Last month I posted a brief summary of the troubles I was having with the de facto standard refractometer correlation for final gravity. Specifically, I found that it under-estimates FGs by, on average, about five “points”. More interesting, or at least more useful, I also found that the degree of the discrepancy is fairly well correlated with the degree of fermentation of the beer. By applying a logarithmic curvefit to the data, I was able to reduce the mean deviation of my (admittedly limited) dataset from 5.1 to 0.1 points, and the standard deviation from 2.2 to 1.2 points. This is comparable to the precision of a consumer-grade hydrometer, and I was pretty satisfied with that, but it wasn’t a particularly elegant solution.
What was really needed was a real, independent three-dimensional surface fit. Fortunately, I found a truly excellent site that did the heavy lifting for me: ZunZun.com. This is one of those transformative “cloud apps” that journalists keep telling us are so revolutionary, and you really ought to check it out. Anyway, equipped with data from twelve FG readings and a little free time today, I decided to see if I could get a reasonable equation together. The data I’ve collected so far is, I think, pretty representative: OGs range from 1.036 to 1.106, FGs from 1.007 to 1.022, and ADFs from 73% to 91%. It turns out there is actually a very good (R2 ≅ 0.98) fit, using a full cubic expansion:
FG = 1.0929176 – 0.0956887RIi + 0.160699RIf + 0.0103753RIi² – 0.00449931RIf² + 0.000585957RIi³ – 0.00911434RIf³ – 0.0165360RIiRIf – 0.00538394RIi²RIf + 0.0128988RIiRIf²
Where RIi and RIf are the initial and final refractive indices, respectively, in wort-corrected degrees Brix. The concern, of course, is that mapping 12 data points to 10 coefficients will result in substantial over-fitting. For the time being, I’m sticking with a simplified cubic form:
FG = 1.0111958 – 0.00813003RIi + 0.0144032RIf + 0.000523555RIi² – 0.00166862RIf² – 0.0000125754RIi³ + 0.0000812663RIf³
The mean deviation is, essentially, zero (10-15), with a maximum of 2.1 points, and the standard deviation is reduced to 0.98 points – still not as good as a quality hydrometer, but quite possibly acceptable to most brewers. In fact, you can reduce the polynomial quite a bit and retain accuracy; I’m reporting the simplified cubic because it can be quickly inserted into existing software by changing the coefficients. A basic linear fit has an SD of 1.1 points, with a maximum deviation of 2.0, and the added advantage of being able to be (quickly) worked out on paper:
FG = 1.00358522 – 0.00123861*RIi + 0.00380186*RIf
It’s worth noting that even this highly simplified equation is superior to the default correlation, although again, this is based on a set of only twelve points. Caveat calculator.
Finally, we come to what I thought was a pretty neat visualization of all this razzmatazz. I plotted the old-school (red), logarithmically corrected (yellow), new cubic fit (green), and simplified linear (blue) correlations against the expected FG values. A perfect fit would result in all the data points lying on the dotted line. As you can see, anything other than the stock correlation provides reasonably good results.
My FG/Attenuation/ABV spreadsheet has been updated to utilize the new correlation, and add the new visuals. If you find it useful and/or accurate, please drop me a line. I’d love to hear more brewers’ experiences with using refractometers to estimate FGs.
Update: 06 Aug 2010
I made a minor adjustment to the spreadsheet, changing the default wort correction factor to 1.02, which is what my own refractometer data support. The current version is now 2.1.
Update: 07 Apr 2011
I’ve tweaked the correlation and posted some results from other brewers, as well as an updated spreadsheet: Refractometer FG Results
fg_calculator_v2.1.ods | fg_calculator_v2.1.xls