To compare two areas, put quadrats at random coordinates: lay two tape measures at right angles and use random numbers for the two distances. You do not choose, so there is no bias. Use plenty of quadrats and take a mean.

number per m² = mean per quadrat ÷ area of one quadrat
population ≈ number per m² × total area

1. 8 quadrats, each 0.5 m × 0.5 m = 0.25 m²; mean 3.5 buttercups per quadrat
2. 3.5 ÷ 0.25 = 14 per m²
3. field 60 m × 30 m = 1800 m², so about 14 × 1800 = 25 200 buttercups

A 0.5 m quadrat is not 0.5 m²: square the side.

Some plants are hard to count, so record percentage cover: in a quadrat of 25 squares, each square is 4%.

Where conditions change steadily (from a wood into a field, up a beach), lay a tape along the change and sample with quadrats at regular distances: a belt transect. Measure an abiotic factor at each point too, such as light with a light meter, and look for a pattern.

A pattern with light does not prove light is the cause: soil moisture and temperature change too.

For animals that move: catch some, mark them harmlessly, release them, then catch again later.

population ≈ (number marked first × number caught second time) ÷ number of marked animals in the second catch

1. 40 woodlice marked; second catch 30, of which 6 are marked
2. 40 × 30 ÷ 6 = 200 woodlice

It assumes the marked animals mix back in evenly, are just as likely to be caught again, and that no animals are born, die, arrive or leave between catches. If marks rub off, fewer marked animals turn up, so the estimate is too high.