Finding and interpreting summary statistics of celebrity earnings

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 and interpret summary statistics of celebrity earnings.  Here's our problem statement: Find the mean, median, mode, and midrange for the data, and then answer the given questions.  Listed below are the highest amounts of net worth in millions of dollars of all celebrities.  What do the results tell us about the population of all celebrities based on the nature of the amounts?  What can be inferred about their precision?

Part A

OK, Part A asks us to find the mean.  And to do this, we're going to take our data and dump it into StatCrunch.  I dumped my data here into StatCrunch.  And now I'm going to resize this window so we can see a little better everything that's going on here.  OK, now in StatCrunch, to get my summary stats, I'm gonna go to Stat --> Summary Stats --> Columns (because my data set is listed in a column here).  In the options window, I'm going to select the column where my data is located.  And then down here under Statistics, notice that we've got multiple statistics that are selected by default.  All we really need for this first part of the problem is to select the mean.  So I'm just going to select mean and get that out.

Now at this point I can go ahead and hit Compute!, and StatCrunch will give me the mean value that I need to put in my answer field here in my assignment.  However, I'm going to be a little smart and look ahead and notice that I'm going to have to do this same procedure to calculate the median, the mode, and the midrange.  So since I'm here already, let's just calculate everything at once, and then we don't have to go back and forth through the menu options multiple times.  So I'm going to calculate multiple statistics all at once.

So we already have the mean value listed.  That's the first one we need to calculate.  The next one is the median.  As I scroll down here, I find the median, and then I hold the Control key down while I select the median.  And now I've got both the mean and the median selected. I do the same thing for the remaining statistics.  The next one is the mode, so I'm going to scroll down here to the bottom, because that's where the mode is, hold the Control key and click on the mode.

And then the midrange --- if you go up and down here through the list, you'll notice there is no midrange.  So that means we need to actually calculate it ourselves.  But to do that we're going to actually select numbers from this list of statistics to help us do that.  There's three ways to calculate the midrange based on the numbers that they give us here, but you only need to select two of them.  So the three from what you need to select two are the range, the min and the max.  You only need to select two of them. I'm actually going to select all three because when we get to that part of the problem, I'll show you the three different ways to calculate this.  And then you can decide which way is going to work best for you.

Notice we have the selections that we've made here are in a certain order.  This is the order in which we selected them.  This is also the order in which they're going to appear in the results window.  So I purposefully selected these statistics in this order because that's the order in which we're going to have to put them in our answer field, and this just makes it easier to go back and forth between the assignment and the results window to put the numbers in so we don't get confused.  Once I hit Compute!, I get my results window, and here are all the statistics that I need to calculate with the mean up front.  So I'm just gonna put that here in my assignment.  Nice work!

Part B

Now, Part B asks for the median.  We already have that calculated here, so all I have to do is just take the number out and stick it in the answer field.  Excellent!

Part C

And Part C asks for the mode.  And it's the same story here, so I'm just going to put that number in here.  Nice work!

Part D

Part D asks for the midrange.  StatCrunch does not calculate the midrange directly.  It's a simple enough calculation.  I don't know why they didn't program it in, but they didn't.  So now we've got to go old school.  And I'll pull up my calculator here, and there's three ways to calculate it.  The way I prefer to calculate it is to take half the range and then add it to the min.  So I'm going to take half the range, which means I divide by two, and then I'm going to add that to the minimum value.  I get 197.5.  This is the mid range.

Now notice I get the same number if I take half the range, which again means dividing by two, and I'm going to subtract this number from the maximum value.  Argh!  I pressed the wrong button on my calculator.  OK, let's try this.  I take half the range, and I'm going to subtract that from the maximum value.  Here we go.  Notice we get the same number out.  I can get the same number out a third way by taking the minimum value and the maximum value and averaging those values together.  So I take 155, add it to 240, and then I take half of that sum.  Notice I get the same number out.  So there's no one right way to do it.  There's multiple paths to the same answer, and that's why I say you need to pick the way that's right for you and then just be consistent with it.  So every time you're asked to find the midrange, you just use the same calculation procedure, and you can just go through and get the answer that you need.  I'm going to put that here in my answer field.  Excellent!

Part E1

And now the first question in Part E asks, "What do the results tell us about the population of all celebrities?"  We've got four answer options here.  So let's take a look at each one of them in turn.  Answer option A says, "Apart from the fact that all other celebrities have amounts of net worth lower than those given, nothing meaningful can be known about the population."  Well, to assess this statement, we need to go back and look at our data set and ask, "What is the data telling us? Where did the data come from?"  The data here are listing the highest amounts of net worth of all celebrities.  So this is a small proportion of all celebrities.  It's not the whole population; it's the people that are at the very top of the list.  These are the people who make the most money.  So most people in the population are going to be making less than this.  And so we can't really get much information about the population since this is just a very small sampling of a select portion of the population.

So it looks like Answer option A makes sense, but before we go ahead and submit this for our answer, let's check the other answer options to make sure we've got the right answer.  Answer option B says, "Apart from the fact that all other celebrities have amounts of net worth lower than those given, the results in Parts A, B and D do not give meaningful results."  OK, that makes sense, because as we just said, we're looking at a small sample of the population that's not representative of the population.  However, “the result from Part C shows that the most common celebrity net worth is equal to the mode”?  Well, that's not true because most of the celebrities are going to be earning far less than the numbers given here in our data set.  So we can't actually say that this is true.  Therefore we're not going to select Answer option B.

Answer option C says, "The results tell us that all celebrities are expected to have amounts of net worth approximately equal to one of the measures of center found in Parts A through D."  Again, that doesn't make any sense because the data that we have here is for a small select portion of the population that's not representative of the population.  So we're not going to select Answer option C.  Answer option D says, "The results tell us that the most common celebrity net worth is the mode, but all other celebrities are expected to have net worths approximately equal to the mean, median, or midrange."  Again, this is nonsense because this sample does not represent the whole population.  It's only the highest numbers from the list of the net worths of all the celebrities.  So again, we're not going to select Answer option D.  It looks like Answer option A is the one we want.  Excellent!

Part E2

And now the last question here in both Part E and the problem asks, "Based on the nature of the amounts, what can be inferred about the precision?"  Well, again, let's look at these values here --- or these answer options, rather --- and see how they fare out.  So Answer option A says, "Since no information is given, nothing can be said about the precision of the given values."  Well, let's look up here at our data set.  We look at the individual data values, and do you notice a pattern?  As you look at each one of these, you should notice that most of these are ending in 5.  There's a couple of them here that end in 0.

So it looks like we have a set number for the last digit in the data set.  It's either going to be a zero or a five, and that suggests that these numbers have been rounded to the nearest $5 million amount.  So "since no information is given, nothing can be said about the precision of the given values"?  No, I wouldn't say that's true.  We can actually look here and say that, yes, something's going on with this data set.  So we could --- we actually probably could say something about the precision.  It looks like there's some rounding going on.  So let's not select Answer option A.

Answer option B says, "The values are all whole numbers, so they appear to be accurate to the nearest whole number."  Well, that's not really true because if we were accurate to the nearest whole number, why is it that the only last digit that we see is either a zero or a five?  Why don't we see any other numbers?  And you would think that you would see other numbers in the data set, but we only see zeros or fives.  So it's probably not rounded to the nearest whole number.  It's rounded to the nearest $5 million value.

Answer option C says, "Since celebrity information is public, these values can be assumed to be unrounded."  Well, what is the public status?  If it's public or private, what does that have to do with whether or not the numbers are rounded or not?  Nothing.  There's nothing that ties those two together.  And so Answer option C is just absolute nonsense.

Now we get Answer option D says that the values end all in zero and five as they appear to be rounded estimates.  This is exactly right.  This is what we've observed by evaluating all the other answer options.  This appears to be the correct one, so we're going to select it. Excellent!

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