Minimum values needed to be significant at the level shown (one-tailed test).

Product Moment CoefficientSample
Level		Spearman's Coefficient
0.100.050.0250.010.0050.050.0250.01
0.80000.90000.95000.98000.990041.0000——
0.68700.80540.87830.93430.958750.90001.00001.0000
0.60840.72930.81140.88220.917260.82860.88570.9429
0.55090.66940.75450.83290.874570.71430.78570.8929
0.50670.62150.70670.78870.834380.64290.73810.8333
0.47160.58220.66640.74980.797790.60000.70000.7833
0.44280.54940.63190.71550.7646100.56360.64850.7455
0.41870.52140.60210.68510.7348110.53640.61820.7091
0.39810.49730.57600.65810.7079120.50350.58740.6783
0.38020.47620.55290.63390.6835130.48350.56040.6484
0.36460.45750.53240.61200.6614140.46370.53850.6264
0.35070.44090.51400.59230.6411150.44640.52140.6036
0.33830.42590.49730.57420.6226160.42940.50290.5824
0.32710.41240.48210.55770.6055170.41420.48770.5662
0.31700.40000.46830.54250.5897180.40140.47160.5501
0.30770.38870.45550.52850.5751190.39120.45960.5351
0.29920.37830.44380.51550.5614200.38050.44660.5218
0.29140.36870.43290.50340.5487210.37010.43640.5091
0.28410.35980.42270.49210.5368220.36080.42520.4975
0.27740.35150.41330.48150.5256230.35280.41600.4862
0.27110.34380.40440.47160.5151240.34430.40700.4757
0.26530.33650.39610.46220.5052250.33690.39770.4662
0.25980.32970.38820.45340.4958260.33060.39010.4571
0.25460.32330.38090.44510.4869270.32420.38280.4487
0.24970.31720.37390.43720.4785280.31800.37550.4401
0.24510.31150.36730.42970.4705290.31180.36850.4325
0.24070.30610.36100.42260.4629300.30630.36240.4251
0.20700.26380.31200.36650.4026400.26400.31280.3681
0.18430.23530.27870.32810.3610500.23530.27910.3293
0.16780.21440.25420.29970.3301600.21440.25450.3005
0.15500.19820.23520.27760.3060700.19820.23540.2782
0.14480.18520.21990.25970.2864800.18520.22010.2602
0.13640.17450.20720.24490.2702900.17450.20740.2453
0.12920.16540.19660.23240.25651000.16540.19670.2327
Formulas reproduced from the official Pearson Edexcel IAL Mathematics & Further Mathematics Formula Books (2018 spec). Pearson and Edexcel are trademarks of Pearson Education Limited. WhateverGo is an independent study tool with no affiliation.