# Help in understanding n-th root in geometric mean?

#### oly

##### New Member
I'm pretty much of a layman in stats, but I've had to calculate the geometric mean, now and then in the past, in my former job as a reporter.

Now, I'd like to understand the concept better in order to instruct others in the technique. The one thing that I've always found confusing has been which number to use as the n-th root in the calculation. Stuff I've found on the Web has been helpful in some areas but not very clear on this point. So I'm hoping someone here can clarify it all for me.

Most importantly, I would like to know the rationale for choosing the correct n-th root number in calculating the geometric mean.

As an exercise, I'm trying to figure out the average per-year change in a certain type of incident (robberies) that occurred between 1993 and 2003 in one small Illinois municipality. Here are the numbers and what i've done so far:

YEAR NO.
1993--432
1994--420
1995--573
1996--556
1997--507
1998--327
1999--335
2000--329
2001--258
2002--303
2003--282

There are actually 11 values in this distribution. But the formula I'm using is (282/432)^(1/10), assuming that in the first year, there is no growth (Am I right in assuming this?). This calculation yields a geometric mean of 0.958244943.

In addition, using MS Excel, I took the above numbers and created a third column showing rates of increase and decrease, as follows.

YEAR NO. CHANGE
1993--432--N/A
1994--420--0.972222222
1995--573--1.364285714
1996--556--0.970331588
1997--507--0.911870504
1998--327--0.644970414
1999--335--1.024464832
2000--329--0.982089552
2001--258--0.784194529-
2002--303--1.174418605
2003--282--0.930693069

Using Excel's GEOMEAN spreadsheet function I selected this third column and, after hitting [enter], got the same answer for the geometric mean of 0.958244943, which is an average 4.18 percent annual decrease in robberies.

OK, so here are my questions. First, have I done this all correctly? Second, is the n-th root in the geometric-yield formula always one less than the total number of values in the distribution and WHY???

And third, how do I use the geometric mean I've calculated to check and verify my incident numbers in this case? I've taken the 1993 number of 432, multiplied it by the geometric mean of 0.958244943 and done that with the resulting product nine more times. I don't get the 2003 number of 282 robberies--I get 290.