# What is the intuitive sense behind the purpose and mechanics of Sufficient Statistics?

The definition of a sufficient statistic is: Let $X_1,...,X_n$ be a random sample from a distribution indexed by a parameter $\theta$. Let $T$ be a statistic. Suppose that, for every $\theta$ and every possible value $t$ of $T$, the conditional joint distribution of $X_1,...,X_n$ given that $T=t$ depends only on $t$ but not on $\theta$. Then, $T$ is a sufficient statistic for parameter $\theta$.

I feel like I know several pieces of the puzzle (like the factorization theorem) to understanding sufficient statistics but do not have the overall theory down.

My main questions are:

1) Why do they say that $T$ is a sufficient statistic for parameter $\theta$? If $\theta$ were the population mean of a normal distribution, say $\mu$, does it mean that anytime we want to find the probability of, say, $X_1,...,X_n$ occurring in a certain way, that we don't need the value of the mean of the population?

2) In real-life, why do we want to use a sufficient statistic? Is seems that just calculating the statistic shouldn't be that much work (such as the sum of X's) so why do we need it?

Thanks!

1. No. What they say is if $X_1^\prime,\dots,X_n^\prime$ is another random sample from the same population as the original data $X_1,\dots,X_n$, it contains an equal amount of probabilistic information about $\theta$. Therefore, we can "recover the data" if we retain $T$ and discard $X_1,\dots,X_n$. That’s why $T$ is "suﬃcient".
2. Data reduction. If $T$ is suﬃcient, the "extra information" carried by $X$ is worthless as long as $θ$ is concerned. It is then only natural to consider inference procedures which do not use this extra irrelevant information. This leads to the Suﬃciency Principle: Any inference procedure should depend on the data only through suﬃcient statistics.