Important Statistics Formulas

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					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!

				
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posted:7/31/2011
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