Statistics: Difference between revisions

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To convert a normal distribution to a Z distribution
To convert a normal distribution to a Z distribution
   Z = (x - μ) / δ
   Z = (x - μ) / δ
Standard deviation is the root of the sum of the square errors:
  &delta; = ( &Sigma; (i - &mu;)<sup>2</sup> )<sup>.5</sup>


== Binomial Distribution ==
== 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 * p<sup>k</sup>(1-p)<sup>n-k</sup>
  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 * .5<sup>2</sup>(1-.5)<sup>6-2</sup> = 15 * .25 * .5<sup>4</sup> = 15 * 0.015625 = 0.234375
== Hypothesis Testing == 
Null hypothesis h<sub>0</sub>. Alternate hypothesis H<sub>A</sub> or H<sub>1</sub>
Single Tail Test
  H<sub>0</sub> : &mu;<sub>1</sub> = &mu;<sub>2</sub>
  H<sub>A</sub> : &mu;<sub>1</sub> > &mu;<sub>2</sub>
Two Tailed Test
  H<sub>0</sub> : &mu;<sub>1</sub> = &mu;<sub>2</sub>
  H<sub>A</sub> : &mu;<sub>1</sub> ? &mu;<sub>2</sub>
=== Chi-Squared ===
Given:
{| class="wikitable"
|
|DF1
|DF2
|...
|DFn
|-
|Observed
|O<sub>1</sub>
|O<sub>2</sub>
|...
|O<sub>n</sub>
|-
|Excpected
|E<sub>1</sub>
|E<sub>2</sub>
|...
|E<sub>n</sub>
|}
x<sup>2</sup><sub>c</sub> = &Sigma; (O<sub>i</sub> - E<sub>i</sub>)<sup>2</sup> / E<sub>i</sub>
x<sup>2</sup><sub>c</sub> 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
{| class="wikitable"
| 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...
  &phi; = ( x<sup>2</sup>/n )<sup>0.5</sup>
=== 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:
{| class="wikitable"
|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:
{| class="wikitable"
|a
|b
|-
|c
|d
|}
  x<sup>2</sup> = (b-c)<sup>2</sup> / (b+c)
=== Proportions ===
For H<sub>0</sub>: p = &pi;<sub>0</sub>, H<sub>1</sub> p > &pi;<sub>0</sub>
  Z = (p-&pi;<sub>0</sub>) / [ (&pi;<sub>0</sub>(1-&pi;<sub>0</sub>)) / n ]<sup>0.5</sup>
* p: proportion observed
* &pi;<sub>0</sub>: proportion expected (being tested)
* n : sample size
* Z is single tailed
For H<sub>0</sub>: p<sub>1</sub> = p<sub>2</sub>, H<sub>1</sub>: p<sub>1</sub> ? p<sub>2</sub>
  Z = (p<sub>1</sub> - p<sub>2</sub>) / [ (p<sub>p</sub>(1-p<sub>p</sub>))*(1/n<sub>1</sub> + 1/n<sub>2</sub>) ]
* p<sub>p</sub>: 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 - &mu;) / s
* x: sample mean (really should be x-bar)
* &mu;: population mean
* s: sample standard deviation
* n: sample size
=== Confidence Interval ===
  CI = x &plusmn; ( t <sub> &alpha;/2 , df</sub>)(s / n<sup>0.5</sup>)
* x: mean (x bar)
* t: this needs to be [http://easycalculation.com/statistics/t-distribution-critical-value-table.php looked-up] in a table
* s: standard deviation
* n: sample size


[[Category:Notes]]
[[Category:Notes]]

Latest revision as of 18:46, 15 October 2013

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
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