Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to create a variance sampling distribution probability distribution table.  Here's our problem statement: Three randomly selected households are surveyed.  The number of people in the households are 2, 4, and 9.  Assume that samples of size n = 2 are randomly selected with replacement from the population of 2, 4, and 9.  Listed below are the nine different samples.  Complete Parts A through C.

OK, Part A says, "Find the variance of each of the nine samples. Then summarize the sampling distribution of the variances in the format of a table representing the probability distribution of the distinct variance values."  Well, a lot of students, when they first see a problem that looks like this, the first impulse is to freak out because they have no clue what it is they're supposed to be doing.  But let's walk you through this.  It's telling you what to do if you just take everything a step at a time.

So the first thing to do, it says, find the variance of each of the nine samples.  Now I prefer to work these types of problems in Excel.  But I've had a number of you who've requested to see this in StatCrunch.  So I'm going to work this problem in StatCrunch.  That means the first thing I need to do here is dump my data into StatCrunch.  So let's go ahead and do that.  So here's my data here in StatCrunch.  And I'll resize this window so we can see a little better what's going on here.

And now again, the first thing we need to do here, it says the find the variance for each of the nine samples.  Well, notice here your samples are in rows.  So the first sample you've got went to the first house, and there's two people in the household.  Then you went to a second house and there's two people in that household.  So there's your first sample.  And you did that nine times to get nine samples here.

So the variance for each of these samples --- variance is a sample stat.  So we're just going to go to Stat --> Summary Stats --> Rows (because here our sample information is in rows).  We don't want to include the sample number in our calculation.  So I'm going to select just x1 and x2.  And then down here we're looking for the variance.  So we don't need all the default selections there; just the variance to select.  I hit Compute!, e voila!  Here are my sample variances listed in the order in which they appeared in the table.

In order to make the probability distribution table, however, the next step says, "Summarize these variances in the format of a table."  So in order to get that probability distribution table out, we need to order these values here.  And the easiest way to do that is come up here and just hit on the little button up here.  And voila!  Notice how it reordered everything from lowest to highest, smallest to greatest.  If I hit that button again, it'll toggle to greatest to lowest, and it just keeps toggling back and forth every time I press that button.

So now I've got everything I need to make my table.  S-squared is the variance, and these are the different variance values here.  So the first variance value is listed as a zero.  So that's how I'm going to select from the dropdown here.  The next number that appears on the table is a 2.  So that's going to be the next number here.  And then just so on and so forth on down the line.

Now to get the probabilities for each of those variance values, again, look at my table. Remember probability is the part over the whole.  So what's the part that has zero?  Well, how many zeros do I have?  I've got 3 zeros out of 9 values totals.  So there's nine samples total, so that's the whole.  So 3 over 9 is my probability, but 3 over 9 reduces to one third.  So that's what I'm going to put in my answer field.  Next, how many twos do we have in our table?  We've got 2 number twos in the table out of 9 total.  So that's 2 over 9.  And the same thing with 12 and a half.  And I've got the same thing with 24.5.  Nice work!

OK, now Part B is saying that we need to compare the population variance to the mean of the sample variances and then choose the correct answer below.  So first, let's get these values here.

The population variance?  Well, to get that in StatCrunch, we're going to have to put the population here into StatCrunch because all it loaded was the actual samples.  We need the population.  So here in the problem statement, it says that the population is 2, 4, and 9.  So I'm going to come back here in StatCrunch, and I'm going to label this population.  So I know what I'm looking at --- two, four and nine.  And now I can come up here to Stat --> Summary Stats --> Columns, select that population column, get the variance for that value, hit Compute!, and here's my population variance, which in this case is 13.

I need to compare that to the mean of the sample variances.  Well, the sample variances are right here, but I can't calculate in StatCrunch any sample statistic unless those numbers are here in the data table.  So I could copy these numbers over into the data table, or I could do the much easier route by going back to my options window and come down here to this box, and I check it for Store in data table.  What this does when you check this box is instead of putting the results in a separate window like we normally see, it'll put the results in the data table.  We can then run calculations off of those numbers in StatCrunch.

So notice here how the values that were in the table have now been put here from that separate window into the data table.  Now I can just go to Stat --> Summary Stats --> Columns, and select that new column.  I want the mean of the sample variances, so I must select the mean here for my sample statistic, and out comes the value that I'm going to be comparing with.

Now, technically, you know in an ideal world, these two numbers should be the same, because the variance is an unbiased estimator.  However, because our sample size here is fairly small, you may seem some disparity with these numbers.  So even though the problem statement is telling you to calculate each of these numbers and then compare them to choose the correct answer, if you read through each of these answer options, you're going to find that none of them apply to these specific numbers.  So the best way to answer this question is to have memorized that list of biased and unbiased estimators and then answer accordingly.

Here we're talking about variances.  Variance is an unbiased estimator.  So the sample of --- the mean of the sample distribution should be the same as the population parameter.  So in this case, we want to select the one where they're the same, so "equal to the mean of the sample variances."  Well done!

Part C asks us a similar question but in a different way.  "Do the sample variances target the value of the population variance."  And again, you want to not just look at these numbers here, because your sample size for purposes of these problems is going to be pretty small.  So you want to not look at the numbers so much as you want to have that list of biased and unbiased estimators memorized.

Variance is an unbiased estimator, so it's going to target --- the sample is going to target the population, and therefore it makes a good estimator.  So this is the answer option that best reflects that.  Excellent!

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 was boring or just doesn't want to help you learn stats, go to aspiremountainacademy.com, where you can learn 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.

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to apply a nonstandard normal distribution to quality control.  Here's our problem statement: The weights of a certain brand of candies are normally distributed with a mean weight of 0.8596 grams and a standard deviation of 0.0512 grams.  A sample of these candies came from a package containing 445 candies, and the package label stated that the net weight is 380.1 grams.  If every package has 445 candies, the mean weight of the candies must exceed 0.8541 grams for the net contents to weigh at least 380.1 grams.

Part A says, "If one candy is randomly selected, find the probability that it weighs more than 0.8541 grams."  Well, to do this, I'm going to use the distribution calculators in StatCrunch, because here it says that my weights are normally distributed.  So I want that normal distribution calculator.  Alternatively, you can use the z-score tables to calculate this out, but hey, it's the 21st century.  Let's join it.  Alright, so I'm going to open up StatCrunch.  Not that I need StatCrunch per se, but the distribution calculators that I want to use are actually here inside StatCrunch.  I'll resize this window so we can see better what's going on here.

OK, so to get my calculator, I go to Stat --> Calculators --> Normal, because again it --- the problem statement said that our weights are normally distributed.  So here's my normal calculator in StatCrunch.  The nice thing about this is that I don't have to do any adjustment with the z-scores like I would with the tables.  I just go ahead and put the mean and the standard deviation here into my calculator and StatCrunch handles all that conversion stuff for me.  Again, it's nice to join the 21st century.

So the mean weight that we get from the problem statement is up here, 0.8596, so I'm going to stick that here in the mean field.  The standard deviation from the problem statement is 0.0512.  So I stick that there.  And I want the probability that we have a weight more than 0.8541, so 0.8541, and I want to get more than.  And there's my probability, calculated out for me.  Well done!

Now Part B asks us the same question, only this time instead of one candy being randomly selected, we're randomly selecting 445 candies.  So everything's pretty much going to remain the same here in StatCrunch, except we need to make an adjustment for our standard deviation.

Remember that when you've got more than one that you're selecting and you're looking for that probability, you have to make an adjustment to your standard deviation because the standard deviation is a biased estimator.  So you've got to make some adjustments to account for that bias.  We're not going to make any adjustments with the mean value because the mean value is an unbiased estimator.  So there's no bias to account for with this.  So we're going to leave the mean value alone.

We have to make an adjustment with the standard deviation, and the adjustment we make is to divide the standard deviation by the square root of whatever our new sample size is, in this case, 445.  So I take my calculator out here, and I'm going to actually perform that operation.  The standard deviation from the problem statement, 0.0512, and I'm going to divide you by the square root of the 445.  There's my new standard deviation.

So I could transfer all these numbers manually and be here, but I'm prone to transcription errors.  So I'm just going to take this, and I'm just going to copy that number from the calculator, come here to StatCrunch, select everything in the field, and Ctrl+V on my keyboard will paste that bad boy number right in there for me.  So I don't have to worry about transcription errors.  And now I just press Compute! again, and voila!  There's the new probability.  Nice work!

Now Part C asks, "Given these results, does it seem that the candy company is providing consumers with the amount claimed on the label?"  Well, the result you really want to focus in is the one we just got done calculating because each package contains the 445 candies that we just calculated the probability for.

And look, that probability is really close to 1.  So it's almost certain that we're going to be providing what's on the label.  So then are, you know --- is the candy company providing consumers with the amount claimed on the label?  The answer's going to be yes, because your probability is just about 1.  And so, yes, because the probability of getting a sample mean when 445 candies are selected is not exceptionally small because, look, it's almost 1, which means it's exceptionally large.  I check my answer.  Excellent!

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 want to help you learn stats, go to aspiremountainacademy.com, where you can learn 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.

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to find probability using the multiplication rule.  Here's our problem statement: Multiple choice questions each have five possible answers, one of which is correct.  Assume that you guess the answers to three such questions.

OK, Part A says, "Use the multiplication rule to find the probability of WCC, where C denotes a correct answer and W denotes a wrong answer."  To help us with our calculations, I'm going to pull up Excel, and I'm gonna run my calculations here in Excel.  So we're asked to find the probability of WCC, where the first question we answer is a wrong answer, and then we get a correct answer, and then we get a correct answer.  And then we're going to combine those probabilities together to get our final probability.

OK, now what's the probability of getting a wrong answer?  Well, there's five possible answers.  Only one's correct, so there's four that are wrong.  So the part over the whole --- four divided by five will give us 80%.  In fact, let's come over here and actually center things out.  So we got 80% for that probability.

Now if you can't do that in your head, you can always do it here in Excel.  Just type the equal sign, then put the part (4) divided by the whole (which is 5), press Enter, and voila!  I get the same answer that comes out.  So 80%.  To get a correct answer is 1 out of 5, which is going to be 20%.  And you can --- alternatively, if you want to calculate that out just like we did before, and you'll get the same number.

So the probability of WCC is the probability of getting the first one wrong and the probability of getting the second one correct and the probability of getting the third one correct.  So, and, when we're using probabilities, means multiplication.  So we're just going to multiply those three numbers together.

So there's two ways I can do that. I can press the equals sign, and then I can select each one of these cells in turn and multiply them together like you see here.  Press Enter, and there's my value.  The other way to do this is with the product function in Excel.  So to use that, I'm going to select --- take equal, then spell out product, open parentheses, I'm going to select these three cells that I want to put into the product, close the parentheses, Enter.  And notice I get the same number coming out.  So whichever way works for you, go for it.

Now notice here it says type an exact answer.  What that means is it wants a fraction.  What we have here as a decimal, so how do we get the fraction from the decimal?  How do we convert that out?  Well, if we notice here with the first question that we answered, there's five possible selections and the same for the second and the same for the third.  If I just copy this formula down, that means I've got 125 total possible outcomes in my sample set.  So out of 125, I've got 0.2% so I can actually multiply those together because the 125 is the whole.  When I multiply that by the part over the whole, I get just the part, just 4.  So the exact answer is going to be 4 divided by 125.  So I come over here --- 4 divided by 125.  Fantastic!

Now Part B says, "Beginning with WCC, make a complete list of the different possible arrangements of two correct answers and one wrong answer.  Then find the probability for each entry in the list."  We can see they have already listed that out for you because you've got the remaining answer spaces here to fill next to each possible outcome.  So they've made the list for you here.

But if we just --- instead of going back through all of this calculation again, which you can do if you want, or we can use a shortcut if we remember one of the properties of multiplication is that when you're multiplying different numbers together, the order doesn't matter.  You're going to come out with the same thing.  So if I take 1 times 2 times 3, I get six.  But if I take 1 times 3 times 2, I get six.  If I take 2 times 1 times 3, I get six.  3 times 1 times 2 is six.  It doesn't matter what the order is.  So we're going to get the same number come out for these combinations as we did for the earlier ones.  So I'm just going to put in the same number for each of these answer fields.  Good job!

Now Part C says, "Based on the proceeding results, what is the probability of getting exactly 2 correct answers when 3 guesses are made?"  And you may be tempted based on what we did with the last part to just put in the same answer that we gave here, and you would be incorrect to do that.  These are possible outcomes.  What this question is asking is for getting any one of these possible outcomes.  So it could be that we get this outcome here --- the first one wrong and the second two correct.  Or it could be that we get the first two correct and the last one wrong.  Or it could be that we get the second one wrong with the first and third correct.  We're going to get one of those outcomes: the first or the second or the third.

Or, when we're dealing with probability, means we add.  Or means addition.  So we're going to add 4 over 125 to itself three times.  And when we do that, we're going to get the answer that comes out here in our answer field.  So 4 plus 4 plus 4 is 12, so I want 12 over 125.  And notice I can't reduce that any further.  So there's my exact answer.  Nice work!

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 want to help you learn stats, go to aspiremountainacademy.com, where you can learn 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.

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to find probability using combinations.  Here's our problem statement: When testing for current in a cable with 10 color coded wires, the author used a meter to test three wires at a time.  How many different tests are required for every possible pairing of three wires?

OK, so we've got a total amount that we're taking a portion at a time.  So we could be calculating permutations, or we could be calculating combinations.  The way we tell the difference is we ask the key question: Is the order important?  Does the order matter?  Here the order does not matter.

You know, if you pick, say, the red wire, the white wire, and the blue wire, and you hook them up to the meter, you're going to get the same result from the meter if you select the red, the white, and, you know --- what was that other color I selected?  Yeah.  Was it black?  Yeah, I just had a brain fart. Anywho, yeah, you just — it doesn't matter what the order is, you hook them up to the same three wires, you hook them up to the meter electrically, they're going to perform the same.  So there is no difference with the order.  The order is therefore not important.  Because the order's not important, we're going to use combinations.

Now typically you're going to be selecting combinations with your calculator.  You'll punch it out on your calculator.  We don't have --- the calculator that I have here on my computer doesn't have a combination or permutation function.  So I'm going to a website, calculator.net, and let's bring that down a little bit so you can see here calculator.net.  And there's all kinds of calculators here.  And the one I'm selecting is for permutations and combinations.  The nice thing about this is that it just punches everything out for you.  You're probably not going to be able to use this on a test, but if you learn how to use your calculator, it'll perform the same function for you.  You just got to get the same numbers here.

The total amount that we're testing —  there's 10 wires total, so we'll put 10 here.  And then the amount in each subset — how many are we taking at a time?  Here it says we're testing three at a time, so I'm just going to put 3.  And notice it calculates both the permutations and the combinations for you.  We want combinations because the order is not important.  So we're going to select the 120.  Excellent!

And that's how we do it at Aspire Mountain Academy.  Be sure and leave your comments below and let us know how good a job we did or how we can improve.  And if your stats teacher was boring or just doesn't want to help you learn stats, go to aspiremountainacademy.com, where you can learn 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.

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to find probability from a frequency table for selections with and without replacement.  Here's our problem statement: Use the data in the following table, which lists drive through order accuracy at popular fast food chains.  Assume that orders are randomly selected from those included in the table.  If two orders are selected, find the probability that they are both from Restaurant D.  (a) Assume that the selections are made with replacement.  Are the events independent?  (b) Assume that the selections are made without replacement.  Are the events independent?

OK, so we're asked to find a probability, but we're asked to find it under two different conditions.  Part A is asking us to find the selections made with replacement.  So what we're going to do first is take this frequency table, and I find it easier to find probabilities with frequency tables by working in Excel.  So I'm going to dump this data into Excel and, OK, there's my --- I don't know what it's doing here.  Whoops.  OK.  Let's see here.  I want to resize this.  Alright, let's move you over.  Behave, behave.  What are you doing?  Come back over here.  OK.  Let's bring that down a bit.  All right, so now I can see more of what's going on.

Here's my data in Excel and we're asked to make two orders.  And for each of these orders we're going to need to calculate a part and a whole.  And we take the part divided by the whole, and that gives us a probability for that part.  So there's two selections.  We're going to select the first order and then we're going to select a second order.  So now we just need to grab our numbers to fill in the table and run our calculations.  You could follow along with a calculator if you want, but the beauty about Excel is that being a spreadsheet, it's made for performing calculations inside.  So it's just really easy to just put the calculations here in Excel, as you'll see in a moment.

We're making selections with replacement.  So what we take out, we're going to put back in.  So the part for the first order --- so we're looking at the probability that the selection is made from Restaurant D.  So what's the part for Restaurant D?  The part that contains Restaurant D is right here in the table --- the 149 plus the 16 in which, if you look down here at the bottom, Excel will automatically sum the cells that we select.  So here we've got 165.  The whole is just everything in the table.  So here that gives us 1114.  And then the probability is just going to be the part divided by the whole.  So there's the first probability.

Now the probably for the second order, the selection is being made with replacement.  So this number is not going to change.  The part is still going to be 165.  The whole doesn't change.  It's still 1114.  So we've got the same calculation.  Now I could type it in again, or I could just come back here to this cell that has the formula I want in it.  And notice when I put the cursor over this little rectangle here on the bottom right corner of the selected cell, notice how my cursor changed to a plus sign.  Once it does that, it says it's ready for copying the contents of that cell over to neighboring cells.  So now that I've got that little plus sign for my cursor, I hold down the left button on my mouse, and I can drag that over to the cell I want a copy to, then release the button on my mouse.  And lo and behold, look, it copied the same formula, just transferred it over.  So I don't have to retype it.  That could be really useful in a lot of situations.  So a little free trick there for you for working in Excel if you didn't know that.

Now I've got the probabilities for each of the orders.  We want the probability for everything together.  So we want the probability that both of the orders are from Restaurant D.  That means the probability of the first order is from Restaurant D and the probability of the second order is from Restaurant D.  And means we multiply.  So I'm going to take this first probability and multiply it by the second probability.  And there's my total overall probability.  Put that in bold text if I want to highlight the difference there a little bit.

I need to round to four decimal places, so I can either look at it here and round it or I can actually come here into Excel.  Whoops.  Wrong way.  I just get down until I get to four, and it will round it automatically for me.

Now are the events independent?  Well, the events are independent because there's no connection between the two samplings.  So, you know, what I get for the second sample is not influenced by what I get from the first sample because I'm making selections with replacement.  Everything comes back in.  So yes, the events are independent because choosing the first order does not affect the choice of the second order.  Nice work!

Now Part B asks us to run the same calculations, only this time selections are made without replacement.  So looking at the first order, there's going to be the same numbers that we had previously for the part and the whole.  So that doesn't change.  But now what changes is the second order.  The part for the second order is going to change because what we took out from the first selection is not put back in to be selected again.  That means we've got one less order to select from, so I have to take one away.  Once I press Enter, notice how everything updates for me.  That's the beauty of doing this in Excel.  I don't have to run through the same calculations again.  Once I set it up, the computer updates everything for me automatically.  It's a beautiful thing.

So here's my new probability, and you can see that it's not much different from what we had before.  And that's because our sample size is relatively large.  We've got 1114.  If we had a much smaller sample size, then we would see more of a difference between the probabilities of selecting with and without replacement.  But that difference gets minimized the larger your sample size becomes.

Here we are asked again if the events are independent.  Here the events are not independent because we had to make a change in that second order because it was influenced by the fact we took one out for the first order and did not put it back in.  We're selecting without replacement.  So here are the events are not independent because choosing the first order does affect the choice of the second order.  Well done!

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 want to help you learn stats, go to aspiremountainacademy.com, where you can learn 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.

Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to find a probability from a frequency table and identify disjoint events.  Here's our problem statement: Use the data in the following table, which lists drive through order accuracy of popular fast food chains.  Assume that orders are randomly selected from those included in the table.  If one order is selected, find the probability of getting an order from Restaurant A or an order that is accurate.  Are the events of selecting an order from Restaurant A and selecting an accurate order disjoint events?

OK, the first part of this problem is asking us to find a probability.  And when you're looking at a frequency table, like what we have here, I find it's easier to work the problem in Excel.  So I'm going to go ahead and open my data in Excel.  Here we go.  Oh, that's something else.  OK, here we go with my data here in Excel.  Let's move this over a bit so we can see better what's actually going on.

OK, so here's my data in Excel, and let's list some numbers here.  We could actually punch these numbers that we're going to calculate in a calculator.  But since we're in a spreadsheet and they're made for calculations, we'll just put all of our numbers right here, and that'll just make it easy.  So we were asked for the probability from getting from Restaurant A or we could have the order to be accurate.  OK, and then of course we've got to include events that you might double count.  And then that's going to give us the part that we want to calculate.

Then we got to get the whole, because the probability is just going to be the part divided by the whole.  So now let's go and grab our numbers so we can do our calculations.  First the probability of getting something from Restaurant A.  So the part here --- that's going to be this column here --- so if I select every number in that column, notice down here at the bottom, Excel already calculates the sum for me.  So here I got 368.

If you want, you can also use the functions in Excel.  I'm going to calculate a sum function here.  Select those same cells and I get the same number.  But since it calculates the sum for us, it's easier to just select and then put the number in.  Orders that are accurate will be in this column here, so it's going to be 963.

Double count --- these are the ones that are included in both Restaurant A and the orders that are accurate.  So where those two, where the row and the column intersect, that's going to be the number that's actually double counted, which here is the 335 because we counted this when we were counting Restaurant A, but we also counted this when we're counting the orders that were accurate.  So that number got double counted.  We got to subtract that out.

Now we're ready to calculate the part for our probability calculation.  So I press the equal sign, come up here, select Restaurant A, I'm going to add it to the orders that are accurate, and then I want to subtract out, the amount that's double counted.  I press Enter, and there's my part.

The whole is just everything in the table.  So I select every number in the table and the outcome is 1098.  Then the probability is just going to be the part divided by the whole.  And there's my probability.  Notice the answer field here doesn't list a percent sign, which means the answer is wanted in decimal form.  And we're instructed to round to three decimal places as needed.  So I'm just going to take that number, round it to three decimal places.  Fantastic!

And now the second part of the problem asks, "Are the events of selecting an order from Restaurant A and selecting an accurate order disjoint events?"  Well, disjoint means mutually exclusive.  In other words, there is no overlap or intersection between those two events.  They're not occurring at the same time.  So the easiest way is to just look at your table.  Are there any areas of overlap?

And the answer is yes.  Remember that we had to take out the 335 that was double counted.  The reason why it was double counted was because it's in both of those events.  It's an accurate order and it's from Restaurant A.  So it's an area of overlap.  And because of that, the events are not disjoint.  So here we're going to say the events are not disjoint because it is possible to receive both an accurate order and get an order from Restaurant A.  Fantastic!

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 want to help you learn stats, go to aspiremountainacademy.com, where you can learn 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.

Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to find and interpret measures of variation.  Here's our problem statement: Find the range, variance, and standard deviation for the given sample data, if possible.  If the measures of variation can be obtained for these values, do the results make sense?  By all just conducted experiments, determine whether a deficiency of carbon dioxide in the soil affects the phenotypes of peas.  Listed below are the phenotype codes, where 1 = smooth yellow, 2 = smooth green, 3 = wrinkled yellow, and 4 = wrinkled green.

OK, the first part of this problem is asking us for the range of the sample data.  And we can get that well enough.  I just come here and open my data into StatCrunch.  I'm going to move and resize this window so we can see better what's going on.  OK, now we've got the data in StatCrunch.

Getting the range is really easy. Go to Stat --> Summary Stats --> Columns.  Here in the options window, I'm going to select the column where my data is located.  And then we don't need all of the default selections here for the statistics.  All we're looking for here from the promise statement, there's the range, the variance, and standard deviation.  So I'm going to select just those three: the range, and then to select the other two I'm going to hold down the Ctrl button on my keyboard while I select standard deviation and the variance.

Notice that the order in which my statistics are listed here in this window is the same as the order that they will appear in the results window.  I press Compute!, and see, here are my results with those statistics in the same order in which they were selected.  The range here is 3.  We didn't really need to use StatCrunch to get that.  If you just look at the data here, the range is the maximum value minus the minimum value.  Well, from the way the data is set up here, the different categories for your data, 4 is the largest it could be, 1 is the smallest it could be, and 4 minus 1 is three.  So we could have done this ourselves.  But for some data sets, StatCrunch actually is very useful, because the numbers don't come out so easy like that to where you could do them in your head.  Anywho, excellent!  Got through that good.

Now the second part of the problem asks for the standard deviation, which we can get here from StatCrunch.  So I'm going to put that answer here in my answer field.  Good job!

And now the third part asks for the same thing with the variance, which again, StatCrunch gives that to us.  So I'm going to put that answer here in my answer field.  Fantastic!

Now the last part of this problem asks us, "Do the results make sense?"  Well, measures of variation, like what we've calculated here, really only make sense when you're dealing with quantitative data.  So if you're looking at categorical data, you're not going to make sense.  And from the way that the data is described here in the problem statement, we have categorical data, because the numbers here aren't being used to represent a quantity; rather they are being used to represent names or labels for different categories.  And so therefore we have categorical data.

So let's go through our different answer options to see which of these answer options best matches that reasoning.  The first answer, Option A, says "The measures of variation do not make sense because the standard deviation cannot be greater than the variance."  Well, it is true; the measures of variation don't make sense, but it's not because of any number of being greater than the other.  It's because of the nature of the data, not the values that are described there in the dataset.

Answer option B says, "While the measure of variation can be found, they do not make sense because the data are nominal. They don't measure or count anything."  Well, that's pretty close to what we were reasoning.  So let's mark that.  But before we select our Check Answer button, let's go ahead and check the remaining answers to see if they are any better.

Answer option C says, "It makes sense that the measures of variation cannot be calculated."  Well, that's obviously wrong because we calculated them over here in StatCrunch.  Answer option D says, "The measures of variation makes sense because the data is numeric."  Well, the measures of variation don't make sense, so it's not going to be answer option D.  We're going to stick here with answer option B.  Nice work!

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 you're stats teacher is boring or just doesn't want to help you learn stats, go to aspiremountainacademy.com, where you can learn 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.

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, median, mode, midrange, and outliers of a dataset. Here's our problem statement: Find the mean, median, mode and midrange for the data, and then answer the given question. Listed below are the amounts in dollars it costs for marriage proposal packages at different sports venues. Are there any outliers?

OK, Part A asks us to find the mean. The easiest way to do that is to dump the data into StatCrunch. Then you use StatCrunch to calculate the mean for us, so I'm going to do that here. OK, my data is now here in StatCrunch. I'm going to resize this window so we can see a little bit better everything that's going on.

OK, now I'm ready to calculate the mean. And to do that, I come up to Stat --> Summary Stats (because I'm looking for a summary statistic; the mean is an example of a summary statistic) --> Columns (because my data is listed there in a column). First, I tell StatCrunch where to find the data, so I select the column with the data. And then notice how we've got some statistics here that are already preselected by default. I don't need all of them. All I want is the mean value, so I'm just going to select the mean, hit Compute!, and there's my mean value. Good job!

Now Part B asks us to find the median. I could go through those menu options again in StatCrunch, or I could just come up here to the Options button and then press Edit. It takes me right back to the options window where I was previously, and I just switch from the mean down to the median.

Now if I'm smart. I'm going to look here and look ahead and see I'm going to have to calculate the mode. I don't have to calculate the mid range, so let's go ahead and stick the mode in here as well so we don't have to go back and recalculate this. The mode is actually towards the bottom of the list. So to select more than one, I'm going to have to press the Ctrl key and then select Mode. If I hit the Shift key, that's going to select everything in between the median and the mode. And I just want these two statistics to come out.

Notice the order in which they appear here is going to be the same order in which they appear in the results window. So when I hit Compute!, you see here they are in the same order. The mean was first and then the mode. So now I can just stick my answer here in the answer field. Well done!

Now Part C asks us to find the mode, which we can just come over here to the window. And we don't have to recalculate that; we've already calculated it. Fantastic!

Find the mid range. StatCrunch actually has no set function for the mid range. I don't know why they don't have it. I mean, it's easy enough to calculate it and code it in, but they just haven't done it. So what we have to do is go about it in a roundabout way.

So I'm going to go back to my options window in StatCrunch. And I'm going to come up here, and I'm going to select two different statistics. The first statistic I'm going to select is the range, and the next I'm going to select as the min. I also select the max so I can demonstrate a second way to calculate the midrange. So now I'm going to hit Compute!

So the midrange is the value that's in the middle of the range between the minimum and the maximum value. So there's different ways to get this out, but you're going to need at least two of these numbers here. And then I'm going to come down here and get my calculator out because we're going to need it. For me, the easiest way to calculate the midrange is just take half the range and then add it to the min. So I can go 2461, which is my range, divided by two. Then I'm going to add that to the minimum value, which here is 39. So I get 1269.5.

I can get the same number if I take half the range and then subtract it from the max value. So again, taking half the range, now I'm going to subtract that from the max value. And see? I get the same number. A third way to calculate it is to just average the min and max values out. So if I want to, I could take 39 plus 2,500, then average that out, and I get 1269.5; it's the same number. Okay, so whichever of these ways is you find easiest for you, that's the way you should go about calculating it. And then just do the same thing every time you're asked to find the midrange. Good job!

Now Part E asks us to identify any outliers in the dataset. Well, you could look at the data, and you can get a feel that there's going to be a couple of outliers at least here at the end because these numbers here in the triple digits are, you know, so far away from the rest of the dataset. But this 39 here -- it looks like it might be an outlier as well.

The best way to determine outliers from the dataset is to make a graphical representation. And a simple one where you've got a dataset here where you've only got one variable, it should just use a box plot. So if I come back here to StatCrunch and go up to Graph --> Box Plot, select the column with my data, and I want to draw the box horizontally, because then that way it represents more of a number line there on the axis. And it's easier for me to figure out what's going on. I press Compute!, and here's my box plot.

And it looks like that 39 value is not an outlier. But these other two data points that you see represented with the dots are outliers. So 1500 and 2500 are outliers. So I'm going to come back here and select the answer that best says that. Excellent!

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 want to help you learn stats, go to aspiremountainacademy.com, where you can learn 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.