Finding the mean and standard deviation for a probability distribution

Intro

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to find the mean and standard deviation for a probability distribution using StatCrunch. Here's our problem statement: Five males with an x-linked genetic disorder have one child each.  The random variable x is the number of children among the five who inherit the x-linked genetic disorder.  Determine whether a probability distribution is given.  If a probability distribution is given, find its mean and standard deviation.  If a probability distribution is not given, identify the requirements that are not satisfied.

Here we're given a table listing the values and our probability distribution, so the random variable x is 0, 1, 2, 3, 4, 5, and then corresponding probabilities are in the other column of the table.

Part 1

Our first question reads, “Does the table show a probability distribution?  Select all that apply.”  To check the values in our probability table to see if we have a probability distribution, we must ascertain that three requirements are met.  The first requirement is that each of the probability values in the probability table must be a probability value, which means they must be between 0 & 1 inclusive.  The second requirement is that each of the values for random variables must have a probability associated with it.  And the third requirement is that the probabilities in the table must add up to equal 1.  So let's check these requirements off.

First, the probabilities that are listed in our table have to be between 0 & 1 inclusive.  And we can see here that each one of these values in the probability column is between 0 & 1 inclusive.  So there's our first requirement.

The second requirement is that a probability value must be assigned to each random variable.  And we can see that each random variable has an associated probability with it.  Notice that in this case for random variables x=2 and x=3, we have the same probability value.  That's OK.  The associations don't need to be unique; you could have the same probability value for more than one random variable.  There just needs to be a probability value assigned to each random variable.  And we have that here, so there's the second requirement.

The third requirement is that each of the probabilities added together must equal 1.  You could punch these out in your calculator, or you could load it up into StatCrunch and get the sum.  Here I actually find it easier to get the sum by loading this into Excel.  So if I dump this table into Excel — and we have to wait a little while for the program to open up because someone is sleeping inside my computer — excellent!

OK, here we go.  So now the table is here in my — hold on, hold still, hold still — so now we have the table here in Excel.  And the easy way to get through this is we simply select the values in the probability column.  And notice how some statistics are calculated down here at the bottom of my window in Excel.  One of those statistics is the sum.  So here we see the sum equals 1, and it's exactly 1.  This means that we have a probability distribution, because the third of the three requirements has now been met.

You have to be careful with this.  It has to be 1 exactly.  So if the sum reads 1.000 something, or 0.999 something, it's not 1 exactly, and therefore it's not a probability distribution.  The numbers have to add up to equal 1 exactly.  So here we have a probability distribution.

If I wanted to calculate the sum in StatCrunch, that's easy enough to do.  Let's just load the table into StatCrunch.  We’ll move this window over a little bit.  So here in StatCrunch, if I go up to Stat –> Summary Stats –> Columns, select my probability, and then down here under Statistics I'm looking for the sum.  And here we see the sum is 1 exactly.

I think StatCrunch is a little bit more clunky; there's a little bit more clicking to do with the mouse.  It's much easier to just load it into Excel, select the cells that you want to add up together, and then it automatically does it for you there at the bottom.  It's a little quicker in Excel.  But if you prefer StatCrunch, there I've showed you how to do it.

Either way, we get the same answer: We do have a probability distribution.  So I'm going to select that from my answer options.  Good job!

Part 2

Now the second part of our problem says, “Find the mean of the random variable x.  Select the correct choice below and, if necessary, fill in the answer box to complete your choice.”  You can do this in either Excel or Statcrunch.  For this particular part of the problem, it's easier to work in StatCrunch, so that's what I'm going to do.

We have the data already loaded here into StatCrunch, so now all we need to do is get the distribution into a calculator.  To do that, we click Stat –> Calculators and then go down here to Custom.  The distribution that we have is not identified as being a particular distribution.  It's just a general probability distribution, and so we're going to select this Custom option at the bottom of the list.

Here in our options window, the values will be the random variable (which is the x), the weights will be our probabilities.  Press Compute!, and here comes my distribution calculator.

And look here!  The mean and the standard deviation are already calculated for me.  So I'm asked for the mean here.  I'll just put that number into the answer field.  Good job!

Part 3

Finally we're asked, “Find the standard deviation of a random variable x.  Select the correct choice below and, if necessary, fill in the answer box to complete your choice.”  We have a standard deviation already calculated here.  We're asked to round to one decimal place, so that would be here 1.1.  Put that in my answer field.  Good job!

And now a final word of caution about the Custom calculator in StatCrunch.  StatCrunch will not check to see if you have a probability distribution before it calculates mean and standard deviation inside the Custom calculator.  So when you use the Custom calculator, it will just take whatever data it gives you and calculate mean and standard deviation for you.  It won't check to see if you have a probability distribution.

So you have to run the check first to see if you have a probability distribution before you use the Custom calculator.  If you just rush off and use the Custom calculator, then these answer options here for Parts 2 and 3 where the table does not show a probability distribution may in fact be the right answer.  You don't want to go selecting the wrong answer, so make sure you check to see that you have a probability distribution before you start taking mean and standard deviation values from your data.

And that's how we do it at Aspire Mountain Academy.  Be sure to leave your comments below and 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.