**Each day, literally hundreds of measurements are made in laboratories using several types of test methods or standards. This article explains why the South African Bureau of Standards (SABS) laboratories (including the National Electrical Test Facility (NETFA)) use a particular rounding-off procedure in their test results. This can have implications in determining whether a product passes or fails particular tests.**

Numerical measurements involve numbers in any format prescribed by the method or standard. These results will then be issued in several types of reports depending on its application, e.g. test reports, research reports, publications, etc. Raw data, in the form results from the physical measurements in the laboratory, cannot be left unprocessed in a final report, as all results and all measurements have some uncertainty. The following article shows all the possible influences on the outcome of a numerical measurement.

**Significant figures**

Any numeral (including zero) which is necessary to describe a number fully is significant. When a laboratory analyst performs measurements and notes results, not all of the digits are significant. In the fields of science and engineering, only the numbers that have significance are reported or used. When determining the significance of a number there are a few rules to be followed:

- Non-zero numbers are always significant.
- All zeros between significant numbers are also significant.
- Zeros in the decimal part of the number following numerals are also significant.
- Zeros in the decimal part of a number that are not preceded by other significant numerals are not significant.
- Trailing zeros in an integer are not significant as they are only a place holder or convention.
- Rounding off should only occur in the final phase of data processing and analysis.

Number |
Ascertain to the nearest |
Significant figures |
Number of significant figures |

0,01750 | 0,00001 | 1, 7, 5 & 0 | 4 |

0,0175 | 0,0001 | 1, 7 & 5 | 3 |

20,0175 | 0,0001 | 2, 0, 0, 1, 7 & 5 | 6 |

3455,0 | 0,1 | 3, 4, 5, 5 & 0 | 5 |

3,455 x 10^{3} |
1 | 3, 4, 5 & 5 | 4 |

*An example of why significant number are important*

A test officer once needed a cube of metal which had to have a mass of 83 g. He knew the density of the metal was 8,67 g/m*l*, which told him the cube’s volume. Believing significant figures were invented just to make life difficult for analysts and had no practical use in the real world, he calculated the volume of the cube to be 9,573 m*l*. He thus determined that the edge of the cube had to be 2,097 cm. He took his plans to the machine shop where a colleague had had the same type of work done the previous year. The shop foreman said, “Yes, we can make this according to your specifications – but it will be expensive.”

“That’s OK,” replied the test officer. “It’s important.” He knew his colleague had paid R35, and he had been given R50 out of the budget to get the job done.

He returned the next day, expecting the job to be done. “Sorry,” said the foreman. “We’re still working on it. Try next week.” Finally the day came, and the test officer got his cube. It looked very, very smooth and shiny and beautiful in its velvet case. Seeing it, our hero had a premonition of disaster and became a bit nervous. But he summoned up enough courage to ask for the bill. “R500, and cheap at the price. We had a terrific job getting it right to three significant figures after the decimal comma – we had to make three before we got one right”[1].

**Precision of measurements or values**

Precision is the condition of being accurate by a degree of refinement in measurement and is highly significant for analysts in testing laboratories, since it is the number of significant digits used to make a calculation. Precision is a description of random errors, a measure of statistical variability. Accuracy has two definitions: More commonly, it is a description of systematic errors, a measure of statistical bias; as these cause a difference between a result and a “true” value. The International Standards Organisation (ISO) calls this “trueness” (see Fig. 1) [2].

**Rules of rounding off**

It is popularly believed and often taught that rounding-off is as simple as rounding up for values of 5 to 9; and rounding down for values between 1 and 4. This rule might cause some incorrect outcomes because it is biased to the upper range 5 to 9, giving false positive outcomes. A case in point are medical results, where false positives can be obtained if statistically inaccurate rounding-off policies are followed.

When you round-off, you change the value of the number, and in the case where you have many numbers which are rounded-off, you will have four out of nine numbers which will be rounded down and five out of nine which will be rounded up, which causes the average to be biased upwards.

The consequence of this method is that an average or mean of the data set will be higher than the initial measurement, which is not acceptable. Statistically, it can be changed so that 50% of the data is rounded down and 50% is rounded up, resulting in a more accurate average, and thus a more accurate way of working.

This is achieved by achieved the following a set of rules:

- For numbers between 0 and 4 as well as numbers between 6 and 9 apply the well known rule by rounding down and up respectively e.g. 1,24 will be rounded down to 1,2 if two significant figures are required.
- When the value is exactly between two possible rounded off values the following rules apply:
- Increase the last digit to be retained by one if it is odd, e.g. 1,35 becomes 1,4 for two significant figures, and
- Leave it unchanged if it is even, e.g. 1,25 becomes 1,2 if two significant figures are required.

Rounding-off should always be done as a one-step process by examining the parts which follow the last digit to be retained as a whole. If the rounding-off is done in steps, the result maybe incorrect, for example:

Take the number 3,21475 and perform step-by-step rounding-off to three significant numbers. This will result in the following:

*Step 1:*3,2148*Step 2:*3,215*Step 3:*3,22

Whereas, by applying the direct method of rounding-off, the number would be 3,21.

Original number |
Rounding off to the nearest |
Rounded off number |

6,4500 | 0,1 | 6,4 |

5,500 | 1 | 6 |

0,04500 | 0,01 | 0,04 |

**Uncertainty of measurement (UoM)**

In accordance with the requirement of ISO/IEC FDIS 17025, the laboratory shall clearly define the decision rule when a customer requests a statement of conformity to a specification or standard if not already imbedded in the specification or standard. When compliance to a specification is made, it should be clear which probability of the expanded uncertainty has been used by a statement of the following:

“The reported uncertainties of measurement are based on a standard uncertainty multiplied by a coverage factor of k = 2, which provides a level of confidence of approximately 95%.”

There is a possibility that a test result becomes inconclusive as a result of the application of the UoM to the test result.

There are three possible outcomes when applying UoM to the test results:

- In the case where the specification limit is not breached by the measurement result plus the UoM with a 95% coverage probability: (Fig 2-(iv)).
*Outcome:*Compliance

- In the case where the specification limit is exceeded by the measurement result minus (plus) the UoM with a 95% coverage probability (Fig 2-(i)).
*Outcome:*Non-Compliance

- In the case where the result plus (minus) the UoM with a 95% coverage probability overlaps the limit, it is not possible to state compliance or non-compliance (Fig 2-(ii and iii)).
*Outcome:*Inconclusive

However, it is required from the test laboratory to make a conclusion on this inconclusive outcome if not specified in the product’s specification. (case (ii and iii) of Fig. 2).

*Statements of conformance*

ILAC-G8:03/2009 provides guidance on the decision rule on inconclusive results (with addition of UoM) specific for national or other regulations (Refer to clause 2.7). In a case where the result is within the pass criteria and the outcome can be stated as compliance, that is case 2 in Fig. 3.

**Therefore:**

- Case 1 results are conclusive and conformity statement is a pass.
- Case 2 results are inconclusive, but conformity statement is a pass (as most test results are below the upper limit).
- Case 3 results are inconclusive, but conformity statement is a fail (as most of the test results are above upper limit).
- Case 4 results are conclusive results and conformity statement is a fail.

This decision rule shall apply for upper as well as lower limit requirements.

**Conclusion**

Over a large set of data the round-to-even rule tends to reduce the total rounding error, with (on average) an equal portion of numbers rounding up as rounding down. This generally reduces skewing of the result upwards. The process of rounding-off makes the result less precise, and this why percentages will not always add up to a hundred. Ideally, data should be rounded appropriately, not too much and not too little. The incorrect application of UoM to results may change the outcome to such extent that it not a true reflection of the performance of a sample tested.

**References**

[1] http://dbhs.wvusd.k12.ca.us/SigFigs/SigFigRules.html

[2] ISO Definition in ISO 50000

[3] ILAC-G8:03/2009

Contact Bjørn Buyst, SABS, Tel 012 428 6996, bjorn.buyst@sabs.co.za

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