Ralph Lucas
12-03-2011, 07:16 AM
I have a universe of points whose mean and median values are 50, and which has a broadly bell-shaped distribution of values between 0 and 100.
Local areas of the universe have means well away from 50.
In each area of the universe, comprising of 1,000 points or so, I have samples of between 1 and 100 points.
I want to produce a map, as best I can, of the known variation of local means within the universe. Unknown areas to be represented as having a value of 50.
Where I have one sample point, the right value for the map is clearly 50 - I have no evidence otherwise. When I have 100 sample points the probability that the sample average is the local average is high enough, given the nature of the measurements, to be assumed to be unity.
It would suit me to be able to bias the measured average of intermediate sample sizes towards the mean, depending on the sample size. I realise that in doing this I lose the ability to discriminate between an area that has a measured value of 52 and 99 data points and one that has a measured value of 70 but only 3 data points, but that suits my purposes well enough. I am only interested in mapping proven variation.
There are plenty of functions with the right sort of shape to use for biassing, but is there one that has some roots in statistical theory, and would thus be fairer than my choosing a shape based on gut feel?
Local areas of the universe have means well away from 50.
In each area of the universe, comprising of 1,000 points or so, I have samples of between 1 and 100 points.
I want to produce a map, as best I can, of the known variation of local means within the universe. Unknown areas to be represented as having a value of 50.
Where I have one sample point, the right value for the map is clearly 50 - I have no evidence otherwise. When I have 100 sample points the probability that the sample average is the local average is high enough, given the nature of the measurements, to be assumed to be unity.
It would suit me to be able to bias the measured average of intermediate sample sizes towards the mean, depending on the sample size. I realise that in doing this I lose the ability to discriminate between an area that has a measured value of 52 and 99 data points and one that has a measured value of 70 but only 3 data points, but that suits my purposes well enough. I am only interested in mapping proven variation.
There are plenty of functions with the right sort of shape to use for biassing, but is there one that has some roots in statistical theory, and would thus be fairer than my choosing a shape based on gut feel?