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I am trying to better understand what it means to be a "sufficient statistic".

"In statistics, a statistic is sufficient with respect to a statistical model and its associated unknown parameter if "no other statistic that can be calculated from the same sample provides any additional information as to the value of the parameter" . "

Source: https://en.wikipedia.org/wiki/Sufficient_statistic

Based on this definition, here is how I conceptualize it:

In a unimodal distribution, the "mean" provides more information about the unimodal distribution - relative to the amount of information provided by the "mean" in a biomodal distribution. Thus in this case, the "mean" is more of a sufficient statistic in a unimodal distribution compared to a biomodal distribution.

For example, in a distribution of basketball player heights that is unimodal - the "mean" would well describe the average height of a basketball player. But if you were to have a distribution of heights corresponding to penguins, ostriches and giraffes - and this distribution was not unimodal, it is more likely that the "mean" would not sufficiently characterize the height of the average measurement in this sample.

Is this interpretation correct?

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    $\begingroup$ Does this answer your question? Sufficient statistic, specifics/intuition problems $\endgroup$
    – Xi'an
    Sep 23 at 12:43
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    $\begingroup$ Sufficiency has a very clear definition in terms of Fisher information. It also very rarely takes place, the "very rarely" being wrt the range of statistical models. $\endgroup$
    – Xi'an
    Sep 23 at 12:44
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The term 'sufficient statistic' is pretty theoretical (i.e., it would not typically be taught in applied stats classes) and I am not sure you are hoping for a theoretical answer.

But, what you have above (about normal and binomial) is a bit off. Although it is a theoretical term, the general idea of it is not that complicated (actually way simpler probably than you are thinking about it).

This is the nutshell example (with small loss of preciseness to make the explanation simple). If your goal is to estimate the mean of a population for any distribution, and you have a random sample $y_1,y_2,\ldots,y_n$ from that distribution, then a (one of many) sufficient statistic for estimating the population mean the sum of the $n$ numbers and the sample size $n$. All that means is that if you know those two numbers (the sum + sample size) then you do not need to know any other information about the $n$ numbers to estimate the population mean.

As a second example, if your goal is to estimate the variance of any population (= distribution), then if you know the values of $n$, the sum of $y_1,y_2,\ldots,y_n$ and also the sum of the squared values $y_1^2,y_2^2,\ldots,y_n^2$ then you do not need to know any other information about the $n$ numbers to estimate the population variance. This is because the sample variance can be written as a function of $\sum_{i=1}^ny_i$ and $\sum_{i=1}^ny_i^2$ and $n$.

Again, if you are looking for a more theoretical answer, this is not it.

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