Science & Research
Volume III - 4.3 Data Handling and Presentation
Section 4 - Basic Statistics and Presentation
|EFFECTIVE DATE: 10/01/2003||REVISED: 1-31-13|
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:
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|
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").
220.127.116.11 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:
18.104.22.168 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:
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
A calculator yields an answer of 4.975740998 = 5.0, since one of the measurements has only two significant figures.