Statistics
- mean
- x_bar = 1/n Σ xi
- 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 HA 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 |
x2c = Σ (Oi - Ei)2 / Ei
x2c 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 |
also...
φ = ( x2/n )0.5
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
Given:
| a | b |
| c | d |
x2 = (b-c)2 / (b+c)
Proportions
For H0: p = π0, H1 p > π0
Z = (p-π0) / [ (π0(1-π0)) / n ]0.5
- p: proportion observed
- π0: proportion expected (being tested)
- n : sample size
- Z is single tailed
For H0: p1 = p2, H1: p1 ? p2
Z = (p1 - p2) / [ (pp(1-pp))*(1/n1 + 1/n2) ]
- pp: Pooled proportion. Total passes over population.
- Z is two tailed
This is useful for paralytics, when comparing two different populations.
Student's T Test
t = n^.5 * (x - μ) / s
- x: sample mean (really should be x-bar)
- μ: population mean
- s: sample standard deviation
- n: sample size
Confidence Interval
CI = x ± ( t α/2 , df)(s / n0.5)
- x: mean (x bar)
- t: this needs to be looked-up in a table
- s: standard deviation
- n: sample size