Metrology Monday! #61 A Refresher on Significant Digits

Here is another requested topic, one that I had the chance to brush up on a few years ago when working on the revision of ILAC P14.? I am sure that everyone has had training on this at one time or another, but it is something that can be easily forgotten, understanding significant digits when reporting measurements and uncertainty.

The first thing that I want to share on this topic is that some publications use the term significant digits and some use the term significant figures, but I have not been able to find a publication that indicates that there is a difference, so I will use the term significant digits in this article.

So why do we care about significant digits?? As I noted back in post #6, we need to employ proper significant digits because reporting a measurement uncertainty with too many digits infers a level of knowledge in the uncertainty estimate that does not realistically exist.

I recall having a discussion with a Fluke engineer that wanted a very aggressive uncertainty specification for a product.? The uncertainty that he was advocating for was less than the resolution of the device.? His reasoning was that “he got extra digits by averaging” which would justify an uncertainty a lot less than the resolution.? What he forgot about was the rules associated with significant digits.

ILAC P14:09/2020 has requirements about the number of significant digits which can be reported in an uncertainty statement. Section 5.3 states “U (the expanded uncertainty) shall be reported at most, to two significant digits.”

The ISO Guide to the Expression of Uncertainty in Measurement states in 7.2.6 “The numerical values of the estimate y and its standard uncertainty uc(y) or expanded uncertainty U should not be given with an excessive number of digits. It usually suffices to quote uc(y) and U [as well as the standard uncertainties u(xi) of the input estimates xi] to at most two significant digits, although in some cases it may be necessary to retain additional digits to avoid round-off errors in subsequent calculations.”

A significant digit is any digit in a number that is necessary to define a numerical value of a quantity. Significant digits indicate the precision of a measurement, observation, or numerical value. If a number is exact, all its digits are significant. If a number is inexact, count only the nonzero digits. The exception is that the zero digits may be significant only if:

1) Zeroes placed before other digits are not significant; 0.039 has two significant digits.

2) Zeroes placed between other digits are always significant; 7009 kg has four significant digits.

3) Zeroes placed after other digits but behind a decimal point are significant; 5.90 has three significant digits.

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Here are some fundamental rules about significant digits:

???????? Digits of a number are called significant digits if the corresponding number is considered to lie within the error limits of the last digit(s).

???????? Consider the number 401 000. Here, 401 contains three significant digits, but it is not known if the right-most three zeros are significant or are just used to indicate the order of magnitude. It is recommended to indicate that distinction in the following way:

???????? 401 × 103 three significant digits

???????? 401.0 × 103 four significant digits

???????? 401.00 × 103 five significant digits

???????? 401.000 × 103 six signifiant digits

???????? All digits after a decimal sign are considered to be significant

???????? Rounding is only carried out in one (the last) step, to avoid errors from rounding prematurely.

?Significant Digits in Multiplication, Division, Trig. Functions, etc.

????????? In a calculation involving multiplication, division, trigonometric functions, etc., the number of significant digits in an answer should equal the least number of significant digits in any one of the numbers that are to be multiplied, divided etc.

Significant Digits in Addition and Subtraction

???????? When quantities are being added or subtracted, the number of decimal places (not significant digits) in the answer should be the same as the least number of decimal places in any of the numbers being added or subtracted.

Keep at least one digit in intermediate answers/calculations and any number less than 1 must have a leading zero (e.g. 0.123 mV).

Next week I will cover rounding rules to complete this topic and share a few more examples of both significant digits and rounding.? If you see me at the Measurement Science Conference this week, please come up and say hello! #MetrologyMonday #FlukeMetrology ??