- [Instructor] Alejandro was curious if sample median
was an unbiased estimator of population median.
He placed ping pong balls numbered from zero to 32
in a drum and mixed them well.
Note that the median of the population is 16.
He then took a random sample of five balls,
and calculated the median of the sample.
He replaced the balls, and repeated this process
for a total of 50 trials.
His results are summarized in the dotplot below
where each dot represents the sample median
from a sample of five balls.
So this right over here is a sampling distribution
of the sample median.
You have a population of balls.
We know a parameter.
We know that the median of the population is 16.
This right over here is the population.
And then he keeps taking samples of five balls,
so that is five-ball sample right over there.
And then he calculates the statistic,
the median of the sample,
so he calculates the median there, and he does it again.
He calculates the median, and he does it again.
He calculates the median.
And then he does this 50 times.
He does this 50 times.
And so, what you see plotted over here, do you see 50 dots?
Every time he took a sample of five balls,
and calculated the median,
we see that as one of the dots here.
So for example, four times, he calculated a median of 20.
Two times, he calculated a median of 19.
One time, he calculated a median of four.
Now they ask us, based on these results,
does the sample median appear to be a biased
or unbiased estimator of the population median?
So pause this video, and see if you can come up
with an answer to that.
In order to be an unbiased estimator,
and that is what you want your estimator to be,
you want it to be unbiased,
the sampling distribution for that statistic
has to be evenly distributed
about the true parameter for the population.
So for example, in this situation,
the true parameter for the population
is a note that the median of the population is 16,
so that's the thing that you're trying to estimate.
So this is where the median of the population is.
And so, if we look at the sampling distribution,
we see that it is roughly balanced
to the left and to the right of that true parameter.
So I would say that this is looking pretty unbiased
because the sampling distribution
is evenly balanced to the left and the right
of the true median of the population, that true parameter.
If, for some reason,
the distribution looked something like this,
then it would not be unbiased.
Or if the distribution looked something like this,
if the distribution looked like this,
if our sampling distribution was like this,
this would mean that we're consistently underestimating
the true parameter.
And if our distribution looked like this,
this means that our statistic is overestimating
the true parameter.
Let's do another example.
The dotplots below show an approximation
to the sampling distribution
for three different estimators
of the same population parameter.
If the actual value of the population parameter is five,
which dotplot displays the estimator with both low bias
and low variability?
Pause this video,
and see if you can come up with the answer.
So the population parameter
that we're trying to estimate is five.
So let's show that on each of these distributions,
these approximations of the sampling distributions.
Over here, we have our scale.
This should be one, two, three, four, five, six, seven,
and one, two, three, four, five, six, seven.
And so, our population parameter is five,
so that is right over there,
that is right over there,
and that this is right over here.
Which of these statistics seem to be biased?
Let's just start there.
Pause this video.
Which of these look biased to you?
The one that looks clearly biased is statistic C.
When you look at its sampling distribution,
it is consistently to the left of our true parameter,
so it's consistently underestimating our parameter,
so this one right over here is biased.
Now statistic A and statistic B
both look reasonably unbiased.
Remember, these are approximations
of the sampling distribution.
If you want to get closer and closer
to the sampling distribution,
the true sampling distribution,
you would keep taking samples,
and keep calculating the statistic,
and keep plotting them on this distribution right over here.
But for statistic A, it looks reasonably balanced
to the left and the right.
It's not perfectly balanced but it's reasonable.
Remember, these are approximations.
And statistic B is also reasonably balanced
to the left and the right.
Now they don't just say which one has a low bias
but they also say low variability.
I would say both statistic A and statistic B
have relatively low bias
but if we talk about low variability,
that's how much spread do they have,
especially around the true parameter.
And it looks like statistic A has much lower spread
than statistic B right over here.
In statistic B, you've calculated statistics
that are too way from the true parameter
while in statistic A, you've calculated a statistic
of certain samples that are roughly half away
from the true parameter.
So the statistic that has both low bias and low variability,
that would be statistic A.