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# Rounding Numerical Values Up or Down with respect to the number of Significant Digits

B

#### BertPeerboom

Hello,

I would like to discuss the following:

the upper limit of a chemical in a certain matrix is 20.
The measured value is 20.3
2 significant digits have to be reported.

Do I round off before or after the desicion weather the value is within limits?

Rounding off before: the result is within limits -> NOT OK
Rounding off after: the result is not within limits -> OK

Bert Peerboom

#### tacom

##### Involved In Discussions
Re: Rounding Numerical Values Up or Down with respect to the number of Significant Di

Hello,

I would like to discuss the following:

the upper limit of a chemical in a certain matrix is 20.
The measured value is 20.3
2 significant digits have to be reported.

Do I round off before or after the desicion weather the value is within limits?

Rounding off before: the result is within limits -> NOT OK
Rounding off after: the result is not within limits -> OK

Bert Peerboom
Upper limit is 20 (2 significant figure) not 20,0 (3 significant figure) or 20,00 (4 significant figure). So, the result (20,3) should be given 2 significant figure (20).
İf limit value is ≤ 20 , the result is within limits
TACO

#### v9991

Trusted
Re: Rounding Numerical Values Up or Down with respect to the number of Significant Di

on the contrary...refer the document right from the agency...

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.

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”).
I personally believe that it must have additional criteria. (not only reporting criteria/requirements); i.e., capability of the method or instrument (technology)
for eg., in your example, if the instrument/technology can provide an sensitivity/accuracy of 0.01 or 0.1 (one additional digit than that is required for reporting) you ought to take that fact into account, before rounding/reporting the final value.

W

#### WKHANNA

Re: Rounding Numerical Values Up or Down with respect to the number of Significant Di

Sorry if this is a silly question, I'm just trying to learn.
What about the measurement system error?
If the measured value is actually lower than the true actual value, then the reported value would be that much more beyond the UL spec.

B

#### BertPeerboom

Re: Rounding Numerical Values Up or Down with respect to the number of Significant Di

Upper limit is 20 (2 significant figure) not 20,0 (3 significant figure) or 20,00 (4 significant figure). So, the result (20,3) should be given 2 significant figure (20).
İf limit value is ≤ 20 , the result is within limits
TACO
Thanks TACO

#### Statistical Steven

##### Statistician
Staff member
Super Moderator
Re: Rounding Numerical Values Up or Down with respect to the number of Significant Di

Hello,

I would like to discuss the following:

the upper limit of a chemical in a certain matrix is 20.
The measured value is 20.3
2 significant digits have to be reported.

Do I round off before or after the desicion weather the value is within limits?

Rounding off before: the result is within limits -> NOT OK
Rounding off after: the result is not within limits -> OK

Bert Peerboom
I always use the specification to set the significant figures for reporting.

If the specification is less than 20 then 20.3 is acceptable (reported as 20)
If the specification is less than 20.0 then 20.3 is unacceptable (reported as 20.3)
If the specification is less than 20.00 then 20.3 is unacceptable (reported as 20.30)

That is how I see it

W

#### WKHANNA

Re: Rounding Numerical Values Up or Down with respect to the number of Significant Di

OK, that make sense to me.
Thanks, Steven!

#### v9991

Trusted
Re: Rounding Numerical Values Up or Down with respect to the number of Significant Di

I always use the specification to set the significant figures for reporting.
If the specification is less than 20 then 20.3 is acceptable (reported as 20)
thats valid from straight forward logic,

howerver, look at it from perspective of
case- 1) what if your method/instrument is capable of accurately reporting "".00"" 2 decimals(i.e., 1 more decimal than requirement of specification...), would you still accept the result as passing?
case-2 ) this also depends upon the test parameter you are considering; consider a difference between impurities & assay/moisture content; once/if you have scenario of case-1, you would still need to report that additional decimal which is demonstrable...and have to fail it;

most commonly, in pharma atleast, we bring in above two factors into play (case-2 is most common inclusion in the rounding examples)
hope that clarifies.

#### Statistical Steven

##### Statistician
Staff member
Super Moderator
Re: Rounding Numerical Values Up or Down with respect to the number of Significant Di

thats valid from straight forward logic,

howerver, look at it from perspective of
case- 1) what if your method/instrument is capable of accurately reporting "".00"" 2 decimals(i.e., 1 more decimal than requirement of specification...), would you still accept the result as passing?
case-2 ) this also depends upon the test parameter you are considering; consider a difference between impurities & assay/moisture content; once/if you have scenario of case-1, you would still need to report that additional decimal which is demonstrable...and have to fail it;

most commonly, in pharma atleast, we bring in above two factors into play (case-2 is most common inclusion in the rounding examples)
hope that clarifies.
This is exactly the issue....

Your measurement system should always be capable of reading to at least one extra decimal place then the specification. The specification DICTATES the level of precision required.

So for case 1....the measurement device is just that, it reports the measurement. The specification determines if that reading is acceptable.

So for case 2...if you have a requirement of moisture content of less than 5% then a moisture reading of 5.499999999999999999999999999% is in specification. If you need MORE precision than the specification would be less than 5.0% so that a reading of 5.04999999999999999999999% is acceptable.

In the pharma industry your significant figures and GDP procedure will determine the rules for your organization.

A

#### allan-M

Re: Rounding Numerical Values Up or Down with respect to the number of Significant Di

Also check out ASTM standard E 29 – 04 "
An American National Standard Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications"

It might be similar to the fda rule

Allan