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Is there any way to calculate a confidence interval (or otherwise gauge the reliability of a sample) when you have a sample size of one and you don't know the population standard deviation?

I work at a farm. We sample our soil yearly for analysis and fertilize recommendations. For each relatively homogeneous field, the standard procedure is to take a number of samples, mix them together, and send a total of 1 lb of soil to the lab. If the field is 5 acres, that would be 1 lb of soil out of $\approx$ 10 millions.

The results would say that we have say 3 lbs of phosphorus per acre. Fertilizer recommendations would then be something like: For < 2 lbs phosphorus per acre, add 75 lbs of rock phosphate; for 2-4 lbs per acre, add 65 lbs; etc...

I've never been totally comfortable with this procedure, as I can't figure out how to put a confidence interval around the results. Is there a way to do so?

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Yes, there is such a confidence interval (assuming normality), but it will be very wide. I don't have ready access to the reference, but if no one supplies it, I will give it tomorrow. – soakley Apr 6 '14 at 14:28

For a normal population and a sample point $x$, a 90% confidence interval is given by $$x \pm 4.84 \ \left| x \right| $$ For 95% confidence, the interval is $$x \pm 9.68 \ \left| x \right| $$

The reference is: "An Effective Confidence Interval for the Mean with Samples of Size One and Two," Wall, Boen, and Tweedie, $\it{The \ American \ Statistician},$ 2001, vol. 55, Iss. 2, pp. 102-105

If you have data from previous years, you might do better trying to estimate the variance using historical data.

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Interesting. Many thanks. The problem with using prior year data is that the population mean will certainly change from year to year, depending on how much and what kind of fertilizer is used, what crops are grown on the soil, and so forth. What I need to do is call my soil testing lab and ask if they have ever run a series of multiple samples on the same field taken in the same year. – user1148928 Apr 9 '14 at 1:49

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