Finding measures of variation before and after adding a data point

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 measures of variation before and after adding a data point.  Here's our problem statement: Use the magnitudes of Richter scale of the 120 earthquakes listed in the accompanying data table.  Use technology to find the range, variance, and standard deviation.  If another value (7.0) is added to those listed in the data set, do the measures of variation change much?

Part 1

OK, the first part here is asking us to find the range.  To do that, I'm going to take my data set here and dump it into StatCrunch here.  And let's resize this window so we can see a little bit better what's going on.  There we go.  Now my data's here in StatCrunch.

So to find the range, I'm simply calculating a statistic for data that I have in my column.  And to do that, I go to Stat –> Summary Stats –> Columns.  I select the column where my data is located, and then here under Statistics, I'm just gonna select the statistic that they're asking for — in this case, the range.  Then I press Compute! and there's my range.  I'm asked to round to three decimal places, which in this case is no big deal.  Well done!

Part 2

Now the next part asks me to find the standard deviation.  I could go through these menu options again, or I could just go up here to Options in my results window, and then in the drop down menu click Edit, and it takes me right back to the same options window where I can just select the new standard deviation that I want to calculate.  Notice in the list here, there are actually two variances and standard deviations.  OK, there’s two at the top, and then there's two here at the bottom (they're listed as unadjusted).  The unadjusted variety is for populations.  Here we're calculating a standard deviation for the sample, so we want to make sure that we select the standard deviation up here at the top, which we have.  I just press Compute!.  I'm asked to round to three decimal places.  Fantastic!

Part 3

The next part asks me to calculate the variance, so I go back to my options window, switch over to the variance.  I'm asked to round to three decimal places again.  Nice work!

Parts 4-6

Now we're asked — with the extra data value — what is the range?  Well, notice how we're gonna go through the same — it looks like we're gonna go through the same three statistics to calculate, but this time we're adding in an extra data value, which from the problem statement is 7.0.  So first I need to come back over here to my data set, and I'm gonna scroll down here to the bottom, and then right here at the bottom I'm going to stick in that extra data point.

Now I go back to my options window, and I know I'm gonna have to calculate the same numbers again, so I'm just going to select them in the same order: the range, standard deviation, and the variance.  This time I don't have to keep going back and forth; I get everything I need for the rest of the problem here in one handy dandy results window.  And so I'm just going to take the numbers there off the results window.  And I'm asked again to round to three decimal places.  Good job!

Part 7

Now the last part of this problem asks, “Do the measures of variation change much with the extra data value?”  If I look on these drop down boxes for the blanks that I'm supposed to fill in, notice how we're using five percentage points as the boundary between what's statistically significant and what's not.  So to determine whether the statistics that I've calculated up here fall into that, I'm just gonna pull up my calculator.

And here we're just gonna start with the range because that's what was asked for first.  So what's the change in the range from a percentage point of view?  Well, I'm going to take the difference between the two and then subtract it and — excuse me, divide — by the original amount.  So I'm going to start with 5.86, subtract out 3.6, and then I'm going to divide that by the original amount (3.6), and I get around 62%, which is well over 5%.  So this first one is going to be more than five percentage points.

I'm going to repeat the same calculation for the remaining statistics.  So the next one they have here is the variance.  So we go with 0.589, subtract out the original value point (0.427), and then divide it by the original amount.  Again, we're well over 5%, so that's going to be more here.  I'll do the same thing with my standard deviation values: 0767 minus 0.653, and then I'm going to divide that by 0.653.  And again, we're well over 5%.

So if they were within 5%, we could say they didn't really change much.  But they all changed by well more than 5%, so there are significant differences by adding in that extra data point.  So here in this last blank, we're gonna select “all of them change.”  Excellent!

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