Statistics for Experimenters

Definitions

Accuracy

A measure of how close the experimental result is to the "true" value

Precisison

A meauser of close the result is determined without knowing the true vaule

Precision is often used to predict the accuracy of a quantity to be measured (you don't know the answer before doing the experiment)

Random Error

The error in a result due to the finite precision of an experiment

A measure of the statistical fluctuations which result after repeated experimentation

Systematic Error

Reproducable inaccuracies introduced by faulty equipment, calibration, or technique.

Probability Distributions

Binomial distribtuion

random, independent process with two possible outcomes

best example is a coin toss, its either heads or tails

mean ([math]\displaystyle{ \mu }[/math]) = number of tries [math]\displaystyle{ n }[/math](coin flips) * probability of success[math]\displaystyle{ p }[/math] (head, 1/2)

standard deviation([math]\displaystyle{ \sigma }[/math]) = [math]\displaystyle{ np(1-p) }[/math]

Poisson Distribution

standard deviation ([math]\displaystyle{ \sigma }[/math]) = root of the mean ([math]\displaystyle{ \sqrt{\mu} }[/math])

use in counting experiments

the distribtuion approximates the Binomial Distribution for the special case when the probability of the event occuring is small.

In the cosmic ray telescope experiment the mean number of detected cosmic rays is much smaller than the number of cosmic rays passing by.

The probability distribution is given as

[math]\displaystyle{ P(k,\mu) = \frac{\mu^{k}e^{-\mu}}{k!} }[/math]

where

[math]\displaystyle{ k }[/math] = number of occurances

[math]\displaystyle{ \mu }[/math] = mean

Gaussian/Normal Distribution

Full WIdth at Half Max (FWHM) = width of the distribution at half the value of the maximum probabilty (distibution peak) = [math]\displaystyle{ \Gamma }[/math]

standard deviation ([math]\displaystyle{ \sigma }[/math]) = [math]\displaystyle{ \frac{\Gamma} {2 \sqrt{2 \ln 2}} = \frac{\Gamma} {2.354} }[/math]

error = [math]\displaystyle{ 0.675 \sigma }[/math]