Finding and interpreting measures of variation

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 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.

Part 1

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.

Part 2

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!

Part 3

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!

Part 4

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!

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