7/2/08
Calculating Geometric Mean Using
Logarithms (Logs)
By Richard G. Weigand, CET
INTRO
There are two widely recognized methods for calculating geometric mean: the Square
Root Method and the Log Method. Class II wastewater operators learn the simpler
square root (
) method, but it only works if you are averaging 2 samples, 4
samples, 8 samples, 16 samples, 32 samples, 64 samples, etc. Otherwise, the log
method must be used. So if you sample fecal coliform 3 or 5 or 6, 7, 9, 10, etc. times
per month, use the log method described below. A review of how to use the square
root method is on page 4.
GETTING STARTED
You must have a scientific calculator that has a LOG function key on it. These
calculators are now very affordable. Remember the only time you use geometric
mean is when you are averaging all the fecal coliform results from the month in order
to complete a DMR.
HERE’S HOW
Let’s learn using some examples:
1.
The lab sends you these 3 results from last month: 250 col/100 mL;
1,500 col/100 mL; 1,200 col/100 mL.
A)
Calculate the log of each number by entering the number, then
pressing the LOG key:
250 LOG = 2.3979
(four places to the right of the decimal
is accurate enough!)
1,500 LOG = 3.1761
1,200 LOG = 3.0792
B)
Let’s average the three logs by adding them, then dividing them by 3.
2.3979 + 3.1761 + 3.0792 = 8.6532 = 2.8844 average log
3
7/2/08
C)
Now we must reverse the log process by taking the “anti-log” to get
the final answer. Different calculators label the anti-log in different
ways. Usually it is labeled 10x and is located above the LOG key.
Since it is the “second function” for the LOG key, we must push the 2nd
key first.
Often (such as with Texas Instrument brand calculators), the second
function key is yellow and is located in the upper left hand corner of the
keypad.
Here is our example:
2.8844 2nd LOG (actually 10x)
766.3
So the Geometric Mean is 766.3. Since we can’t count part of a FC
bacteria, always round this number up to the next whole number –767.
That’s what goes on the DMR (plus 2 excursions!).
2.
Lab results are: 65; 10; 2,300; 120;
A)
2
Calculate the log of each:
65 LOG = 1.8129
10 LOG = 1.0000
2,300 LOG = 3.3617
120 LOG = 2.0792
2 LOG = 0.3010
B)
Average the logs:
1.8129 + 1.0000 + 3.3617 + 2.0792 + 0.3010 = 8.5548
5
1.7110 average log
C)
Take the anti-log:
1.7110 2nd LOG (actually 10x) = 51.4
-
Report 52
=
7/2/08
NOTE
 If you have a “less than” result, just use the number.
2 use 2;
5 use 5;
10 use 10.
 If a zero is reported for some reason, use 1 in the calculation; then call the lab
and tell them they are not calculating their results correctly. There should
never be a zero.
 If you get a “greater than” result, just use the number.
2,000 use 2,000;
5,400 use 5,400.
 The log method will also work on 2, 4, 8, 16, 32 . . . etc. sample results.
3)
Lab results are:
A)
5; 110; 260; 4,250
Calculate the logs:
5 LOG = 0.6990
110 LOG = 2.0414
260 LOG = 2.4150
4,250 LOG = 3.6284
B)
Average the logs:
0.6990 + 2.0414 + 2.4150 + 3.6284 = 8.7838
4
2.1960 average log
C)
=
Take the antilog:
2.1960 2nd LOG = 157.04
-
Report 158
Here are a few practice problems with the answers below.
4)
65
125
20
5)
265
2
1,400
6,500
6)
2
10
110
5
7)
5
20
5,850
5)
264
6)
21
7)
84
Answers
4)
55
Still have questions? Call the ETC at (304) 372-7878.