a sufficient statistic
When you have a big pile of data but only care about one parameter, it would be wonderful to compress the whole dataset into a few numbers without losing any information that matters for that parameter. A sufficient statistic is exactly such a lossless summary: a function of the data that captures everything the data can tell you about the unknown parameter — once you know it, the raw data have nothing more to add.
The precise idea is captured by conditioning. A statistic T(data) is sufficient for a parameter theta if the conditional distribution of the full data given T does not depend on theta at all. In words: once you are told the value of T, the leftover randomness in the data is just noise unrelated to theta, so the data carry no extra clue about theta beyond T. The practical test is the Fisher-Neyman factorization criterion: T is sufficient exactly when the likelihood factors as p(data given theta) = g(T(data), theta) times h(data), where the parameter theta touches the data only through T. For n independent Bernoulli trials, the total number of successes is sufficient for the success probability — the specific order of heads and tails is irrelevant. For normal data with unknown mean and variance, the sample sum and sum of squares together are sufficient.
Sufficiency is the formal backbone of data reduction. It tells you a maximum-likelihood estimate or a Bayesian posterior can depend on the data only through a sufficient statistic, so you may store and transmit just T and lose nothing. The Rao-Blackwell theorem even shows that any estimator can be improved (its variance reduced) by conditioning on a sufficient statistic. The honest caveat: sufficiency is always RELATIVE to a specific model. The success count is sufficient for the Bernoulli success probability, but if you secretly doubt the trials are identically distributed, the discarded order suddenly matters again — throwing away 'irrelevant' detail is only safe if your model is truly the right one.
Flip a coin 100 times to estimate its bias. The data is a length-100 sequence of H's and T's, but the count of heads alone is sufficient: HHTTH... and THHTH... with the same total of, say, 58 heads carry identical information about the bias. You can throw away the order, keep just '58 out of 100', and lose nothing relevant to estimating p.
A sufficient statistic squeezes the data to its informative core — for one specific model.
Sufficiency is always relative to an assumed model; the discarded detail is only truly irrelevant if that model is correct, so a sufficient statistic can hide the very evidence that would reveal the model is wrong.