Weighing and measuring a coin
Weight and diameter are the first two things you record. They do not tell you everything, but they tell you a great deal, and they cost nothing to check.
What you need
A precision balance reading to the centigram is enough to begin with; to the milligram, you see the differences better. Callipers, even plastic ones, give you the diameter to a tenth of a millimetre. The two together cost less than the cheapest coin in the catalogue.
Weigh the coin on its own, without capsule or paper. Measure the diameter in several places: ancient flans are rarely perfect circles, and the gap between the largest and smallest diameter is already telling you something about how the flan was prepared.
What to compare it with
Every denomination has an expected weight range, which varies by mint and by year. A denarius from the early third century runs around three grams, an antoninianus of Caracalla around five, a reformed aurelianus around four, a tetrarchic nummus between eight and ten depending on the date of issue.
Those ranges are wide, and rightly so: ancient coins were struck al marco, that is, by cutting a fixed number of flans from a pound of metal. The average weight was held to, the individual specimen was not. A slightly light coin is nothing to worry about; a coin twice too light or twice too heavy is.
What a discrepancy means
A markedly lower weight can come from heavy wear, from corrosion that has eaten into the metal, or from a flan deliberately clipped. A markedly higher weight, with a normal diameter, raises a question about the metal itself.
The ratio of weight to diameter often says more than either taken alone. A coin that is broad and light is probably thin: either a late issue at reduced module, or something else. That is the moment to look at the edge.