Statistics for Experimenters: Difference between revisions
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;Systematic Error | ;Systematic Error | ||
: Reproducable inaccuracies introduced by faulty equipment, calibration, or technique. | |||
=Probability Distributions= | =Probability Distributions= | ||
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: standard deviation (<math>\sigma</math>) = root of the mean (<math>\sqrt{\mu}</math>) | : standard deviation (<math>\sigma</math>) = root of the mean (<math>\sqrt{\mu}</math>) | ||
: use in counting experiments | : use in counting experiments | ||
: the distribtuion approximates the Binomial Distribution for the special case when | : 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. | : 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>P(k,\mu) = \frac{\mu^{k}e^{-\mu}}{k!}</math> | |||
where | |||
: <math>k</math> = number of occurances | |||
: <math>\mu</math> = mean | |||
;Gaussian/Normal Distribution | ;Gaussian/Normal Distribution | ||
: Full WIdth at Half Max (FWHM) = width of the distribution at half the value of the maximum probabilty (distibution peak) = <math>\Gamma</math> | : Full WIdth at Half Max (FWHM) = width of the distribution at half the value of the maximum probabilty (distibution peak) = <math>\Gamma</math> | ||
:standard deviation (<math>\sigma</math>) = <math>\frac{\Gamma} {2.354}</math> | :standard deviation (<math>\sigma</math>) = <math>\frac{\Gamma} {2 \sqrt{2 \ln 2}} = \frac{\Gamma} {2.354}</math> | ||
:error = <math>0.675 \sigma</math> | :error = <math>0.675 \sigma</math> | ||
Latest revision as of 21:26, 20 September 2007
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]