Mathematics Formulae & Statistical Tables for A-Level, Cheat Sheet of Statistics

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ADVANCED SUBSIDIARY GENERAL CERTIFICATE OF EDUCATION

ADVANCED GENERAL CERTIFICATE OF EDUCATION

MATHEMATICS

LIST OF FORMULAE

AND

STATISTICAL T ABLES

(List MF1)

MF

CST

January 2007

Pure Mathematics

Mensuration

Surface area of sphere = 4 π r^2

Area of curved surface of cone = π r × slant height

Trigonometry

a^2 = b^2 + c^2 − 2 bc cos A

Arithmetic Series

u n = a + ( n − 1 ) d

S n = 12 n ( a + l ) = 12 n { 2 a + ( n − 1 ) d }

Geometric Series

u n = ar n −^1

S n =

a ( 1 − r n )

1 − r

S ∞ =

a

1 − r

for | r | < 1

Summations n ∑ r = 1

r^2 = 16 n ( n + 1 )( 2 n + 1 )

n ∑ r = 1

r^3 = 14 n^2 ( n + 1 )^2

Binomial Series

n

r

n

r + 1

n + 1

r + 1

( a + b ) n^ = a n^ + (

n

) a n −^1 b + (

n

) a n −^2 b^2 +... + (

n

r

) a n − r^ br^ +... + b n^ ( n ∈ ),

where (

n

r

) = n C r =

n!

r !( n − r )!

( 1 + x ) n^ = 1 + nx +

n ( n − 1 )

x^2 +... +

n ( n − 1 )... ( n − r + 1 )

1.2.3... r

x r^ +... (| x | < 1, n ∈ )

Logarithms and exponentials

e x^ ln^ a^ = a x

Complex Numbers

{ r (cos θ + i sin θ)} n^ = r n (cos n θ + i sin n θ)

e i^ θ^ = cos θ + i sin θ

The roots of  n^ = 1 are given by  = e

2 π k i

n , for k = 0, 1, 2,... , n − 1

Vectors

The resolved part of a in the direction of b is

a.b

| b |

The point dividing AB in the ratio λ : μ is

μ a + λ b

Vector product: a × b = | a | | b | sin θ n ˆ =

i a 1 b 1

j a 2 b 2

k a 3 b 3

a 2 b 3 − a 3 b 2

a 3 b 1 − a 1 b 3

a 1 b 2 − a 2 b 1

If A is the point with position vector a = a 1 i + a 2 j + a 3 k and the direction vector b is given by

b = b 1 i + b 2 j + b 3 k , then the straight line through A with direction vector b has cartesian equation

x − a 1

b 1

y − a 2

b 2

 − a 3

b 3

The plane through A with normal vector n = n 1 i + n 2 j + n 3 k has cartesian equation

n 1 x + n 2 y + n 3  + d = 0, where d = − a.n

The plane through non-collinear points A , B and C has vector equation

r = a + λ ( b − a ) + μ( c − a ) = ( 1 − λ − μ) a + λ b + μ c

The plane through the point with position vector a and parallel to b and c has equation r = a + s b + t c

The perpendicular distance of ( α, β, γ ) from n 1 x + n 2 y + n 3  + d = 0 is

∣ n 1 α^ +^ n 2 β^ +^ n 3 γ^ +^ d

( n^21 + n^22 + n^23 )

Matrix transformations

Anticlockwise rotation through θ about O : (

cos θ − sin θ

sin θ cos θ

Reflection in the line y = (tan θ) x : (

cos 2 θ sin 2 θ

sin 2 θ − cos 2 θ

Differentiation

f( x ) f′^ ( x )

tan kx k sec 2 kx

sin−^1 x

( 1 − x^2 )

cos−^1 x −

( 1 − x^2 )

tan−^1 x

1 + x^2

sec x sec x tan x

cot x − cosec^2 x

cosec x − cosec x cot x

sinh x cosh x

cosh x sinh x

tanh x sech 2 x

sinh−^1 x

( 1 + x^2 )

cosh−^1 x

( x^2 − 1 )

tanh−^1 x

1 − x^2

If y =

f( x )

g( x )

then

d y

d x

f ′^ ( x )g( x ) − f( x )g′^ ( x )

{g( x )}^2

Integration ( + constant; a > 0 where relevant)

f( x ) ^ f( x ) d x

sec^2 kx

k

tan kx

tan x ln |sec x |

cot x ln |sin x |

cosec x − ln |cosec x + cot x | = ln ∣∣tan 12 x ∣∣

sec x ln |sec x + tan x | = ln ∣∣tan(^12 x + 14 π)∣∣

sinh x cosh x

cosh x sinh x

tanh x ln cosh x

( a^2 − x^2 )

sin−^1 (

x

a

) (| x | < a )

a^2 + x^2

a

tan−^1 (

x

a

( x^2 − a^2 )

cosh−^1 (

x

a

) or ln{ x +

( x^2 − a^2 )} ( x > a )

( a^2 + x^2 )

sinh−^1 (

x

a

) or ln{ x +

( x^2 + a^2 )}

a^2 − x^2

2 a

ln

a + x

a − x

a

tanh−^1 (

x

a

) (| x | < a )

x^2 − a^2

2 a

ln ∣∣

x − a

x + a

 u d v

d x

d x = uv − ^ v

d u

d x

d x

Area of a sector

A = 12  r^2 d θ (polar coordinates)

A = 12 ^ ( x

d y

d t

− y

d x

d t

) d t (parametric form)

Numerical Mathematics

Numerical integration

The trapezium rule: 

b a

y d x ≈ 12 h {( y 0 + yn ) + 2 ( y 1 + y 2 +... + yn − 1 )}, where h =

b − a

n

Simpson’s Rule: 

b a

y d x ≈ 13 h {( y 0 + y n ) + 4 ( y 1 + y 3 +... + y n − 1 ) + 2 ( y 2 + y 4 +... + yn − 2 )},

where h =

b − a

n

and n is even

Numerical Solution of Equations

The Newton-Raphson iteration for solving f( x ) = 0: xn + 1 = x n −

f( x n )

f′^ ( x n )

Probability & Statistics

Probability

P( A ∪ B ) = P( A ) + P( B ) − P( A ∩ B )

P( A ∩ B ) = P( A )P( B | A )

P( A | B ) =

P( B | A )P( A )

P( B | A )P( A ) + P( B | A ′^ )P( A ′^ )

Bayes’ Theorem: P( Aj | B ) =

P( Aj )P( B | Aj )

ΣP( A i )P( B | Ai )

Discrete distributions

For a discrete random variable X taking values x i with probabilities p i

Expectation (mean): E( X ) = μ = Σ x i p i

Variance: Var( X ) = σ^2 = Σ( x i − μ)^2 p i = Σ x^2 i p i − μ^2

For a function g( X ): E(g( X )) = Σ g( x i ) p i

The probability generating function of X is G X ( t ) = E( t X^ ), and

E( X ) = G′ X ( 1 )

Var( X ) = G′′ X ( 1 ) + G′ X ( 1 ) − {G′ X ( 1 )}^2

For Z = X + Y , where X and Y are independent: G Z ( t ) = G X ( t )G Y ( t )

Standard discrete distributions

Distribution of X P( X = x ) Mean Variance P.G.F.

Binomial B( n , p ) (

n

x

) p x ( 1 − p ) n − x^ np np ( 1 − p ) ( 1 − p + pt ) n

Poisson Po( λ ) e−^ λ^

λ x

x!

λ λ e^ λ^ ( t −^1 )

Geometric Geo( p ) on 1, 2, … p ( 1 − p ) x −^1

p

1 − p

p^2

pt

1 − ( 1 − p ) t

Continuous distributions

For a continuous random variable X having probability density function f

Expectation (mean): E( X ) = μ = ^ x f( x ) d x

Variance: Var( X ) = σ^2 =  ( x − μ)^2 f( x ) d x = ^ x^2 f( x ) d x − μ^2

For a function g( X ): E(g( X )) = ^ g( x )f( x ) d x

Cumulative distribution function: F( x ) = P( X ≤ x ) = 

x −∞

f( t ) d t

The moment generating function of X is M X ( t ) = E(e tX^ ) and

E( X ) = M′ X ( 0 )

E( X n ) = M( Xn )( 0 )

Var( X ) = M′′ X ( 0 ) − {M′ X ( 0 )}^2

For Z = X + Y , where X and Y are independent: M Z ( t ) = M X ( t )M Y ( t )

Standard continuous distributions

Distribution of X P.D.F. Mean Variance M.G.F.

Uniform (Rectangular) on [ a , b ]

b − a

1

2 ( a^ +^ b )^

1

12 ( b^ −^ a )

2 e

bt − e at

( b − a ) t

Exponential λ e−^ λ^ x^

λ − t

Normal N( μ, σ^2 )

e−

1

2 (^

x − μ

2

μ σ^2 e^ μ t +

1 2 σ (^2) t 2 Expectation algebra

Covariance: Cov( X , Y ) = E(( X − μ X )( Y − μ Y )) = E( XY ) − μ X μ Y

Var( aX ± bY ) = a^2 Var( X ) + b^2 Var( Y ) ± 2 ab Cov( X , Y )

Product moment correlation coefficient: ρ =

Cov( X , Y )

σ X σ Y

If X = aX ′^ + b and Y = cY ′^ + d , then Cov( X , Y ) = ac Cov( X ′, Y ′^ )

For independent random variables X and Y

E( XY ) = E( X )E( Y )

Var( aX ± bY ) = a^2 Var( X ) + b^2 Var( Y )

Sampling distributions

For a random sample X 1 , X 2 ,... , Xn of n independent observations from a distribution having mean μ

and variance σ^2

X is an unbiased estimator of μ, with Var( X ) =

σ^2

n

S^2 is an unbiased estimator of σ^2 , where S^2 =

Σ( Xi − X )^2

n − 1

For a random sample of n observations from N( μ, σ^2 )

X − μ

n

∼ N(0, 1)

X − μ

S /

n

∼ tn − 1 (also valid in matched-pairs situations)

If X is the observed number of successes in n independent Bernoulli trials in each of which the

probability of success is p , and Y =

X

n

, then

E( Y ) = p and Var( Y ) =

p ( 1 − p )

n

For a random sample of n x observations from N( μ x , σ x^2 ) and, independently, a random sample of

n y observations from N( μ y , σ y^2 )

( X − Y ) − ( μ x − μ y )

σ x^2

n x

σ y^2

n y

∼ N(0, 1)

If σ x^2 = σ y^2 = σ^2 (unknown) then

( X − Y ) − ( μ x − μ y )

{ S^2 p (

n x

n y

∼ tn

x + n^ y −^2

where S^2 p =

( n x − 1 ) S^2 x + ( n y − 1 ) S y^2

nx + n y − 2

CUMULATIVE BINOMIAL PROBABILITIES

n^

p

x^

n^

p

x^

n^

p

x^

n^

p

x^

CUMULATIVE BINOMIAL PROBABILITIES

n^

p

x^

n^

(^10) p^

x^

n^

(^12) p^

x^

CUMULATIVE BINOMIAL PROBABILITIES

n^

(^18) p^

x^

n^

(^20) p^

x^

CUMULATIVE BINOMIAL PROBABILITIES

n^

(^25) p^

x^

CUMULATIVE POISSON PROBABILITIES

THE NORMAL DISTRIBUTION FUNCTION

If Z has a normal distribution with mean 0 and

variance 1 then, for each value of , the table gives

the value of Φ(), where

Φ() = P( Z ≤ ).

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

 0 1 2 3 4 5 6 7 8 9 1 2^3 4 5 6 7 8

ADD

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  such that

P( Z ≤ ) = p.

p 0.75 0.90 0.95 0.975 0.99 0.995 0.9975 0.999 0.  0.674 1.282 1.645 1.960 2.326 2.576 2.807 3.090 3.

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

• λ 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.
• x = 0 0.9900 0.9802 0.9704 0.9608 0.9512 0.9418 0.9324 0.9231 0. - 1 1.0000 0.9998 0.9996 0.9992 0.9988 0.9983 0.9977 0.9970 0. - 2 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0. - 3 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.
  • λ 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.
• x = 0 0.9048 0.8187 0.7408 0.6703 0.6065 0.5488 0.4966 0.4493 0. - 1 0.9953 0.9825 0.9631 0.9384 0.9098 0.8781 0.8442 0.8088 0. - 2 0.9998 0.9989 0.9964 0.9921 0.9856 0.9769 0.9659 0.9526 0. - 3 1.0000 0.9999 0.9997 0.9992 0.9982 0.9966 0.9942 0.9909 0. - 4 1.0000 1.0000 1.0000 0.9999 0.9998 0.9996 0.9992 0.9986 0. - 5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0. - 6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.
  • λ 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.
• x = 0 0.3679 0.3329 0.3012 0.2725 0.2466 0.2231 0.2019 0.1827 0.1653 0. - 1 0.7358 0.6990 0.6626 0.6268 0.5918 0.5578 0.5249 0.4932 0.4628 0. - 2 0.9197 0.9004 0.8795 0.8571 0.8335 0.8088 0.7834 0.7572 0.7306 0. - 3 0.9810 0.9743 0.9662 0.9569 0.9463 0.9344 0.9212 0.9068 0.8913 0. - 4 0.9963 0.9946 0.9923 0.9893 0.9857 0.9814 0.9763 0.9704 0.9636 0. - 5 0.9994 0.9990 0.9985 0.9978 0.9968 0.9955 0.9940 0.9920 0.9896 0. - 6 0.9999 0.9999 0.9997 0.9996 0.9994 0.9991 0.9987 0.9981 0.9974 0. - 7 1.0000 1.0000 1.0000 0.9999 0.9999 0.9998 0.9997 0.9996 0.9994 0. - 8 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0. - 9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.
  • λ 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.
• x = 0 0.1353 0.1225 0.1108 0.1003 0.0907 0.0821 0.0743 0.0672 0.0608 0. - 1 0.4060 0.3796 0.3546 0.3309 0.3084 0.2873 0.2674 0.2487 0.2311 0. - 2 0.6767 0.6496 0.6227 0.5960 0.5697 0.5438 0.5184 0.4936 0.4695 0. - 3 0.8571 0.8386 0.8194 0.7993 0.7787 0.7576 0.7360 0.7141 0.6919 0. - 4 0.9473 0.9379 0.9275 0.9162 0.9041 0.8912 0.8774 0.8629 0.8477 0. - 5 0.9834 0.9796 0.9751 0.9700 0.9643 0.9580 0.9510 0.9433 0.9349 0. - 6 0.9955 0.9941 0.9925 0.9906 0.9884 0.9858 0.9828 0.9794 0.9756 0. - 7 0.9989 0.9985 0.9980 0.9974 0.9967 0.9958 0.9947 0.9934 0.9919 0. - 8 0.9998 0.9997 0.9995 0.9994 0.9991 0.9989 0.9985 0.9981 0.9976 0. - 9 1.0000 0.9999 0.9999 0.9999 0.9998 0.9997 0.9996 0.9995 0.9993 0. - 10 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9998 0. - 11 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0. - 12 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.
  • λ 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.
• x = 0 0.0498 0.0450 0.0408 0.0369 0.0334 0.0302 0.0273 0.0247 0.0224 0. - 1 0.1991 0.1847 0.1712 0.1586 0.1468 0.1359 0.1257 0.1162 0.1074 0. - 2 0.4232 0.4012 0.3799 0.3594 0.3397 0.3208 0.3027 0.2854 0.2689 0. - 3 0.6472 0.6248 0.6025 0.5803 0.5584 0.5366 0.5152 0.4942 0.4735 0. - 4 0.8153 0.7982 0.7806 0.7626 0.7442 0.7254 0.7064 0.6872 0.6678 0. - 5 0.9161 0.9057 0.8946 0.8829 0.8705 0.8576 0.8441 0.8301 0.8156 0. - 6 0.9665 0.9612 0.9554 0.9490 0.9421 0.9347 0.9267 0.9182 0.9091 0. - 7 0.9881 0.9858 0.9832 0.9802 0.9769 0.9733 0.9692 0.9648 0.9599 0. - 8 0.9962 0.9953 0.9943 0.9931 0.9917 0.9901 0.9883 0.9863 0.9840 0. - 9 0.9989 0.9986 0.9982 0.9978 0.9973 0.9967 0.9960 0.9952 0.9942 0. - 10 0.9997 0.9996 0.9995 0.9994 0.9992 0.9990 0.9987 0.9984 0.9981 0. - 11 0.9999 0.9999 0.9999 0.9998 0.9998 0.9997 0.9996 0.9995 0.9994 0. - 12 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9999 0.9998 0. - 13 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0. - 14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.
  • λ 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4. CUMULATIVE POISSON PROBABILITIES
• x = 0 0.0183 0.0166 0.0150 0.0136 0.0123 0.0111 0.0101 0.0091 0.0082 0. - 1 0.0916 0.0845 0.0780 0.0719 0.0663 0.0611 0.0563 0.0518 0.0477 0. - 2 0.2381 0.2238 0.2102 0.1974 0.1851 0.1736 0.1626 0.1523 0.1425 0. - 3 0.4335 0.4142 0.3954 0.3772 0.3594 0.3423 0.3257 0.3097 0.2942 0. - 4 0.6288 0.6093 0.5898 0.5704 0.5512 0.5321 0.5132 0.4946 0.4763 0. - 5 0.7851 0.7693 0.7531 0.7367 0.7199 0.7029 0.6858 0.6684 0.6510 0. - 6 0.8893 0.8786 0.8675 0.8558 0.8436 0.8311 0.8180 0.8046 0.7908 0. - 7 0.9489 0.9427 0.9361 0.9290 0.9214 0.9134 0.9049 0.8960 0.8867 0. - 8 0.9786 0.9755 0.9721 0.9683 0.9642 0.9597 0.9549 0.9497 0.9442 0. - 9 0.9919 0.9905 0.9889 0.9871 0.9851 0.9829 0.9805 0.9778 0.9749 0. - 10 0.9972 0.9966 0.9959 0.9952 0.9943 0.9933 0.9922 0.9910 0.9896 0. - 11 0.9991 0.9989 0.9986 0.9983 0.9980 0.9976 0.9971 0.9966 0.9960 0. - 12 0.9997 0.9997 0.9996 0.9995 0.9993 0.9992 0.9990 0.9988 0.9986 0. - 13 0.9999 0.9999 0.9999 0.9998 0.9998 0.9997 0.9997 0.9996 0.9995 0. - 14 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9999 0.9999 0. - 15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0. - 16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.
  • λ 5.00 5.50 6.00 6.50 7.00 7.50 8.00 8.50 9.00 9.
• x = 0 0.0067 0.0041 0.0025 0.0015 0.0009 0.0006 0.0003 0.0002 0.0001 0. - 1 0.0404 0.0266 0.0174 0.0113 0.0073 0.0047 0.0030 0.0019 0.0012 0. - 2 0.1247 0.0884 0.0620 0.0430 0.0296 0.0203 0.0138 0.0093 0.0062 0. - 3 0.2650 0.2017 0.1512 0.1118 0.0818 0.0591 0.0424 0.0301 0.0212 0. - 4 0.4405 0.3575 0.2851 0.2237 0.1730 0.1321 0.0996 0.0744 0.0550 0. - 5 0.6160 0.5289 0.4457 0.3690 0.3007 0.2414 0.1912 0.1496 0.1157 0. - 6 0.7622 0.6860 0.6063 0.5265 0.4497 0.3782 0.3134 0.2562 0.2068 0. - 7 0.8666 0.8095 0.7440 0.6728 0.5987 0.5246 0.4530 0.3856 0.3239 0. - 8 0.9319 0.8944 0.8472 0.7916 0.7291 0.6620 0.5925 0.5231 0.4557 0. - 9 0.9682 0.9462 0.9161 0.8774 0.8305 0.7764 0.7166 0.6530 0.5874 0. - 10 0.9863 0.9747 0.9574 0.9332 0.9015 0.8622 0.8159 0.7634 0.7060 0. - 11 0.9945 0.9890 0.9799 0.9661 0.9467 0.9208 0.8881 0.8487 0.8030 0. - 12 0.9980 0.9955 0.9912 0.9840 0.9730 0.9573 0.9362 0.9091 0.8758 0. - 13 0.9993 0.9983 0.9964 0.9929 0.9872 0.9784 0.9658 0.9486 0.9261 0. - 14 0.9998 0.9994 0.9986 0.9970 0.9943 0.9897 0.9827 0.9726 0.9585 0. - 15 0.9999 0.9998 0.9995 0.9988 0.9976 0.9954 0.9918 0.9862 0.9780 0. - 16 1.0000 0.9999 0.9998 0.9996 0.9990 0.9980 0.9963 0.9934 0.9889 0. - 17 1.0000 1.0000 0.9999 0.9998 0.9996 0.9992 0.9984 0.9970 0.9947 0. - 18 1.0000 1.0000 1.0000 0.9999 0.9999 0.9997 0.9993 0.9987 0.9976 0. - 19 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9995 0.9989 0. - 20 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9996 0. - 21 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0. - 22 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0. - 23 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0. - 24 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.
  • λ 10.00 11.00 12.00 13.00 14.00 15.00 16.00 17.00 18.00 19. CUMULATIVE POISSON PROBABILITIES
• x = 0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0. - 1 0.0005 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0. - 2 0.0028 0.0012 0.0005 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0. - 3 0.0103 0.0049 0.0023 0.0011 0.0005 0.0002 0.0001 0.0000 0.0000 0. - 4 0.0293 0.0151 0.0076 0.0037 0.0018 0.0009 0.0004 0.0002 0.0001 0. - 5 0.0671 0.0375 0.0203 0.0107 0.0055 0.0028 0.0014 0.0007 0.0003 0. - 6 0.1301 0.0786 0.0458 0.0259 0.0142 0.0076 0.0040 0.0021 0.0010 0. - 7 0.2202 0.1432 0.0895 0.0540 0.0316 0.0180 0.0100 0.0054 0.0029 0. - 8 0.3328 0.2320 0.1550 0.0998 0.0621 0.0374 0.0220 0.0126 0.0071 0. - 9 0.4579 0.3405 0.2424 0.1658 0.1094 0.0699 0.0433 0.0261 0.0154 0. - 10 0.5830 0.4599 0.3472 0.2517 0.1757 0.1185 0.0774 0.0491 0.0304 0. - 11 0.6968 0.5793 0.4616 0.3532 0.2600 0.1848 0.1270 0.0847 0.0549 0. - 12 0.7916 0.6887 0.5760 0.4631 0.3585 0.2676 0.1931 0.1350 0.0917 0. - 13 0.8645 0.7813 0.6815 0.5730 0.4644 0.3632 0.2745 0.2009 0.1426 0. - 14 0.9165 0.8540 0.7720 0.6751 0.5704 0.4657 0.3675 0.2808 0.2081 0. - 15 0.9513 0.9074 0.8444 0.7636 0.6694 0.5681 0.4667 0.3715 0.2867 0. - 16 0.9730 0.9441 0.8987 0.8355 0.7559 0.6641 0.5660 0.4677 0.3751 0. - 17 0.9857 0.9678 0.9370 0.8905 0.8272 0.7489 0.6593 0.5640 0.4686 0. - 18 0.9928 0.9823 0.9626 0.9302 0.8826 0.8195 0.7423 0.6550 0.5622 0. - 19 0.9965 0.9907 0.9787 0.9573 0.9235 0.8752 0.8122 0.7363 0.6509 0. - 20 0.9984 0.9953 0.9884 0.9750 0.9521 0.9170 0.8682 0.8055 0.7307 0. - 21 0.9993 0.9977 0.9939 0.9859 0.9712 0.9469 0.9108 0.8615 0.7991 0. - 22 0.9997 0.9990 0.9970 0.9924 0.9833 0.9673 0.9418 0.9047 0.8551 0. - 23 0.9999 0.9995 0.9985 0.9960 0.9907 0.9805 0.9633 0.9367 0.8989 0. - 24 1.0000 0.9998 0.9993 0.9980 0.9950 0.9888 0.9777 0.9594 0.9317 0. - 25 1.0000 0.9999 0.9997 0.9990 0.9974 0.9938 0.9869 0.9748 0.9554 0. - 26 1.0000 1.0000 0.9999 0.9995 0.9987 0.9967 0.9925 0.9848 0.9718 0. - 27 1.0000 1.0000 0.9999 0.9998 0.9994 0.9983 0.9959 0.9912 0.9827 0. - 28 1.0000 1.0000 1.0000 0.9999 0.9997 0.9991 0.9978 0.9950 0.9897 0. - 29 1.0000 1.0000 1.0000 1.0000 0.9999 0.9996 0.9989 0.9973 0.9941 0. - 30 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9994 0.9986 0.9967 0. - 31 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9993 0.9982 0. - 32 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9996 0.9990 0. - 33 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9995 0. - 34 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0. - 35 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0. - 36 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0. - 37 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0. - 38 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1. - p 0.75 0.90 0.95 0.975 0.99 0.995 0.9975 0.999 0. P( T ≤ t ) = p
• v = 1 1.000 3.078 6.314 12.71 31.82 63.66 127.3 318.3 636. - 2 0.816 1.886 2.920 4.303 6.965 9.925 14.09 22.33 31. - 3 0.765 1.638 2.353 3.182 4.541 5.841 7.453 10.21 12. - 4 0.741 1.533 2.132 2.776 3.747 4.604 5.598 7.173 8. - 5 0.727 1.476 2.015 2.571 3.365 4.032 4.773 5.894 6. - 6 0.718 1.440 1.943 2.447 3.143 3.707 4.317 5.208 5. - 7 0.711 1.415 1.895 2.365 2.998 3.499 4.029 4.785 5. - 8 0.706 1.397 1.860 2.306 2.896 3.355 3.833 4.501 5. - 9 0.703 1.383 1.833 2.262 2.821 3.250 3.690 4.297 4. - 10 0.700 1.372 1.812 2.228 2.764 3.169 3.581 4.144 4. - 11 0.697 1.363 1.796 2.201 2.718 3.106 3.497 4.025 4. - 12 0.695 1.356 1.782 2.179 2.681 3.055 3.428 3.930 4. - 13 0.694 1.350 1.771 2.160 2.650 3.012 3.372 3.852 4. - 14 0.692 1.345 1.761 2.145 2.624 2.977 3.326 3.787 4. - 15 0.691 1.341 1.753 2.131 2.602 2.947 3.286 3.733 4. - 16 0.690 1.337 1.746 2.120 2.583 2.921 3.252 3.686 4. - 17 0.689 1.333 1.740 2.110 2.567 2.898 3.222 3.646 3. - 18 0.688 1.330 1.734 2.101 2.552 2.878 3.197 3.610 3. - 19 0.688 1.328 1.729 2.093 2.539 2.861 3.174 3.579 3. - 20 0.687 1.325 1.725 2.086 2.528 2.845 3.153 3.552 3. - 21 0.686 1.323 1.721 2.080 2.518 2.831 3.135 3.527 3. - 22 0.686 1.321 1.717 2.074 2.508 2.819 3.119 3.505 3. - 23 0.685 1.319 1.714 2.069 2.500 2.807 3.104 3.485 3. - 24 0.685 1.318 1.711 2.064 2.492 2.797 3.091 3.467 3. - 25 0.684 1.316 1.708 2.060 2.485 2.787 3.078 3.450 3. - 26 0.684 1.315 1.706 2.056 2.479 2.779 3.067 3.435 3. - 27 0.684 1.314 1.703 2.052 2.473 2.771 3.057 3.421 3. - 28 0.683 1.313 1.701 2.048 2.467 2.763 3.047 3.408 3. - 29 0.683 1.311 1.699 2.045 2.462 2.756 3.038 3.396 3. - 30 0.683 1.310 1.697 2.042 2.457 2.750 3.030 3.385 3. - 40 0.681 1.303 1.684 2.021 2.423 2.704 2.971 3.307 3. - 60 0.679 1.296 1.671 2.000 2.390 2.660 2.915 3.232 3.
  • 120 0.677 1.289 1.658 1.980 2.358 2.617 2.860 3.160 3. - ∞ 0.674 1.282 1.645 1.960 2.326 2.576 2.807 3.090 3.