How do we know radiometric dating is accurate
However, by creating a “map” of carbon-14 production rates over time we can take these difficulties into account. There’s a quantity called the “variance”, written “σ” or “Var(X)”, that describes how spread out a random variable is. So, for a die, If you have two random variables and you add them together you get a new random variable (same as rolling two dice instead of one). This property is a big part of why variances are used in the first place.
Still, the difficulties aren’t to be found in the randomness of decay which are ironed out very effectively by the law of large numbers. It’s why, for example, large medical studies and surveys are more trusted than small ones. The average also adds, so if the average of one die is 3.5, the average of two together is 7.
Very radioactive isotopes decay all the time, so their half-life is short (and luckily, that means there won’t be much of it around), and mildly radioactive isotopes have long half-lives.
Now, say the isotope “Awesomium-1” has a half-life of exactly one hour.
So a deviation of 1% is 20,000 standard deviations, which translates to a chance of less than 1 in 10.
If you were to see a 1% deviation in this situation, take a picture: you’d have just witnessed the least likely thing anyone has ever seen (ever), by a chasmous margin.
More recently, the nuclear tests in the 50’s caused a brief spike in carbon-14 production.
If you start with only 2 atoms, then after an hour there’s a 25% chance that both have decayed, a 25% chance that neither have decayed, and a 50% chance that one has decayed.
So with just a few atoms, there’s not much you can say with certainty.
This isn’t a mysterious force at work; there are just more ways to get a 7 (, , , , , ) than there are to get, say, 3 (, ).
The more dice that are rolled and added together, the more the sum will tend to cluster around the average.