5 ms·
I am wary of trying to contribute to this (a little knowledge is a dangerous thing) but hopefully someone can correct my mistakes.(And what better way for me to
by alex_stoddard 17y ago
I am wary of trying to contribute to this (a little knowledge is a dangerous thing) but hopefully someone can correct my mistakes.(And what better way for me to learn...)
A typical controlled science experiment is designed to take measurements of multiple groups where one variable is different between the groups and others are controlled. We wish to see if the variable of interest has an effect.
Therefore the commonest statistical test is to determine if the mean value of the groups are different (t-test for two groups, anova for multiple groups) e.g. does mean blood pressure increase when on a high salt diet.
If I understand the CLT (big if) then the distribution of the _mean_ of a sample is, by the CLT, going to be Gaussian, regardless of the distribution from which the actual measurements are drawn. i.e. for comparing group means it it doesn't matter if my data is sampled from Guassians or not.
Of course that leads to the question of if a significant difference in group means is really relevant in a given context.
- fburnaby 17y agoThis is also only true if the distribution of each sample is the same. If you don't understand the system you're sampling from, or where the variability comes from (maybe its systematic), then all bets are off.
- joe_the_user 17y agoNot even necessarily true then - you also need each sample to have finite mean and variance...
- lutorm 17y ago"If I understand the CLT (big if) then the distribution of the _mean_ of a sample is, by the CLT, going to be Gaussian" Yes, for an infinite number of samples. The rate at which is converges to a Gaussian, though, is strongly dependent on the distribution from which the measurements are drawn.
- joe_the_user 17y agoActually, not even this is not true! The Central Limit Theorem applies only to samples which have finite mean and variance (see: http://en.wikipedia.org/wiki/Central_limit_theorem http://en.wikipedia.org/wiki/Central_limit_theorem). Take a distribution which has infinite variance or mean and you can wind-up instead with one of the fractal distributions which Mandlebrot studied. see http://en.wikipedia.org/wiki/Stable_distribution http://en.wikipedia.org/wiki/Stable_distribution Distributions with infinite variance or mean are more common than one might imagine. Some might argue the stock market would qualify.
- lutorm 17y agoGood point. Distributions with infinite variance are less common in physics, so that's normally not something we have to worry about.
- joe_the_user 17y agoYou need an infinity of samples to be sure. But moreover, in most situations, the other variables cannot be absolutely controlled but rather are only roughly controlled. How roughly depends on the field and is often the rub.
- alex_stoddard 17y agoAbsolutely true, but this is more a criticism of experiment design than statistical methodology. Also it is bias in uncontrolled variables that is most concerning. (More than uncontrolled noise although that will reduce the power to detect real differences).