Is it true that for an unbiased estimator, the mean of the sampling distribution is very close to, but not always equal to, the true value of the parameter being estimated?

My textbook says that "An unbiased statistic will sometimes fall above or below the true value of the parameter ... because its sampling distribution is centered at the true value, however, there is no systematic tendency to overestimate or underestimate the parameter"; but doesn't this contradict the definition that the expected value of an unbiased estimator is equal to the value of the parameter?

A sample problem:

At a university, students spend an average of 46 minutes per day on Facebook. If all possible SRS of students of size $n=7$ are chosen and a sampling distribution of the sample means is constructed, we would expect the center of that sampling distribution to be ____ 46 minutes.

Would you say exactly equal, or only approximately?


The idea behind sampling is that we look at an affordable set of observations to make projections about the whole population, which is of course not affordable to collect data on.

In the question, there is a hypothetical scenario, i.e. the case when ALL the samples of size $n=7$ are collected and then the sampling distribution is analyzed. In that case, the center of this distribution will be EXACTLY equal to the true value. We know that some of the samples will overestimate, some of them will underestimate. And the reason that we are interested in this hypothetical scenario is to figure out how we can quantify these errors and make a judgement about the validity of our projections. You will never collect all of the $n=7$ samples, you will collect one, but asking the question of what would have happened if we were to collect all of those samples would give you a measure as to whether you would reject some projections or not.


By definition, an estimator $X$ for a parameter $\theta$ is unbiased if and only if the mean of the sampling distribution for $X$ is exactly $\theta$. All the textbook is saying is that sometimes the estimated value you get will be above $\theta$ and sometimes below, but the average will be $\theta$.


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