# What is an estimator and how to construct it?

The definition of an estimator "rule that tells how to calculate an estimate " as given here is not clear to me. If I make measurements of some quantity, say age in a group of N people, my outcomes will be $$x_1, \dots, x_N$$ where $$x_i$$ represents the age of $$i$$-th person. One can now talk about the mean as the quantity that gives us "rough idea" about the age of this group.

I want to understand how is the notion of estimator relevant to a situation like above (if at all it is)? What the estimator actually is and how to construct it?

• stats.stackexchange.com/questions/47728/…
– whuber
Commented Feb 16, 2022 at 19:17
• There are many ways to answer this question. Can you be more specific? Judging from how you express your sample using notation, it's not difficult to express the sample mean $\bar{x}_n = \sum_{i=1}^n x_i/n$ and that it converges to the true mean $\mu$ by SLLN. Tukey's book "Data Analysis and Statistics" defines an estimator as "a function of the observations" which "can be expressed arithmetically or as a process such as sorting the data and taking the sample median". (it turns out the median can be expressed algorithmically as well) Commented Jul 22 at 17:31
• One illuminating case is considering the sample mode for a continuous distribution like participant age. You can plot the data in a histogram and choose the highest bar, but in that case the the choice of bins affects what you call the "mode" and other binnings can produce massively discrepant esitmates. Commented Jul 22 at 17:32