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## AP®︎/College Statistics

### Unit 9: Lesson 3

Biased and unbiased point estimates# Sample statistic bias worked example

AP.STATS:

UNC‑3 (EU)

, UNC‑3.H (LO)

, UNC‑3.H.5 (EK)

, UNC‑3.I (LO)

, UNC‑3.I.1 (EK)

Determining bias of sample statistics based on approximate sampling distributions example.

## Want to join the conversation?

- is sample median as good a estimator as sample mean? or one is better than the other?(3 votes)
- Yes because it helps you determine if it's bias or unbias(2 votes)

- What about the mean of the sampling distribution of medians ? shouldn't it be a better estimation of bias ?(3 votes)
- So will the term "sampling distribution" be only applied to the population parameter when all the samples are indicated? For example, will the dot plot above just be an approximation of the sampling distribution and not the sampling distribution itself?(3 votes)
- agree it will be above just be an approximation of the sampling distribution(1 vote)

- if i don't know the parameter, how do i know if the distribution is biased or not? and if i already know the parameter, why would i need to estimate it?(2 votes)
- How much skewed should it be to become biased?(1 vote)
- Does the unbiased estimator refer to the mean or the sampling distribution in general? For example, would I say "the mean of this sampling distribution is an unbiased estimator" or would I say "this sampling distribution is unbiased" or is mean always an unbiased estimator?(1 vote)
- I don't think I understand what it means to have a biased estimator. What could cause such biase?(1 vote)
- The bias of an estimator is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called unbiased. In statistics, "bias" is an objective property of an estimator.

Does this help?(1 vote)

## Video transcript

- [Instructor] We're told
Alejandro was curious if sample median was an unbiased estimator of population median. He placed ping pong balls
numbered from zero to 32 so I guess that would be
what, 33 ping pong balls in a drum and mixed them well. Note that the median of the
population is 16, alright? The median number of course
yes in that population is 16. He then took a random sample of five balls and calculated the median of the sample. So we have this population of balls. He takes a, we know the
population parameter. We know that the population median is 16 but then he starts taking
a sample of five balls so n equals five and he
calculates a sample median, sample median, and then he replaced the balls
and repeated this process for a total of 50 trials. His results are summarized
in the dot plot below where each dot represents
the sample median from a sample of five balls. So he does this, he
takes these five balls, puts them back in then he does it again then he does it again and
every time he calculates the sample median for that sample and he plots that on the dot plot so, and he will do this for 50 samples and each dot here represents
that sample statistic so it shows that four times
we got a sample median, in four of those 50 samples,
we got a sample median of 20. In five of those sample medians, we got a sample median of 10 and so what he ends up
creating with these dots is really an approximation
of the sampling distribution of the sample medians. Now, to judge whether it is a
biased or unbiased estimator for the population median, well, actually, pause the video, see if you can figure that out. Alright, now let's do this together. Now, to judge it, let's think about where the
true population parameter is, the population median. It's 16, we know that and
so that is right over here, the true population parameter. So if we were dealing with a biased, a biased estimator for
the population parameter then as we get that, our approximation of the
sampling distribution, we would expect it to be somewhat skewed. So for example, if the sampling, if this approximation of
the sampling distribution looked something like that then we'd say, okay, that looks like a biased estimator or if it was looking something like that, we'd say, okay, that looks
like a biased estimator but if this approximation
for our sampling distribution that Alejandro was constructing where we see that roughly
the same proportion of the sample statistics came out below as came out above the true parameter and it doesn't have to be exact but it seems roughly the case,
this seems pretty unbiased and so to answer the question
based on these results, it does appear that the sample median is an unbiased estimator
of the population median.