[wcoenen]

With 80% efficiency, you have to burn 1/0.8=1.25 units of fuel to get one fuel unit worth of useful heat. With 97% efficiency it's 1/.97≈1.03 units.

So if fuel prices and the amount of useful heat usage stayed the same, then your bill should be 1.03/1.25≈82% of what it was before. If your bill is "less than half" then something else must have changed.

[7786655]

He switched from burning oil to burning natural gas. According to the table, gas is way cheaper.

[lsllc]

Yes, I switched from an oil burner (#2 heating oil) to a high efficiency natural gas boiler. The other key ingredient is that the oil boiler was 20 years old and did not have "cold start" (so it always keeps itself at temp even in the summer).

Any modern boiler (gas or oil) will almost certainly be cold start, so it only fires when there's a call for heat. This alone probably saves a gallon of oil a day at least.