Alcohol by Volume from Gravity

Fermentation converts sugar into roughly equal molar parts ethanol and carbon dioxide. The CO₂ leaves, so the wort loses mass; and the ethanol that stays behind is lighter than the water it is dissolved in, so the wort loses density twice over. The drop in gravity from original to final therefore encodes how much alcohol was made, and $ABV = (OG - FG) \times 131.25$ is the straight line drawn through that relationship.

The 131.25 is not derived from anything. It is a fitted number, and the fit is to gravity in SPECIFIC GRAVITY units — feed it gravity points and the answer is off by a factor of a thousand, feed it degrees Plato and it is off by roughly four. It is also a straight line fitted to a curve, which means it is right in the middle of its range and wrong at both ends. Below about 4% it reads slightly low; above about 10% it reads noticeably high, and by 15% it can overstate by half a percentage point or more. Mead and barleywine brewers who use it and then wonder why their beer does not taste as strong as the arithmetic claims have found the edge of the fit.

Better forms exist and are worth knowing about. The polynomial published by the Alcohol and Tobacco Tax and Trade Bureau, and the alternative fit that divides by $(1.775 - SG)$, both track the curve where the linear version wanders. All of them share one limitation that no arithmetic can fix: they infer alcohol from density, and anything else that changes density changes the answer. Add sugar mid-fermentation, blend two batches, or top up with water, and the gravity readings no longer describe one clean fermentation. The only measurement that is genuinely independent of all that is distillation, or an NIR analyser, which is why every commercial brewery that has to print a number on the label owns one.

Two practical cautions. Take both readings at the same temperature or correct for it, because a hydrometer calibrated at 20 °C reads low in warm wort and the error goes straight into the difference. And be honest about ABV versus ABW: alcohol by volume is about 1.25 times alcohol by weight, and the older North American convention of quoting beer by weight is why some historical strength figures look surprisingly modest.