Finding the mean of a frequency 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 of a frequency distribution.  Here's our problem statement: Find the mean of the data summarized in the given frequency distribution.   Compare the computed mean to the actual mean of 58.2 degrees.

So we have a table here listing temperatures and frequency counts.  So let's go ahead and click this icon to the right.  We're gonna open this in StatCrunch.  And let's just move our window over here a little bit.  There!  Now that we've resized our window, we can see everything.

Part 1: Find the class midpoints

So we have our data here in StatCrunch, but notice these are frequency counts.  And the way we can do this really easy in StatCrunch requires us to have first midpoints for each of the classes.  What we have listed here are actually the lower and upper class limits for each of our bins, so that's not going to help us with StatCrunch.

So we have to actually calculate out here in another column what the actual frequency midpoints are going to be.  So there's my calculator.  I'm just going to take the average of my upper and lower class limits for that first class there.  So 44 plus 40 is 84 divided by 2 is 42.  So my first midpoint is 42.  I'm gonna do it again for the next class.  Punch that out — 49 plus 45 divided by 2 is 47.

Now if you notice, 47 is five more than 42, and that's because 45, the lower limit for the second class, is five more than 40, which is the lower limit for the first class.  Or you can compare the upper limits.  You can see they're also separated by five.  So I can just go ahead and just add five to each one of these midpoints here to get the rest for my table.  If you want, you could actually calculate the averages like we were doing with the first two bins, but you'll come out with these numbers here.

Part 1: Find the distribution mean

And now we're ready to find the mean of the frequency distribution.  StatCrunch makes this super easy!  You don't have to use that obnoxious formula that's in your textbook!  So now that we've got the midpoints and we've got the frequency counts, let's go up to Stat, I'm gonna click on Calculators, and then go down here to the bottom where it says Custom.

Values are going to be the midpoints that we just calculated.  I'm gonna select that.  Column weights are the frequency counts, so I select that column.  And then I just hit Compute! and voila!  Look here — the mean value 52.869565.  We want to the nearest tenth, so I round that to 52.9.  Well done!

Part 2: Interpret the mean values

And now, the second part of the problem: Which of the following best describes the relationship between the computed mean and the actual mean?  So if we look at our answer options here, the computed mean is either close or not close to the actual mean, and then the difference is being compared to five percent, so we’re either more than five percent or less than five percent.

So if I come back here to my calculator — clear that out — let's compute the actual difference so we know we're looking at.  52.9 is the calculated or computed mean, 52.8 — excuse me, 52.8 degrees listed here in the problem statement is the actual mean.  So we're gonna take the actual mean 52.8 — excuse me, 58.2 — subtract out the computed mean (52.9).  This is 5.3.  Divide by the actual 52.8, multiply by 100, and as you can see, we're at 9.1%, which is greater than that 5% threshold.  So therefore we're gonna conclude that it's not close to the actual mean because we're more than five percent off.  Check our answer.  Nice work!

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