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I have 4 data points which I am using to interpolate a query point using bilinear interpolation. Each of the 4 data points is obtained from the average of several observations (typically 10-16 for each data point). So each has its own standard deviation as well. I am currently estimating the standard deviation of the interpolated query point using standard uncertainty propagation (variance formula - https://en.wikipedia.org/wiki/Propagation_of_uncertainty). I realize this is an (possibly crude) approximation, but it seems to work well for my application. (Please feel free to suggest better methods.)

My question is, once I have estimated the standard deviation of the interpolated value, what is the appropriate way to estimate 95% confidence interval of the interpolated value? I know how to compute confidence intervals for each of the known data points (using standard error, number of observations and t-distribution) but not for the interpolated point.

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