Electric heat is 100% efficient. Natural gas is anywhere between 80-97% efficient depending on your model of furnace. Convert therms of gas[0] and kWh[1] to BTUs, and multiply by efficiency and price to determine which is cheapest[2]. Usually natural gas wins.
For emissions it’s more complicated. You need to look up the energy source for your grid[3], it will vary a bit seasonally, and factor in transmission losses for electric heat. You can then calculate the emissions per BTU of natural gas heat vs. the emissions of the grid to give you a BTU of electric heat. For most grids, natural gas still wins.
Where this gets complicated is if you use a heat pump, either a “mini split” (common in Europe and some new apartments) or a ground source (“geothermal”) heat pump. These have a seasonally adjusted efficiency rating[4], usually between 2 and 4. This means per unit of electrical energy the unit is moving 2-4 units of heat (depending on model and exterior conditions). You can take the electric heating number from above and divide by this number to figure out the cost and emissions for a heat pump. Usually a heat pump will win on emissions and lose on price (including installation costs) because natural gas is very very cheap.
0 - A “therm” of gas is the traditional measuring and billing unit of natural gas in the United States, it’s equivalent to roughly 100,000 BTUs. Your bill will probably be in therms, but cubic feet are trivially convertible to BTUs too.
1 - 1 kWh is 3,412 BTUs
2 - You can either compare per BTU costs, or your estimated monthly cost based on your heating need. The latter is important if you have tiered or seasonal electricity pricing. To figure this out take a sample monthly gas bill from winter, subtract your summer gas bill (if you have gas hot water) and use the calculations above to determine your heating needs in BTUs. This lets you calculate your electricity needs in kWh and therefore costs.
3 - It’s important that this is a seasonally correct. Most grids get dirtier as more polluting and less efficient power plants are turned on for summer AC, often coal power plants in most regions. Therefore the emissions per kWh will vary on a month by month basis for a given grid.
4 - If we’re being very precise, a non seasonally adjusted number would be best here. An air sourde Heat pump will struggle to extract heat from very cold exterior air, reducing its efficiency. A ground source one will be more efficient, given that ground water is warmer than the air typically.
On a somewhat recent decently high gas usage month that emulated a cold winter (forgot to turn off the pool heater from spa mode on the whole pool circulation mode for a few days...oof) my $/CCF (Centi-Cubic Foot) was 1.22. Converting $/CCF to $/MMBTU, divide by 1.037 to get ~$1.176/MMBTU.
Electricity is usually billed at kWh pricing. 1MMBTU = ~293kWh. I pay ~$0.086/kWh, so 293 * .086 = $25.198/MMBTU.
Obviously, you're not getting 100% efficiency with the natural gas furnace. I think the one I have in my attic which is almost a decade old is rated at like 85% efficient. 90% efficient furnaces are available, some of them don't even require a metal chimney as the resulting air isn't even that hot.
But even at only 85% efficiency, you're still taking more than 10x less dollars per heat.
This math changes when thinking about buying new equipment and other factors. I haven't looked at equipment pricing so it would be hard for me to say how expensive it would be compared to alternatives. This can also vary a good bit from your local market. Energy is stupid cheap in my area, natural gas is a good bit cheaper here than a good chunk of the US while electricity prices are about median, maybe a bit lower.
What calculations are these? How do I figure out if it's worth it?