If a trail is listed as 1500m of elevation change, what exactly does that mean?
For example, if the trail goes up 500m, then down 200m, then up 100m, then down 300m, then up 400m, is that considered a 1500m (500+200+100+300+400) elevation change trail?
Or would it be 500m elevation change, since the end is 500m above the start?
There are three different ways of describing elevation gain/change for hikes.
The least useful – and sadly a very common – method is simply to subtract the starting elevation from the ending elevation. This should be called the net elevation gain and mathematically is identical to the sum of the gains, 1000m in your example, minus the sum of the losses, 500m, yielding a net gain of 500m. Since many hikes are round-trip hikes, returning to the starting point, the net gain for the entire hike is always zero. The lazy guidebook writer or trail website contributor will use this method but it has limited value.
The most useful method is to provide the total gains and total losses for any hike. For a one-way hike, your example has a total gain of 1000m and a total loss of 500m. For a round-trip – and assuming the return trip follows the same route – your total gain is 1500m and the total loss is 1500m. Yes, the overall net gain is zero but you have gone up and down a lot. Trail descriptions that go to the trouble of summing all the ups and downs are very beneficial.
Of intermediate use is the "wikipedia" method that Charlie cites: only quote the total gains and ignore the total losses. For a one-way hike, your example has a total gain of 1000m. For the return trip, assuming the same route, you now need to sum what were losses (500m) but are now gains. So your round-trip has a total gain of 1500m. The only use for this method that I can think of is for backcountry skiing where the uphill is hard work and the downhill is all fun and games.
Of course, if the net gain (1000m) and total gain (500m) are both given, then the total loss (500m) is found by simple subtraction.
You almost have it, you don't count the downhills.
In running, cycling, and mountaineering, cumulative elevation gain refers to the sum of every gain in elevation throughout an entire trip. It is sometimes also known as cumulative gain or elevation gain, or often in the context of mountain travel, simply gain. Elevation losses are not counted in this measure. Cumulative elevation gain, along with round-trip distance, is arguably the most important value used in quantifying the strenuousness of a trip.
So in your example, it would be 500+100+400 = 1000m of elevation gain but a change in elevation of 500m.
Sometimes on say the Keyhole route up Longs Peak there is very little difference between the elevation gain and elevation change since you are going up the whole time. Personally, when I do it, I focus on just once hill at a time and so the difference in that situation is negligible.
I use total elevation change -- where down is also counted as well as up. I find that downhill (especially steep downhill) also slows down horizontal travel, so when estimating time for a trail, I count 1000 ft elevation change up or down as the equivalent of an extra mile horizontally.
This is an oversimplification. Down doesn't slow you as much, generally, and you substitute sore knees for out of breath. Exception: Steep descents on irregular footing -- slopes of large scree are an example -- can be slower going down than going up.
Indeed there is a 'magic slope' a couple percent grade, going down, that seems to be just enough to overcome friction, and you cruise seemingly without effort.
To count total elevation change: Count the number of contour intervals your route crosses or touches. This will be an underestimate. Even a 100 foot interval hides a lot of topographic nuance. However contour counting is quick, and it's fairly consistent, which is all you really care about.
