Creating in StatCrunch a probability distribution table for a sampling distribution of the medians

Intro

Howdy!  I am Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to create in StatCrunch a probability distribution table for a sampling distribution of the medians.  Here's our problem statement: Three randomly selected households are surveyed.  The numbers of people in the households are 2, 3, and 10.  Assume that samples of size n = 2 are randomly selected with replacement from the population of 2, 3, and 10.  Listed below are the nine different samples.  Complete Parts A through C.

Part A

Part A says, “Find the median of each of the nine samples, then summarize the sampling distribution of the medians in the format of a table representing the probability distribution of the distinct median values.”  When most students see this type of problem, they freak out because they don't know what in the world they're looking at, and they have no idea how to approach this.  Once you understand what's going on with this, it's pretty easy to solve these types of problems.  Let's run through that.

I mentioned earlier that we were going to solve this problem in StatCrunch.  So here in the little icon next to the data set that they give me in the problem statement, I'm going to click on that, and I'm going to open this in StatCrunch.  Now I actually find it a little bit easier to solve these types of problems in Excel, but that's because I'm very much more versed in Excel than I am in StatCrunch.  I have more experience in Excel than StatCrunch, so I'm more comfortable with it there.  But I said we're going to solve this problem in StatCrunch, so that's what we're going to do.  I’ll have to make another video at another time to show you how to solve this problem in Excel if you're interested.

So here we have our data in StatCrunch, and I'm going to resize this window to make things a little easier for us.  Now, here we have our data in StatCrunch.  To start with, we need to calculate the medians for each of the different samples.  So here we have our samples listed in rows where the first sample we get was 2 and 2.  And the second sample we get is 2 and 3.  We're picking from these population values that are stated here in our problem statement — 2, 3, and 10.  We have a total of 9 samples.

Now what we need to do is calculate the median, because this is the statistic we were asked to calculate — a sampling distribution of the medians — so we're going to calculate the median value for each of these different samples.  And to do that, I'm going to put my cursor over here in the next available column, and then I'm going to go up here to Stat –> Summary Stats —> Rows, because my data for the sample is listed in rows.

Here I'm going to select the actual columns where my sample data is located.  Notice that's in the x1 and x2 column. I can select more than one column by holding the Shift key or the Control (Ctrl) key on my keyboard while I click the mouse.  The difference between the two is if you hold down the Shift key it will select everything in between the first selection you made and the second selection you made.  If you hold down the Ctrl key, then that's selecting just the individual selections by themselves regardless of what's in between.  Here that makes no difference because the two columns we want are right next to each other.  The statistic we're asked to calculate is the median, so I'm going to scroll down here and select Median and then come down and hit Compute!  Now I've got all the median values that I need to construct my probability table.

When we're constructing the probability table, we're going to ignore this first column for Row because it's just listing the different rows here, and we're actually going to sort the column for the medians because it makes it easier to construct our probability table.  To sort the rows, all I'm going to do here is click on this little arrow at the top right next to Median, and notice how the values sorted themselves automatically.  If I click this again, it sorts it again in reverse order.  We actually want this from smallest to largest, so I'm going to sort it in that order.

Now I have everything I need to construct my probability table.  Here in my answer fields, the way I'm going to do that is by recognizing what a probability is.  A probability is nothing more than the part divided by the whole.  So we look at our first value here, which is a 2.  That means I'm going to go over here and from the drop-down menu on this first selection, I'm going to select 2.  What's the probability of getting a 2 for my median value?  Well, how many 2s are there here in my Median column?  There's only one.  This is why I sorted it, because it makes it easier to count.  There's only one, so the part that I have is going to be 1 divided by the whole — which there's nine rows total, so that's going to be 1 over 9.

The next value I see in my in my Median column is 2.5, so down here under the drop-down menu selection, I'm going to select 2.5.  I have two of those, so that means my probability is 2 divided by 9, because there's two for the part and then there's nine for the whole.  I continue on in that way to complete the probability table.  The next value was 3, and there's only one of those, so its probability is 1 over 9.  The next value is 6, and there's two of those, so that probability is 2 divided by 9.  The next value is 6.5, and there's two of those, so that probability is 2 divided by 9.  And the last value is 10, and its probability is 1/9.

Notice that all the probabilities in my table are either 1/9 or 2/9.  If, for example, I had 3/9, I would need to simplify that to 1/3 before I check my answer because, the way this is constructed, the answers are expected to be in simplest terms.  I don't even need to worry about reducing anything, because everything's already reduced as far as it can go with 1/9 and 2/9.  So I check my answer.  Well done!

Part B

Now Part B asks, “Compare the population median to the mean of the sample medians.  Choose the correct answer below.”  OK, so the first thing we need to do is find the population median and the mean of the sample median so that we can compare them.

To get the population median, we're going to have to put those numbers here into StatCrunch.  So back here in StatCrunch, I'm going to label this next column Population so that I know what I'm looking at when I'm trying to select between different columns.  And here I'm going to put in the values from the population, which from our problem statement listed here is 2, 3, and 10.  Now I'm looking for the median of the population, so I do that by coming up to Stat –> Summary Stats –> Columns.  Select Population, and then down here for the Statistics, select Median.  Here we have a population median of 3.

I need to compare that with the mean of the sample medians.  Well, the sample median values are over here in this table.  But in order to calculate a mean value, I have to get these numbers into the data table in StatCrunch.  If I wanted, I could just type these in, but there's a much easier way to get them in.  Here in my results window, if I push the Options button here at the top, and then in the drop down menu I select Edit, I go right back to my options window.  Then I can come down here to the bottom and click on this box for Store in data table.  This will actually put the output not in a results window but in the data table, and that's where we want it.  I press Compute!, and now my median values are there in the data table itself.  Now it's easy to run the calculation I need.  Go to Stat –> Summary Stats –> Columns.  I want the Row Median because that's where I'm going to calculate my data, and then I'm looking for the mean value for those numbers.  Hit Compute!, and the mean of the median values is 5.

So we see that 3 is the population and 5 is for the sample.  So now we can look at our answer options and see which one is the correct one.  Answer Option A says, “The population median is equal to half of the mean of the sample medians.”  Well, 3 is not half of 5, so that's not right.  Answer Option B says, “The population median is equal to the mean of the sample medians.”  3 is not equal to 5, so that's not right.  Answer Option C says, “The population median is not equal to the mean of a sample medians.  It is also not half or double the mean of the sample medians.”  That sounds right, but let's check Answer Option D just to make sure. Answer Option D says, “The population median is equal to double the mean of the sample medians.”  Well, 3 is not twice 5,  so Answer Option C is the correct answer.  Fantastic!

Part C

And now the last part of our problem, Part C, says, “Do the sample medians target the value of the population median?  In general, do sample medians make good estimators of population medians? Why or why not?”  By target, what it means is that the number we get from the sample should approach or equal the number from the population with the more samples that we take.  Here we see that the numbers are not equal; therefore, we're going to say that they're not going to target that.

I actually recommend that students answer these types of questions not by looking at the numbers so to speak, because I actually have seen instances where the numbers are the same but the correct answer is the sample doesn't target the population.  What you need to look at is what's the statistic that you're calculating and is that a biased or unbiased estimator.  Here we're calculating the median.  Medians are biased estimators; they tend not to target the population from the sample.  So here right off the bat, I know Answer Option A and Answer Option D are incorrect, because they say the sample statistic targets the population parameter.  Here that's not the case.

Between Answer Options B and C, the difference is that in Answer Option B it says, “Sample medians do not make good estimators.”  Answer Option C says, “Sample medians make good estimators.”  Because the median is a biased estimator, it doesn't target the population median.  And so therefore, it's not going to make a good estimator.  So the correct answer we want is Answer option B.  Well done!

And that's how we do it at Aspire Mountain Academy.  Be sure to leave your comments below.  Let us know how good a job we did or how we can improve.  And if your stats teacher is boring or just doesn't care to help you learn stats, go to aspiremountainacademy.com, where you can find out more about accessing our lecture videos or provide feedback on what you'd like to see.  Thanks for watching!  We'll see you in the next video.