Mathematics (9709) Formula Sheet

Pure Mathematics

Mensuration

Volume of sphere = $\frac{4}{3}\pi r^3$

Surface area of sphere = $4\pi r^2$

Volume of cone or pyramid = $\frac{1}{3} \times \text{base area} \times \text{height}$

Area of curved surface of cone = $\pi r \times \text{slant height}$

Arc length of circle = $r\theta$ ( $\theta$ in radians)

Area of sector of circle = $\frac{1}{2}r^2\theta$ ( $\theta$ in radians)

Algebra

For the quadratic equation $ax^2 + bx + c = 0$ :

\displaystyle x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

For an arithmetic series:

u_n = a + (n-1)d,

\displaystyle S_n = \frac{1}{2}n(a+l) = \frac{1}{2}n\{2a + (n-1)d\}

For a geometric series:

u_n = ar^{n-1},

\displaystyle S_n = \frac{a(1-r^n)}{1-r} \quad (r \ne 1),

\displaystyle S_{\infty} = \frac{a}{1-r} \quad (|r| < 1)

Binomial series:

$\displaystyle (a+b)^n = a^n + \binom{n}{1} a^{n-1}b + \binom{n}{2} a^{n-2}b^2 + \binom{n}{3} a^{n-3}b^3 + \dots + b^n$ , where $n$ is a positive integer

and $\displaystyle \binom{n}{r} = \frac{n!}{r!(n-r)!}$

$\displaystyle (1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \dots$ , where $n$ is rational and $|x| < 1$

Trigonometry

\displaystyle \tan \theta = \frac{\sin \theta}{\cos \theta}

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

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

\cot^2 \theta + 1 = \mathrm{cosec}^2 \theta

\sin(A \pm B) \equiv \sin A \cos B \pm \cos A \sin B

\cos(A \pm B) \equiv \cos A \cos B \mp \sin A \sin B

\displaystyle \tan(A \pm B) \equiv \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}

\sin 2A \equiv 2 \sin A \cos A

\cos 2A \equiv \cos^2 A - \sin^2 A \equiv 2\cos^2 A - 1 \equiv 1 - 2\sin^2 A

\displaystyle \tan 2A \equiv \frac{2\tan A}{1 - \tan^2 A}

Principal values:

\displaystyle -\frac{1}{2}\pi \leqslant \sin^{-1} x \leqslant \frac{1}{2}\pi,

\displaystyle 0 \leqslant \cos^{-1} x \leqslant \pi,

\displaystyle -\frac{1}{2}\pi < \tan^{-1} x < \frac{1}{2}\pi

Differentiation

\mathbf{f}(x)

\mathbf{f}'(x)

x^n

nx^{n-1}

\ln x

\displaystyle \frac{1}{x}

\mathrm{e}^x

\mathrm{e}^x

\sin x

\cos x

\cos x

-\sin x

\tan x

\sec^2 x

\sec x

\sec x \tan x

\mathrm{cosec} \, x

-\mathrm{cosec} \, x \cot x

\cot x

-\mathrm{cosec}^2 \, x

\tan^{-1} x

\displaystyle \frac{1}{1+x^2}

uv

\displaystyle v\frac{\mathrm{d}u}{\mathrm{d}x} + u\frac{\mathrm{d}v}{\mathrm{d}x}

\displaystyle \frac{u}{v}

\displaystyle \frac{v\frac{\mathrm{d}u}{\mathrm{d}x} - u\frac{\mathrm{d}v}{\mathrm{d}x}}{v^2}

If $x = \mathrm{f}(t)$ and $y = \mathrm{g}(t)$ then $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{\mathrm{d}y}{\mathrm{d}t} \div \frac{\mathrm{d}x}{\mathrm{d}t}$

Integration

(Arbitrary constants are omitted; $a$ denotes a positive constant.)

\mathbf{f}(x)

\int \mathbf{f}(x) \, \mathrm{d}x

x^n

\displaystyle \frac{x^{n+1}}{n+1}

(n \neq -1)

\displaystyle \frac{1}{x}

\ln |x|

\mathrm{e}^x

\mathrm{e}^x

\sin x

-\cos x

\cos x

\sin x

\sec^2 x

\tan x

\displaystyle \frac{1}{x^2 + a^2}

\displaystyle \frac{1}{a} \tan^{-1} \left( \frac{x}{a} \right)

\displaystyle \frac{1}{x^2 - a^2}

\displaystyle \frac{1}{2a} \ln \left| \frac{x-a}{x+a} \right|

(x > a)

\displaystyle \frac{1}{a^2 - x^2}

\displaystyle \frac{1}{2a} \ln \left| \frac{a+x}{a-x} \right|

( |x| < a )

\displaystyle \int u \frac{\mathrm{d}v}{\mathrm{d}x} \, \mathrm{d}x = uv - \int v \frac{\mathrm{d}u}{\mathrm{d}x} \, \mathrm{d}x

\displaystyle \int \frac{f'(x)}{f(x)} \, \mathrm{d}x = \ln |f(x)|

Vectors

If $\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}$ and $\mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k}$ then

\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 = |\mathbf{a}| |\mathbf{b}| \cos \theta

Further Pure Mathematics

Algebra

Summations:

\displaystyle \sum_{r=1}^n r = \frac{1}{2}n(n+1),

\displaystyle \sum_{r=1}^n r^2 = \frac{1}{6}n(n+1)(2n+1),

\displaystyle \sum_{r=1}^n r^3 = \frac{1}{4}n^2(n+1)^2

Maclaurin’s series:

\displaystyle f(x) = f(0) + x f'(0) + \frac{x^2}{2!} f''(0) + \dots + \frac{x^r}{r!} f^{(r)}(0) + \dots

\displaystyle \mathrm{e}^x = \exp(x) = 1 + x + \frac{x^2}{2!} + \dots + \frac{x^r}{r!} + \dots

(\text{all } x)

\displaystyle \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots + (-1)^{r+1} \frac{x^r}{r} + \dots

(-1 < x \leqslant 1)

\displaystyle \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots + (-1)^r \frac{x^{2r+1}}{(2r+1)!} + \dots

(\text{all } x)

\displaystyle \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots + (-1)^r \frac{x^{2r}}{(2r)!} + \dots

(\text{all } x)

\displaystyle \tan^{-1} x = x - \frac{x^3}{3} + \frac{x^5}{5} - \dots + (-1)^r \frac{x^{2r+1}}{2r+1} + \dots

(-1 \leqslant x \leqslant 1)

\displaystyle \sinh x = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \dots + \frac{x^{2r+1}}{(2r+1)!} + \dots

(\text{all } x)

\displaystyle \cosh x = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \dots + \frac{x^{2r}}{(2r)!} + \dots

(\text{all } x)

\displaystyle \tanh^{-1} x = x + \frac{x^3}{3} + \frac{x^5}{5} + \dots + \frac{x^{2r+1}}{2r+1} + \dots

(-1 < x < 1)

Trigonometry

\displaystyle t = \tan \frac{1}{2} x

then:

\displaystyle \sin x = \frac{2t}{1+t^2}

and

\displaystyle \cos x = \frac{1-t^2}{1+t^2}

Hyperbolic functions

\cosh^2 x - \sinh^2 x = 1,

\sinh 2x \equiv 2\sinh x \cosh x,

\cosh 2x \equiv \cosh^2 x + \sinh^2 x

\sinh^{-1} x = \ln(x + \sqrt{x^2 + 1})

\cosh^{-1} x = \ln(x + \sqrt{x^2 - 1})

(x \geqslant 1)

\displaystyle \tanh^{-1} x = \frac{1}{2}\ln \left( \frac{1+x}{1-x} \right)

( |x| < 1 )

Differentiation

\mathbf{f}(x)

\mathbf{f}'(x)

\sin^{-1} x

\displaystyle \frac{1}{\sqrt{1-x^2}}

\cos^{-1} x

\displaystyle -\frac{1}{\sqrt{1-x^2}}

\sinh x

\cosh x

\cosh x

\sinh x

\tanh x

\mathrm{sech}^2 \, x

\sinh^{-1} x

\displaystyle \frac{1}{\sqrt{1+x^2}}

\cosh^{-1} x

\displaystyle \frac{1}{\sqrt{x^2-1}}

\tanh^{-1} x

\displaystyle \frac{1}{1-x^2}

Integration

(Arbitrary constants are omitted; $a$ denotes a positive constant.)

\mathbf{f}(x)

\int \mathbf{f}(x) \, \mathrm{d}x

\sec x

\ln|\sec x + \tan x| = \ln\left|\tan\left(\frac{1}{2}x + \frac{1}{4}\pi\right)\right|

\displaystyle \left( |x| < \frac{1}{2}\pi \right)

\mathrm{cosec} \, x

-\ln|\mathrm{cosec} \, x + \cot x| = \ln\left|\tan\left(\frac{1}{2}x\right)\right|

(0 < x < \pi)

\sinh x

\cosh x

\cosh x

\sinh x

\mathrm{sech}^2 \, x

\tanh x

\displaystyle \frac{1}{\sqrt{a^2 - x^2}}

\displaystyle \sin^{-1} \left( \frac{x}{a} \right)

( |x| < a )

\displaystyle \frac{1}{\sqrt{x^2 - a^2}}

\displaystyle \cosh^{-1} \left( \frac{x}{a} \right)

( x > a )

\displaystyle \frac{1}{\sqrt{a^2 + x^2}}

\displaystyle \sinh^{-1} \left( \frac{x}{a} \right)

Mechanics

Uniformly accelerated motion

v = u + at,

\displaystyle s = \frac{1}{2}(u+v)t,

\displaystyle s = ut + \frac{1}{2}at^2,

v^2 = u^2 + 2as

Further Mechanics

Motion of a projectile

Equation of trajectory is:

\displaystyle y = x \tan \theta - \frac{gx^2}{2V^2 \cos^2 \theta}

Elastic strings and springs

\displaystyle T = \frac{\lambda x}{l},

\displaystyle E = \frac{\lambda x^2}{2l}

Motion in a circle

For uniform circular motion, the acceleration is directed towards the centre and has magnitude

\omega^2 r

\displaystyle \frac{v^2}{r}

Centres of mass of uniform bodies

Triangular lamina: $\frac{2}{3}$ along median from vertex

Solid hemisphere of radius $r$ : $\frac{3}{8}r$ from centre

Hemispherical shell of radius $r$ : $\frac{1}{2}r$ from centre

Circular arc of radius $r$ and angle $2\alpha$ : $\displaystyle \frac{r \sin \alpha}{\alpha}$ from centre

Circular sector of radius $r$ and angle $2\alpha$ : $\displaystyle \frac{2r \sin \alpha}{3\alpha}$ from centre

Solid cone or pyramid of height $h$ : $\frac{3}{4}h$ from vertex

Probability & Statistics

Summary statistics

For ungrouped data:

\displaystyle \bar{x} = \frac{\Sigma x}{n},

standard deviation =

\displaystyle \sqrt{\frac{\Sigma (x - \bar{x})^2}{n}} = \sqrt{\frac{\Sigma x^2}{n} - \bar{x}^2}

For grouped data:

\displaystyle \bar{x} = \frac{\Sigma xf}{\Sigma f},

standard deviation =

\displaystyle \sqrt{\frac{\Sigma (x - \bar{x})^2 f}{\Sigma f}} = \sqrt{\frac{\Sigma x^2 f}{\Sigma f} - \bar{x}^2}

Discrete random variables

\mathrm{E}(X) = \Sigma xp ,

\mathrm{Var}(X) = \Sigma x^2 p - \{ \mathrm{E}(X) \}^2

For the binomial distribution $\mathrm{B}(n, p)$ :

\displaystyle p_r = \binom{n}{r} p^r (1-p)^{n-r} ,

\mu = np ,

\sigma^2 = np(1-p)

For the geometric distribution $\mathrm{Geo}(p)$ :

p_r = p(1-p)^{r-1},

\displaystyle \mu = \frac{1}{p}

For the Poisson distribution $\mathrm{Po}(\lambda)$ :

\displaystyle p_r = \mathrm{e}^{-\lambda} \frac{\lambda^r}{r!} ,

\mu = \lambda ,

\sigma^2 = \lambda

Continuous random variables

\displaystyle \mathrm{E}(X) = \int x f(x) \, \mathrm{d}x ,

\displaystyle \mathrm{Var}(X) = \int x^2 f(x) \, \mathrm{d}x - \{ \mathrm{E}(X) \}^2

Sampling and testing

Unbiased estimators:

\displaystyle \bar{x} = \frac{\Sigma x}{n} ,

\displaystyle s^2 = \frac{\Sigma (x - \bar{x})^2}{n-1} = \frac{1}{n-1} \left( \Sigma x^2 - \frac{(\Sigma x)^2}{n} \right)

Central Limit Theorem:

\displaystyle \bar{X} \sim \mathrm{N} \left( \mu, \frac{\sigma^2}{n} \right)

Approximate distribution of sample proportion:

\displaystyle \mathrm{N} \left( p, \frac{p(1-p)}{n} \right)

Further Probability & Statistics

Sampling and testing

Two-sample estimate of a common variance:

\displaystyle s^2 = \frac{\Sigma(x_1 - \bar{x}_1)^2 + \Sigma(x_2 - \bar{x}_2)^2}{n_1 + n_2 - 2}

Probability generating functions

\mathrm{G}_X(t) = \mathrm{E}(t^X) ,

\mathrm{E}(X) = \mathrm{G}'_X(1) ,

\mathrm{Var}(X) = \mathrm{G}''_X(1) + \mathrm{G}'_X(1) - \{ \mathrm{G}'_X(1) \}^2

The Normal Distribution Function

If Z has a normal distribution with mean 0 and variance 1, then, for each value of z, the table gives the value of Φ(z), where

Φ(z) = P(Z ≤ z).

For negative values of z, use Φ(−z) = 1 − Φ(z).

z	0	1	2	3	4	5	6	7	8	9	1	2	3	4	5	6	7	8	9
z	0	1	2	3	4	5	6	7	8	9	ADD
0.0	0.5000	0.5040	0.5080	0.5120	0.5160	0.5199	0.5239	0.5279	0.5319	0.5359	4	8	12	16	20	24	28	32	36
0.1	0.5398	0.5438	0.5478	0.5517	0.5557	0.5596	0.5636	0.5675	0.5714	0.5753	4	8	12	16	20	24	28	32	36
0.2	0.5793	0.5832	0.5871	0.5910	0.5948	0.5987	0.6026	0.6064	0.6103	0.6141	4	8	12	15	19	23	27	31	35
0.3	0.6179	0.6217	0.6255	0.6293	0.6331	0.6368	0.6406	0.6443	0.6480	0.6517	4	7	11	15	19	22	26	30	34
0.4	0.6554	0.6591	0.6628	0.6664	0.6700	0.6736	0.6772	0.6808	0.6844	0.6879	4	7	11	14	18	22	25	29	32
0.5	0.6915	0.6950	0.6985	0.7019	0.7054	0.7088	0.7123	0.7157	0.7190	0.7224	3	7	10	14	17	20	24	27	31
0.6	0.7257	0.7291	0.7324	0.7357	0.7389	0.7422	0.7454	0.7486	0.7517	0.7549	3	7	10	13	16	19	23	26	29
0.7	0.7580	0.7611	0.7642	0.7673	0.7704	0.7734	0.7764	0.7794	0.7823	0.7852	3	6	9	12	15	18	21	24	27
0.8	0.7881	0.7910	0.7939	0.7967	0.7995	0.8023	0.8051	0.8078	0.8106	0.8133	3	5	8	11	14	16	19	22	25
0.9	0.8159	0.8186	0.8212	0.8238	0.8264	0.8289	0.8315	0.8340	0.8365	0.8389	3	5	8	10	13	15	18	20	23
1.0	0.8413	0.8438	0.8461	0.8485	0.8508	0.8531	0.8554	0.8577	0.8599	0.8621	2	5	7	9	12	14	16	19	21
1.1	0.8643	0.8665	0.8686	0.8708	0.8729	0.8749	0.8770	0.8790	0.8810	0.8830	2	4	6	8	10	12	14	16	18
1.2	0.8849	0.8869	0.8888	0.8907	0.8925	0.8944	0.8962	0.8980	0.8997	0.9015	2	4	6	7	9	11	13	15	17
1.3	0.9032	0.9049	0.9066	0.9082	0.9099	0.9115	0.9131	0.9147	0.9162	0.9177	2	3	5	6	8	10	11	13	14
1.4	0.9192	0.9207	0.9222	0.9236	0.9251	0.9265	0.9279	0.9292	0.9306	0.9319	1	3	4	6	7	8	10	11	13
1.5	0.9332	0.9345	0.9357	0.9370	0.9382	0.9394	0.9406	0.9418	0.9429	0.9441	1	2	4	5	6	7	8	10	11
1.6	0.9452	0.9463	0.9474	0.9484	0.9495	0.9505	0.9515	0.9525	0.9535	0.9545	1	2	3	4	5	6	7	8	9
1.7	0.9554	0.9564	0.9573	0.9582	0.9591	0.9599	0.9608	0.9616	0.9625	0.9633	1	2	3	4	4	5	6	7	8
1.8	0.9641	0.9649	0.9656	0.9664	0.9671	0.9678	0.9686	0.9693	0.9699	0.9706	1	1	2	3	4	4	5	6	6
1.9	0.9713	0.9719	0.9726	0.9732	0.9738	0.9744	0.9750	0.9756	0.9761	0.9767	1	1	2	2	3	4	4	5	5
2.0	0.9772	0.9778	0.9783	0.9788	0.9793	0.9798	0.9803	0.9808	0.9812	0.9817	0	1	1	2	2	3	3	4	4
2.1	0.9821	0.9826	0.9830	0.9834	0.9838	0.9842	0.9846	0.9850	0.9854	0.9857	0	1	1	2	2	2	3	3	4
2.2	0.9861	0.9864	0.9868	0.9871	0.9875	0.9878	0.9881	0.9884	0.9887	0.9890	0	1	1	1	2	2	2	3	3
2.3	0.9893	0.9896	0.9898	0.9901	0.9904	0.9906	0.9909	0.9911	0.9913	0.9916	0	1	1	1	1	2	2	2	2
2.4	0.9918	0.9920	0.9922	0.9925	0.9927	0.9929	0.9931	0.9932	0.9934	0.9936	0	0	1	1	1	1	1	2	2
2.5	0.9938	0.9940	0.9941	0.9943	0.9945	0.9946	0.9948	0.9949	0.9951	0.9952	0	0	0	1	1	1	1	1	1
2.6	0.9953	0.9955	0.9956	0.9957	0.9959	0.9960	0.9961	0.9962	0.9963	0.9964	0	0	0	0	1	1	1	1	1
2.7	0.9965	0.9966	0.9967	0.9968	0.9969	0.9970	0.9971	0.9972	0.9973	0.9974	0	0	0	0	0	1	1	1	1
2.8	0.9974	0.9975	0.9976	0.9977	0.9977	0.9978	0.9979	0.9979	0.9980	0.9981	0	0	0	0	0	0	0	1	1
2.9	0.9981	0.9982	0.9982	0.9983	0.9984	0.9984	0.9985	0.9985	0.9986	0.9986	0	0	0	0	0	0	0	0	0

Critical values for the normal distribution

If Z has a normal distribution with mean 0 and variance 1, then, for each value of p, the table gives the value of z such that

P(Z ≤ z) = p.

p	0.75	0.90	0.95	0.975	0.99	0.995	0.9975	0.999	0.9995
z	0.674	1.282	1.645	1.960	2.326	2.576	2.807	3.090	3.291

Critical Values for the t-Distribution

If T has a t-distribution with v degrees of freedom, then, for each pair of values of p and v, the table gives the value of t such that:

P(T ≤ t) = p.

p	0.75	0.90	0.95	0.975	0.99	0.995	0.9975	0.999	0.9995
v= 1	1.000	3.078	6.314	12.71	31.82	63.66	127.3	318.3	636.6
2	0.816	1.886	2.920	4.303	6.965	9.925	14.09	22.33	31.60
3	0.765	1.638	2.353	3.182	4.541	5.841	7.453	10.21	12.92
4	0.741	1.533	2.132	2.776	3.747	4.604	5.598	7.173	8.610
5	0.727	1.476	2.015	2.571	3.365	4.032	4.773	5.894	6.869
6	0.718	1.440	1.943	2.447	3.143	3.707	4.317	5.208	5.959
7	0.711	1.415	1.895	2.365	2.998	3.499	4.029	4.785	5.408
8	0.706	1.397	1.860	2.306	2.896	3.355	3.833	4.501	5.041
9	0.703	1.383	1.833	2.262	2.821	3.250	3.690	4.297	4.781
10	0.700	1.372	1.812	2.228	2.764	3.169	3.581	4.144	4.587
11	0.697	1.363	1.796	2.201	2.718	3.106	3.497	4.025	4.437
12	0.695	1.356	1.782	2.179	2.681	3.055	3.428	3.930	4.318
13	0.694	1.350	1.771	2.160	2.650	3.012	3.372	3.852	4.221
14	0.692	1.345	1.761	2.145	2.624	2.977	3.326	3.787	4.140
15	0.691	1.341	1.753	2.131	2.602	2.947	3.286	3.733	4.073
16	0.690	1.337	1.746	2.120	2.583	2.921	3.252	3.686	4.015
17	0.689	1.333	1.740	2.110	2.567	2.898	3.222	3.646	3.965
18	0.688	1.330	1.734	2.101	2.552	2.878	3.197	3.610	3.922
19	0.688	1.328	1.729	2.093	2.539	2.861	3.174	3.579	3.883
20	0.687	1.325	1.725	2.086	2.528	2.845	3.153	3.552	3.850
21	0.686	1.323	1.721	2.080	2.518	2.831	3.135	3.527	3.819
22	0.686	1.321	1.717	2.074	2.508	2.819	3.119	3.505	3.792
23	0.685	1.319	1.714	2.069	2.500	2.807	3.104	3.485	3.768
24	0.685	1.318	1.711	2.064	2.492	2.797	3.091	3.467	3.745
25	0.684	1.316	1.708	2.060	2.485	2.787	3.078	3.450	3.725
26	0.684	1.315	1.706	2.056	2.479	2.779	3.067	3.435	3.707
27	0.684	1.314	1.703	2.052	2.473	2.771	3.057	3.421	3.689
28	0.683	1.313	1.701	2.048	2.467	2.763	3.047	3.408	3.674
29	0.683	1.311	1.699	2.045	2.462	2.756	3.038	3.396	3.660
30	0.683	1.310	1.697	2.042	2.457	2.750	3.030	3.385	3.646
40	0.681	1.303	1.684	2.021	2.423	2.704	2.971	3.307	3.551
60	0.679	1.296	1.671	2.000	2.390	2.660	2.915	3.232	3.460
120	0.677	1.289	1.658	1.980	2.358	2.617	2.860	3.160	3.373
∞	0.674	1.282	1.645	1.960	2.326	2.576	2.807	3.090	3.291

Critical Values for the χ²-Distribution

If X has a χ²-distribution with v degrees of freedom then, for each pair of values of p and v, the table gives the value of x such that

P(X ≤ x) = p.

p	0.01	0.025	0.05	0.9	0.95	0.975	0.99	0.995	0.999
v= 1	0.0³ 1571	0.0³ 9821	0.0² 3932	2.706	3.841	5.024	6.635	7.879	10.83
2	0.02010	0.05064	0.1026	4.605	5.991	7.378	9.210	10.60	13.82
3	0.1148	0.2158	0.3518	6.251	7.815	9.348	11.34	12.84	16.27
4	0.2971	0.4844	0.7107	7.779	9.488	11.14	13.28	14.86	18.47
5	0.5543	0.8312	1.145	9.236	11.07	12.83	15.09	16.75	20.51
6	0.8721	1.237	1.635	10.64	12.59	14.45	16.81	18.55	22.46
7	1.239	1.690	2.167	12.02	14.07	16.01	18.48	20.28	24.32
8	1.647	2.180	2.733	13.36	15.51	17.53	20.09	21.95	26.12
9	2.088	2.700	3.325	14.68	16.92	19.02	21.67	23.59	27.88
10	2.558	3.247	3.940	15.99	18.31	20.48	23.21	25.19	29.59
11	3.053	3.816	4.575	17.28	19.68	21.92	24.73	26.76	31.26
12	3.571	4.404	5.226	18.55	21.03	23.34	26.22	28.30	32.91
13	4.107	5.009	5.892	19.81	22.36	24.74	27.69	29.82	34.53
14	4.660	5.629	6.571	21.06	23.68	26.12	29.14	31.32	36.12
15	5.229	6.262	7.261	22.31	25.00	27.49	30.58	32.80	37.70
16	5.812	6.908	7.962	23.54	26.30	28.85	32.00	34.27	39.25
17	6.408	7.564	8.672	24.77	27.59	30.19	33.41	35.72	40.79
18	7.015	8.231	9.390	25.99	28.87	31.53	34.81	37.16	42.31
19	7.633	8.907	10.12	27.20	30.14	32.85	36.19	38.58	43.82
20	8.260	9.591	10.85	28.41	31.41	34.17	37.57	40.00	45.31
21	8.897	10.28	11.59	29.62	32.67	35.48	38.93	41.40	46.80
22	9.542	10.98	12.34	30.81	33.92	36.78	40.29	42.80	48.27
23	10.20	11.69	13.09	32.01	35.17	38.08	41.64	44.18	49.73
24	10.86	12.40	13.85	33.20	36.42	39.36	42.98	45.56	51.18
25	11.52	13.12	14.61	34.38	37.65	40.65	44.31	46.93	52.62
30	14.95	16.79	18.49	40.26	43.77	46.98	50.89	53.67	59.70
40	22.16	24.43	26.51	51.81	55.76	59.34	63.69	66.77	73.40
50	29.71	32.36	34.76	63.17	67.50	71.42	76.15	79.49	86.66
60	37.48	40.48	43.19	74.40	79.08	83.30	88.38	91.95	99.61
70	45.44	48.76	51.74	85.53	90.53	95.02	100.4	104.2	112.3
80	53.54	57.15	60.39	96.58	101.9	106.6	112.3	116.3	124.8
90	61.75	65.65	69.13	107.6	113.1	118.1	124.1	128.3	137.2
100	70.06	74.22	77.93	118.5	124.3	129.6	135.8	140.2	149.4

Wilcoxon Signed-Rank Test

The sample has size n.

P is the sum of the ranks corresponding to the positive differences.

Q is the sum of the ranks corresponding to the negative differences.

T is the smaller of P and Q.

For each value of n the table gives the largest value of T which will lead to rejection of the null hypothesis at the level of significance indicated.

Critical values of T

	Level of significance
One-tailed	0.05	0.025	0.01	0.005
Two-tailed	0.1	0.05	0.02	0.01
n= 6	2	0
7	3	2	0
8	5	3	1	0
9	8	5	3	1
10	10	8	5	3
11	13	10	7	5
12	17	13	9	7
13	21	17	12	9
14	25	21	15	12
15	30	25	19	15
16	35	29	23	19
17	41	34	27	23
18	47	40	32	27
19	53	46	37	32
20	60	52	43	37

For larger values of $n$ , each of $P$ and $Q$ can be approximated by the normal distribution with mean $\frac{1}{4}n(n+1)$ and variance $\frac{1}{24}n(n+1)(2n+1)$ .

Wilcoxon Rank-Sum Test

The two samples have sizes $m$ and $n$ , where $m \leqslant n$ .

$R_m$ is the sum of the ranks of the items in the sample of size $m$ .

$W$ is the smaller of $R_m$ and $m(n + m + 1) - R_m$ .

For each pair of values of $m$ and $n$ , the table gives the largest value of $W$ which will lead to rejection of the null hypothesis at the level of significance indicated.

Critical values of W

	Level of significance
One-tailed	0.05	0.025	0.01	0.05	0.025	0.01	0.05	0.025	0.01	0.05	0.025	0.01
Two-tailed	0.1	0.05	0.02	0.1	0.05	0.02	0.1	0.05	0.02	0.1	0.05	0.02
n	m = 3			m = 4			m = 5			m = 6
3	6	–	–
4	6	–	–	11	10	–
5	7	6	–	12	11	10	19	17	16
6	8	7	–	13	12	11	20	18	17	28	26	24
7	8	7	6	14	13	11	21	20	18	29	27	25
8	9	8	6	15	14	12	23	21	19	31	29	27
9	10	8	7	16	14	13	24	22	20	33	31	28
10	10	9	7	17	15	13	26	23	21	35	32	29

	Level of significance
One-tailed	0.05	0.025	0.01	0.05	0.025	0.01	0.05	0.025	0.01	0.05	0.025	0.01
Two-tailed	0.1	0.05	0.02	0.1	0.05	0.02	0.1	0.05	0.02	0.1	0.05	0.02
n	m = 7			m = 8			m = 9			m = 10
7	39	36	34
8	41	38	35	51	49	45
9	43	40	37	54	51	47	66	62	59
10	45	42	39	56	53	49	69	65	61	82	78	74

For larger values of $m$ and $n$ , the normal distribution with mean $\frac{1}{2}m(m+n+1)$ and variance $\frac{1}{12}mn(m+n+1)$ should be used as an approximation to the distribution of $R_m$ .

Mathematics (9709)Formula Booklet

Pure Mathematics

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Further Pure Mathematics

Algebra

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Hyperbolic functions

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Further Probability & Statistics

Sampling and testing

Probability generating functions

The Normal Distribution Function

Critical values for the normal distribution

Critical Values for the t-Distribution

Critical Values for the χ2-Distribution

Wilcoxon Signed-Rank Test

Critical values of T

Wilcoxon Rank-Sum Test

Critical values of W

Critical Values for the χ²-Distribution