# Understanding the Importance of "Sufficiency" within Statistics

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" . "

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?

• Does this answer your question? Sufficient statistic, specifics/intuition problems Sep 23 '21 at 12:43
• 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. Sep 23 '21 at 12:44

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 $$n$$.