Creating a normal quantile plot to find z-scores

Intro

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 normal quantile plot to find z-scores.  Here's our problem statement: Use the given data values (a sample of female arm circumferences in centimeters) to identify the corresponding z-scores that are used for a normal quantile plot, then identify the coordinates of each point in the normal quantile plot.  Construct the normal quantile plot, then determine whether the data appear to be from a population with a normal distribution.

Part 1

OK, the first part of our problem asks us to list the z-scores for the normal quantile plot.  This is most easily done by constructing a normal quantile plot in StatCrunch, so I'm going to take my data here, click on this icon to the right, and select Open in StatCrunch.  Now that my data is open here in StatCrunch, I'm going to construct my normal quantile plot.  And to do that, I'm going to come up here to the top and select Graph –> QQ Plot.  In the options window, I select the column where my data is located, and then I'm going to check this box down here next to Normal quantile on y axis.

For whatever reason, whoever coded StatCrunch decided the convention to use would be to put the normal quantile on the x axis.  I don't know why they decided that, because everywhere I've seen a normal quantile plot in industry and in my days since, I've always seen the normal quantile on the y axis.  This is the convention that Pearson uses for your assignments, so when you're asked to compare normal quantile plots, construct one, select the proper one, so on and so forth, the normal quantile always appears on the y axis.  So you want to check this box to make sure you're comparing apples with apples.

When I press Compute!, I get this lovely normal quantile plot.  Now I can get the z-scores just by taking the y-coordinate from each of the ordered pairs that appear here in my plot.  To do that, I'm gonna put my cursor over the first one, and notice how I have the Y and the X values for that point listed in a little window.  So all I need to do is just copy those numbers over.  So I'm gonna come over here, put my cursor in the first answer field, and then put my cursor back over here over this first plot and type in that value for the Y — the normal quantile.  And then I'm gonna use my Tab button to tab between different answer field options as I move my cursor to each succeeding point.  And I'm just going to type these in one at a time until I get every single one of them in.  (I think that first one needs adjusting —  it does need adjusting.)  Now I check my answer.  Fantastic!

Part 2

Now the second part of our problem asks us to “identify the coordinates of each point in normal quantile plot.”  Obviously I'm just going to go through the same process that I went through before, only this time in my answer field I'm going to type in ordered pairs.  That means I have to put my x and y values in parenthesis separated by a comma.  So this first one will be 31.9, -0.96.  And I just continue on for each one in succession, and I'm using the Tab button on my keyboard to jump my cursor between answer fields in my assignment.  Notice also that the box from what you're taking your ordered pair coordinates list the Y first and then the X, but the convention with ordered pairs is to list the X first and then the Y.  So make sure you get everything in the right order.   And I need to correct this first point because we're rounding to two decimal places.  Fantastic!

Part 3

The third part of our problem asks us to “construct a normal quantile plot and select the right option from four options given.”  We've already constructed a normal quantile plot, but I find it easier to make the comparison for the right answer if I match the values on my axes for the normal quantile plot that I constructed in StatCrunch with those for my answer options.  Notice here in the answer options, the x axis has a minimum value of 30 and a maximum value of 50, and the y axis has a minimum value of -2 and a maximum value of +2.

So I'm going to change the axes on my normal quantile plot to match.  To do that, I'm going to select this little three-line icon in the lower left-hand corner of my graph.  And when I left click on that icon, I get a menu where I can change the x-axis and the y-axis as well as change some of the graphical display on the graph.  I'm going to select X-axis to change the x-axis first.  I want a minimum value of 30 and a maximum value of 50.  And I'm going to change the y-axis so that it matches also.  And I'm going to reduce the width a little bit so that I get something that more or less matches what I'm seeing in my answer options.

Most students, when they get to this point, they're looking for an exact match.  But the way this is constructed, you just want to look for a general trend.  So look at the general pattern that the points in your graph are making, and find the answer option that has points with a similar pattern.  Here that answer is going to be Answer Option B.  Notice how the points don't exactly match up.  For example, here this first point in the graph we constructed is above the red line, but in the answer option we selected it's below the red line.  And you can see differences if you look at the last two points in the graph as well.  However, we're looking for a general pattern of the points with respect to each other, not with respect to the red line.  So I'm going to select this as my answer option.  Excellent!

Part 4

And now, the last part of the problem asks, “Do the data come from a normally distributed population?”  Well, in our normal quantile plot, what we see are points that more or less conform to that red line.  Yes, it's not exact.  Yes, there's some deviation of the points from the red line.  But for the most part, what we're looking for is a general trend.  So don't get caught up in little details when you're answering these types of questions.  Just look for a general trend.

The general trend here is that the data points are fairly close to the line.  And they're not making any sort of pattern like an S- or sinusoidal pattern that would indicate something other than a normal distribution.  So I'm going to conclude that, yes, these points are representing what is reasonably close to a straight line.  That is Answer Option C.  Nice work!

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