Re: Estimating variance with limited data?

Dan Bolser wrote:
> Hello,
>
> I am a real beginner in stats, so sorry for the potentially naïev
> question (and badterminology)!
>
> I have a series of 'experiments' each with a few observations within the
> range of discreet values...
>
> experiment A, results 1, 2, 1, 4, 3, 1, 4
> experiment B, results 5, 2, 4, 4, 3, 6, 4, 5, 6, 7
> experiment C, results 1, 9, 3, 4
>
> Each 'experiment' has a ranking, and when I plot the mean value for each
> set of results for each experiment, I think I can see a clear trend.
>
> What I think I want to do is to estimate the variance of the results in
> each experiment, so I can see what trends 'fit' the data. I think if I
> give each point (mean) some error bars, I can see if my trend is supported
> or not.
>
>
> I used bootstrapping to estimate the variance of the mean value for each
> experiment. However, I have a problem, some experiments have results that
> only take one value. How do I estimate the variance in these cases?
>
> I want to do that to match my intuition that
>
> experiment X, results 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
>
>
> has less variance than
>
> experiment Y, results 1, 1, 1, 1, 1
>
>
> How should I present the data, given that I want to show an average value
> with a variance?
>
> I think perhaps I can do something tricky like add the mean value for the
> whole results set to each experiment and recalculate the estimated
> variance, but can I do that?
>
>
> If it helps I can send my real data.
>
> Thanks for any clues for how to advance my analysis,
>
> All the best,
> Dan.

Your intuition is confusing "size of effect" with "strength of
evidence". Both X and Y have exactly the same variance -- zero.
It would be wrong to say that X has less variance than Y.
However, because X has more data than Y, the p-value of the
zero for X is smaller than the p-value of the zero for Y,
and the zero for X is therefore stronger evidence than the
zero for Y against a claim that the true variance is some
nonzero value.

In degenerate cases such as this, the point estimates of the
variances of X and Y will both be zero. On the other hand, the
interval estimates will differ, with the upper limit of the
X-variance being lower than the upper limit of the Y-variance.
How you get those intervals depends on the general form of the
distributions from which the data were sampled.

.

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