Statistics

mean

x_bar = 1/n Σ x_i

permutation

nPk = n! / (n-k)!

n: size of the set

k: elements drawn

Normal Distribution

Z distribution is a normal distribution with a mean (μ) of 0 and a standard deviation (δ) of 1. Any normal distribution is described by its μ and δ value.

To convert a normal distribution to a Z distribution

 Z = (x - μ) / δ

Standard deviation is the root of the sum of the square errors:

 δ = ( Σ (i - μ)2 ).5

Binomial Distribution

Similar to normal distribution, but for values that can only be discrete.

Probability of an event happening k times given n trials, where p is the probability of n:

 nCk * pk(1-p)n-k
 nCk = n! / ( k! * (n-k)! )

Example: The probability of two heads in six coin flips:

 6C2 = 6! / ( 2! * (6-2)! ) = 6! / (2!4!) = 720 / (24 * 2) = 15
 15 * .52(1-.5)6-2 = 15 * .25 * .54 = 15 * 0.015625 = 0.234375

Hypothesis Testing

Null hypothesis h₀. Alternate hypothesis H_A or H₁

Single Tail Test

 H0 : μ1 = μ2
 HA : μ1 > μ2

Two Tailed Test

 H0 : μ1 = μ2
 HA : μ1 ? μ2

Chi-Squared

Given:

	DF1	DF2	...	DFn
Observed	O1	O2	...	On
Excpected	E1	E2	...	En

x²_c = Σ (O_i - E_i)² / E_i

x²_c is then used against a reference value in the Critical values table. Look up a comparison value by the degrees of freedom and probability threshold

DF	>0.9	>0.95	>0.975	>0.99	>0.999
1	2.706	3.841	5.024	6.635	10.828
2	4.605	5.991	7.378	9.210	13.816
3	6.251	7.815	9.348	11.345	16.266
4	7.779	9.488	11.143	13.277	18.467
5	9.236	11.070	12.833	15.086	20.515
6	10.645	12.592	14.449	16.812	22.458

Familywise Error Rate

The probability of make one or more false discoveries (Type I errors) among all hypothesis when performing multiple hypothesis tests.

Fisher's Exact Test

Smaller data sets than Chi-squared.

Given:

a	b
c	d

 p = ((a+b)!(c+d)!(a+c)!(b+d)!) / (n!a!b!c!d!)

McNemar's Test for Matched Pairs

A Type of Chi-squared test. Takes 2x2 Chi data but computes X with

 x2 = (b-c)2 / (b+c)