# Important Statistics Formulas by fdh56iuoui

VIEWS: 19 PAGES: 5

• pg 1
```									The Logarithmic Function:
L = log b N
The inverse function is: N = b L
For example:
log 2 8 = 3 since 8 = 2 3
log10 0.01 = −2 since 0.01 = 10 −2
log 5 5 = 1 since 5 = 51
log b 1 = 0 since 1 = b 0

Selected Algebra Topics
Basic Laws of Exponents
Law              Example
m+ n
a a =a
m n
x 5 x −2 = x 3
am                     x5
= a m−n , a ≠ 0         = x2
an                     x3
(a )m n
= a mn        (x )
−2 3
= x −6
(ab )m        = a mb m       (xy )2 = x 2 y 2
m                                       2
⎛a⎞      am                          ⎛x⎞      x2
⎜ ⎟ = m ,b ≠ 0                       ⎜ ⎟ = 2
⎜ y⎟
⎝b⎠      b                           ⎝ ⎠      y
1                                     1
a −m = m , a ≠ 0                      x −3 = 3
a                                     x
a = 1, a ≠ 0                   2(3x ) = 2(1) = 2
0                                   0

a1 = a               (3x )  2 1
= 3x 2

Laws for fractional exponents
Law                      Example
m                                       2
a       n
= n am                     x        3
= 3 x2
n                                      3
a              a                        16
n
=n       ,b ≠ 0             3
=3 8=2
b              b                         2
25 = 5, (not ± 5)
1
a       2
= 2 a1 = a , a ≥ 0
Trigonometric Identities

sin (θ ) =                      csc(θ ) =
a                                  1      c
=
c                               sin (θ ) a

cos(θ ) =                       sec(θ ) =
b                                  1      c
=
c                               cos(θ ) b
sin (θ ) a                      cos(θ ) b
tan (θ ) =         =            cot (θ ) =         =
cos(θ ) b                       sin (θ ) a

sin (− x ) = − sin ( x )
csc(− x ) = − csc( x )
cos(− x ) = cos( x )
sec(− x ) = sec( x )
tan (− x ) = − tan ( x )
cot (− x ) = − cot ( x )

sin 2 ( x ) + cos 2 ( x ) = 1
tan 2 ( x ) + 1 = sec 2 ( x )
cot 2 ( x ) + 1 = csc 2 ( x )
sin ( x ± y ) = sin ( x ) cos( y ) ± cos( x )sin ( y )
cos( x ± y ) = cos( x ) cos( y ) ± sin ( x )sin ( y )
tan ( x ) ± tan ( y )
tan ( x ± y ) =
1 ± tan ( x ) tan ( y )
sin (2 x ) = 2 sin (x ) cos(x )
cos(2 x ) = cos 2 ( x ) − sin 2 (x ) = 2 cos 2 ( x) − 1 = 1 − 2 sin 2 ( x )
2 tan ( x )
tan (2 x ) =
(              )
1 − tan 2 ( x )

sin 2 ( x ) = − cos(2 x )
1 1
2 2
cos 2 ( x ) =     + cos(2 x )
1 1
2 2
⎛ (x − y ) ⎞ ⎛ (x + y ) ⎞
sin ( x ) − sin ( y ) = 2 sin ⎜          ⎟ cos⎜       ⎟
⎝ 2 ⎠ ⎝ 2 ⎠
⎛ (x − y ) ⎞ ⎛ (x + y ) ⎞
cos( x ) − cos( y ) = −2 sin ⎜              ⎟ sin ⎜      ⎟
⎝ 2 ⎠ ⎝ 2 ⎠

Given Triangle abc, with angles A,B,C; a is opposite to A, b is opposite to B, and c is
opposite to C:

a        b        c
Law of Sines:              =        =
sin ( A) sin (B ) sin (C )

Law of Cosines:
c 2 = a 2 + b 2 − 2ab cos(C )
b 2 = a 2 + c 2 − 2ac cos(B )
a 2 = b 2 + c 2 − 2bc cos( A)

(a − b ) = tan (12 ( A − B ))
(a + b ) tan (1 ( A + B ))
Law of Tangents:
2
Important Statistics Formulas:
Parameters:
(ΣX i )
Population mean: μ =
N
Σ( X i − μ ) 2
Population Standard Deviation: σ =
N
Σ( X i − μ ) 2
Population Variance: σ 2 =
N
(X − μ)
Standardized Score: Z =
σ
⎡ 1 ⎤ ⎧⎡ ( X − μ x ) ⎤ ⎡ (Yi − μ y ) ⎤ ⎫
⎪                              ⎪
Population Correlation Coefficient: ρ = ⎢ ⎥ * Σ ⎨⎢ i         ⎥*⎢             ⎥⎬
⎣ N ⎦ ⎪⎣ σ x
⎩            ⎦ ⎢ σ y ⎥⎪
⎣             ⎦⎭

Statistics:
(Σxi )
Sample mean: x =
n
Σ( xi − x ) 2
Sample standard deviation: s =
(n − 1)
Σ( xi − x ) 2
Sample variance: s 2 =
(n − 1)
⎡ 1 ⎤ ⎧ ⎡ ( xi − x ) ⎤ ⎡ ( y i − y ) ⎤ ⎫
⎪                      ⎪
Sample Correlation coefficient: r = ⎢         ⎥ * Σ ⎨⎢ s   ⎥*⎢           ⎥⎬
⎣ (n − 1) ⎦ ⎪⎣  ⎩    x ⎦ ⎢ s y ⎥⎪
⎣           ⎦⎭
1    ⎛ (x − μ)2                        ⎞
Normal Distribution Formula:          exp⎜ −
⎜                                 ⎟
⎟
σ 2π    ⎝   2σ 2                          ⎠
1     ⎛ z2 ⎞
Or             exp⎜ − ⎟
⎜ 2 ⎟
σ 2π    ⎝    ⎠

Simple Linear Regression:
^
Simple linear regression line: y = b0 + b1 x

Regression coefficient: b1 =
[(         )(
Σ xi − x y i − y              )]
(
Σ xi − x   )   2

Regression slope intercept: b0 = y − b1 * x
2
⎛       ^
⎞
Σ⎜ y i − y i ⎟
⎝           ⎠
(n − 2)
Standard error of regression slope: s b1 =
(
Σ xi − x   )
2

Random Variables:
Expected value of X: E ( X ) = μ x = Σ[xi * P( xi )]
Variance of X: Var ( X ) = σ 2 = Σ[xi − E ( x) )] * P( xi ) = Σ[xi − μ x ] * P( xi )
2                  2

Normal Random Variable: z − score = z =
(x − μ )
σ
Expected value of sum of random variables: E ( X + Y ) = E ( X ) + E (Y )
Expected value of difference between random variables: E ( X − Y ) = E ( X ) − E (Y )
Variance of the sum of independent random variables:
Var ( X + Y ) = Var ( X ) + Var (Y )
Variance of the difference between independent random variables:
Var ( X − Y ) = Var ( X ) − Var (Y )

Sampling Distributions:
σ
Standard deviation of the mean: σ x =
n

Standard Error:
s
Standard error of the mean: SE x = s x =
n
Taylor series expansion:

∞
f ( n ) ( x ) Δx n                            f ' ' ( x ) Δx 2 f ' ' ' ( x ) Δx 3
f ( x + Δx ) = ∑                        = f ( x ) + f ' ( x ) Δx +                 +                   +L
n =0            n!                                           2!                3!

∞
f ( n ) ( 0) x n
Maclaurin series expansion:                f ( x) = ∑
n =0          n!
2        3       4      5
x     x    x    x
ex = 1 + x +     +    +    +    +L
2! 3! 4! 5!
x3 x5 x7 x9
sin x = x −     +    −    +    +L
3! 5! 7! 9!
x 2 x 4 x6 x8
cos x = 1 −     +    −    +    +L
2! 4! 6! 8!

```
To top