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B: DMR Calculations
Averages
Data averages recorded on DMRs should produce the arithmetic mean for all parameters with the exception of Fecal Coliform. The average for Fecal Coliform should be calculated as the geometric mean of the values.
Arithmetic Mean
This is the simple, common averaging of a series of numbers. You add a group of numbers together to get the sum. Then you divide the sum by the number of values you added to get the sum. The result is the arithmetic mean or average of the series of numbers.
Example:
5.30
6.21
4.00
5.25
+ 8.72
29.48
29.48 / 5 = 5.896 (round off to 5.90 = average)
This calculation is used for averaging of all parameters except for Fecal Coliform (code number 31616).
Geometric Mean
There are two ways to go about calculating the geometric mean. The two procedures are really just different ways of doing the same thing and either way yields the same result, but both require the use of a scientific calculator. Calculators adequate for performing such operations as the geometric mean can probably be purchased for less than $15.00. With the calculator, follow these steps:
PROCEDURE 1 (we think this is the easier of the two)
1. Multiply all the data values together.
2. Take the "nth" root of the product of the multiplication, where "n" is the number of values multiplied. In other words, if you multiply 4 values and get a result, take the 4th root of the product. This is the geometric mean.
To do this on the calculator, get the product of multiplication, then press (1) the "INV" key, (2) the "yx" key, and (3) the "nth" root number. Labeling of keys and functions will vary with different brands of calculators. Consult your calculators users manual for the specific procedure to perform these functions.
Example: (Using as data the numbers: 50, 100, 150 and 200)
50 x 100 x 150 x 200 = 150,000,000 (1.5x108)
4th root of 150,000,000 = 111 (rounded from 110.66819 - value on calculator)
use only whole numbers when reporting fecal coliform back to top
PROCEDURE 2
1. Add together the base 10 logarithms for the data values.
2. Divide the sum by the number of values added.
3. Take the antilog of the result of step 2. This will again be the geometric mean.
To do this on the calculator, key the data value and press the "log" key. Then press "+" and repeat for all values. After the last logarithm is entered, press "=" to get the sum. Divide by the number of values that were added. Press the "INV" key and then press the "log" key.
Example: (again, using 50, 100, 150 and 200 as data values)
Base 10 logarithm of: 50 is 1.69897
100 is 2.0
150 is 2.1760913
200 is + 2.30103
Sum = 8.1760913
8.1760613 / 4 = 2.0440228
Antilog of 2.0440228 = 111 (rounded from 110.66819 - value on calculator)
* Calculation may be performed in the same manner (although calculation values will be different) using natural logarithms (ln x).
Use of "Less Than" Values
Complications may arise in calculations when dealing with testing results showing values of less than a minimum detection level for the testing method. Current Division policy gives permittees the benefit of doubt all the way to the lowest levels when performing calculations using such "less than" values. When calculating a sum or arithmetic mean for compliance purposes, you may consider a "less than" value as equal to zero. For the calculation of a geometric mean, a "less than" value may be considered to be equal to one. Remember, this procedure pertains only to the results of calculations. You must report individual data values on the DMR exactly as reported to you by your laboratory. For example, a daily result of <1.0 mg/L must be recorded as such, not as ND, LT, 0, etc. If you are doing calculations with "less than" (<) values, here is how they should be handled for compliance purposes:
a) Calculating a sum of results containing less than values
In calculating a sum, such as in adding TKN and NO3-NO2-N results to get Total Nitrogen, "less than" values may be considered to equal zero (0). (Remember that the analyses must be sufficiently sensitive to establish compliance.) If all results for a particular sample are "less than," the sum for the sample would also be zero.
Example: (using 3.2 mg/L NO3-NO2-N and <0.5 mg/L TKN as data values)
3.2
+ 0 (<0.5)
3.2
b) Calculating an arithmetic mean of results containing less than values
Likewise, in calculating the arithmetic mean, "less than" values may be considered to equal zero (0). If all results for a particular parameter during the month are "less than," the average for the month would be zero. The maximum and minimum, however, should be recorded as the "less than" values reported by your lab (<5, etc.).
Example: (using 10, 15, 20 and <5 as data values)
10
15
20
+ 0 (<5)
45
45 / 4 = 11.25 (round off to 11)
c) Calculating a geometric mean of results containing less than values
In calculating the geometric mean, all "less than" values may be considered to equal one (1). This is due to the nature of the calculation; zeros may not be used in the calculation of the geometric mean. If all testing results for Fecal Coliform during a particular month came back as "less than" values, the geometric mean for the month would be one (1). The maximum and minimum for the month, however, should be recorded as the "less than" values.
Example: (using Procedure 1 noted above and 10, 15, 20 and <5 as data)
10
15
20
x 1 (<5)
3000
4th root of 3000 = 7.400828 (record on DMR as 7)
If these procedures are followed, there is never an instance when a "less than" value should be reported as a monthly average. Using these procedures, a discreet value can always be obtained for either the arithmetic or geometric mean. We (the compliance staff) realize that this policy does not necessarily represent good chemistry, but it allows for a standard practice in dealing with this type of data.
Please note that it is a requirement of your permit that you utilize testing methods that can evaluate the discharge to levels low enough so as to demonstrate compliance with permit limits. For example, if you are required to monitor for a parameter with a limit of 50 mg/L, you must utilize a test that can analyze to at least that level. If current laboratory technology will not allow for a parameter to be analyzed to the permit limit, you must utilize the best available method for that parameter. If the analyses using that method show no detection of the parameter in question, you will be considered in compliance. Questions regarding laboratory methods and practices can be answered by the staff of the Division of Water Resources Laboratory Section, by calling (919) 733-3908.
Use of "Greater Than" Values
Such values are only expected (infrequently) in the reporting of Fecal Coliform and (even less frequently) BOD. It is a violation of reporting rules to report a Fecal Coliform value of "Too Numerous To Count" (TNTC). For fecal coliform, "greater than" (> ##,###) values denote at least one, and possibly all volumes of the evaluated sample yielded results outside the method range for accurate counting of the colonies of bacteria (or TNTC). Laboratories should perform enough dilutions to the sample to produce a discreet number as the result of testing. If a "greater than" value is reported, the numeric portion of the value should be sufficiently high so as to make the facility aware of the extent of any problems with disinfection.
For both fecal coliform and BOD, the generation of a "greater than" value usually occurs when characteristics of the effluent differ from what is normally discharged. If laboratories "set up" the test procedures based upon normal conditions, they may not have the capacity to get an accurate measurement of higher fecal coliform or BOD concentrations, hence the reported "greater than" value. If you are suspicious or aware of conditions at your plant that cause you to believe effluent pollutant concentrations are beyond their normal levels, you should inform your laboratory of your concern so modifications to testing procedures can be made.
For calculation purposes only, when you report a "greater than" value, the numeric portion of the value must be used to calculate the average (arithmetic or geometric mean). The actual result of testing must be reported in the daily cell on the DMR.
Conversion from mg/L to lbs/day
Some permits have parameters limited in units of pounds per day (lbs/day). Laboratories report the concentrations (such as mg/L) of sample characteristics as a result of testing. They cannot report the daily load in lbs/day because that total is dependant upon the amount of flow carrying a known concentration. But if the concentration of a pollutant and the daily flow from the facility are known, the daily load in lbs/day can be calculated:
Multiply the concentration (in mg/L) x daily flow (in MGD) x 8.34 (a constant). This formula will yield the result in units of lbs/day. You must be certain to use data values with the noted units of measurement to get the proper result from this particular formula. Other units of measurement (like mg/L or GPD) may be used, but you will need to modify the constant (which is a composite number representing all the unit conversions).
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