Most calculators use a simplified formula for calculation of ABV which is (OG - FG) * 131.25 = %ABV.

The actual calculation is a bit more complex and is described in the paper: Examination of the Relationships Between Original, Real and Apparent Extracts, and Alcohol in Pilot Plant and Commercially Produced Beers published by Anthony J. Cutaia, Anna-Jean Reid and R. Alex Speers in the Journal of the Institute of Brewing.

Calculating ABV from specific gravity

The standard method of estimating ABV is to use the specific gravity. Specific gravity is the relative density of a liquid compared with the density of water. When fermentation occurs, the specific gravity of the liquid changes. You can monitor this change to find the alcohol content of your homebrew.

All formulas for finding the ABV are approximate, as they are found empirically rather than derived. The most recent formula was derived by Cutaia, Reid and Speers in 2009. The first formula from their research relates the measurements of the alcohol content by weight with the original and apparent extract:

ABW =(0.372 + 0.00357 * OE) * (OE - AE)

where:

ABW is the alcohol by weight, measured in percents

OE is the original extract (before fermentation), measured in Plato degrees (?P)

AE is the apparent extract (after fermentation), measured in Plato degrees (?P)

From this alcohol formula, they derived an equation for alcohol by volume:

ABV = ABW * (1.308 * 10^-5 + 3.868 * 10^-3 * AE + 1.275 * 10^-5 * AE^2 + 6.3 * 10^-8 * AE^3 + 1) / 0.7907

The last thing our beer calculator does is finding the exact amount of alcohol in a given volume of alcoholic beverage. For example, you can determine the volume of alcohol in a 500 ml bottle of 4.5% beer. If you are interested in finding this value, use this equation:

alcohol volume = total volume * ABV

Using this calculation, I got the following results based upon your posted examples:

Example 1: OG 1.077 > FG: 1.017 ==> 8.08% (BS = 8.0%)

Example 2: OG 1.102 > FG 1.029 ==> 10.08% (BS = 9.9%)

Given rounding errors, this is not too far off the mark.

I also do not know what the other calculators are using for their computations.