| Other Laboratory Operations Food and Drug Administration |
| DOCUMENT NO.: III-04 | VERSION NO.:1.2 | Section 4 - Basic Statistics and Presentation | EFFECTIVE DATE: 10/01/2003 | REVISED: 06/27/2008 |
4.3 Data Handling and Presentation
In the most general sense, analytical work results in the generation of numerical
data. Operations such as weighing, diluting, etc. are common to almost every
analytical procedure, and the results of these operations, together with instrumental
outputs, are combined mathematically to obtain a result or series of results.
How these results are reported is important in determining their significance.
As a regulatory agency, it is important that we report analytical results in
a clear, unbiased manner that is truly reflective of the operations that go
into the result. Data should be reported with the proper number of significant
digits and rounded correctly. Procedures for accomplishing this
are given below:
4.3.1 Rounding of Reported Data
When a number is obtained by calculations, its accuracy depends on the accuracy
of the number used in the calculation. To limit numerical errors, an extra
significant figure is retained during calculations, and the final answer rounded
to the proper number of significant figures (see next section for discussion
of significant figures).
The following rules should be used:
- If the extra digit is less than 5, drop the digit.
- If the extra digit is greater than 5, drop it and increase the previous
digit by one.
- If the extra digit is five, then increase the previous digit by
one if it is odd; otherwise do not change the previous digit.
Examples are given in the following table:
Calculated Number |
Significant digits
to report |
Number with one extra
digit retained |
Reported rounded number |
79. 35432 |
4 |
79.354 |
79.35 |
99.98798 |
5 |
99.9879 |
99.988 |
32.9653 |
4 |
32.965 |
32.96 |
32.9957 |
4 |
32.995 |
33.00 |
0.0396 |
1 |
0.039 |
0.04 |
105.67 |
3 |
105.6 |
106 |
29 |
2 |
29 |
29 |
4.3.2 Significant Figures
Significant figures (or significant digits) are used to express, in an approximate
way, the precision or uncertainty associated with a reported numerical result.
In a sense, this is the most general way to express "how well" a
number is known. The correct use of significant figures is important in today's
world, where spreadsheets, handheld calculators, and instrumental digital
readouts are capable of generating numbers to almost any degree of apparent
precision, which may be much different than the actual precision associated
with a measurement. A few simple rules will allow us to express results with
the correct number of significant figures or digits. The aim of these rules
is to ensure that the final result should never contain any more significant
figures than the least precise data used to calculate it. This makes intuitive
as well as scientific sense: a result is only as good as the data that is
used to calculate it (or more popularly, "garbage in, garbage out").
4.3.2.1 Definitions and Rules for Significant Figures
- All non-zero digits are significant.
- The most significant digit in a reported result is the
left-most non-zero digit: 359.741 (3 is the most significant digit).
- If there is a decimal point, the least significant digit in
a reported result is the right-most digit (whether zero or not): 359.741 (1
is the least significant digit). If there is no decimal point present, the
right-most non-zero digit is the least significant digit.
- The number of digits between and including the most and least significant
digit is the number of significant digits in the result: 359.741
(there are six significant digits).
The following table gives examples of these definitions:
|
Number |
Sig. Digits |
A |
1.2345 g |
5 |
B |
12.3456 g |
6 |
C |
012.3 mg |
3 |
D |
12.3 mg |
3 |
E |
12.30 mg |
4 |
F |
12.030 mg |
5 |
G |
99.97 % |
4 |
H |
100.02 % |
5 |
4.3.2.2 Significant Figures in Calculated Results
Most analytical results in ORA laboratories are obtained by arithmetic combinations
of numbers: addition, subtraction, multiplication, and division. The proper
number of digits used to express the result can be easily obtained in all
cases by remembering the principle stated above: numerical results are reported
with a precision near that of the least precise numerical measurement used
to generate the number. Some guidelines and examples follow.
Addition and Subtraction
The general guideline when adding and subtracting numbers is that the answer
should have decimal places equal to that of the component with the least number
of decimal places:
21.1
2.037
6.13
________
29.267 = 29.3, since component 21.1 has the least number of decimal places
Multiplication and Division
The general guideline is that the answer has the same number of significant
figures as the number with the fewest significant figures:
56 X 0.003462 X 43.72
1.684
A calculator yields an answer of 4.975740998 = 5.0, since one of the measurements
has only two significant figures.
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