Significant Figures and Units

Overview: In reporting numerical results, it is important to include the correct number of significant digits. While determining the correct number of digits to include is a straightforward process, beginning students often overlook this important detail. Here we outline the rules involved in determining the appropriate number of digits to include when reporting results of calculations and experimental measurements.

Skills:

New terms:


Defining the Terms Used to Discuss Significant Figures

Significant Figures: The number of digits used to express a measured or calculated quantity.
By using significant figures, we can show how precise a number is. If we express a number beyond the place to which we have actually measured (and are therefore certain of), we compromise the integrity of what this number is representing. It is important after learning and understanding significant figures to use them properly throughout your scientific career.

Precision: A measure of how closely individual measurements agree with one another.
Accuracy: Refers to how closely individual measurements agree with the correct or true value.

Digits that are Significant

  1. Non-zero digits are always significant.
  2. Any zeros between two non-zero digits are significant.
  3. A final zero or trailing zeros in the decimal portion ONLY are significant.

Examples:

How many significant figures are in: 1. 12.548, 2. 0.00335, 3. 504.70, 4. 4000
  1. There are 5. All numbers are significant.
  2. There are 3. The zeros are simply placeholders and locate the decimal. They are not trailing zeros. They are not significant.
  3. There are 5. The two zeros are not simply placeholders. One is between two significant digits and the other is a final, trailing zero in the decimal portion. Hence, they are both significant.
  4. This is a bit confusing. It is somewhere between 1 and 4. In order to clarify, we need to convert this to scientific notation. If it were 4 x 103, there is one significant figure. If it were 4.000 x 103, then there are 4 significant figures.

Rules for Using Significant Figures

More Examples:

Addition and Subtraction. 12.793 + 4.58 + 3.25794 = 20.63094

Multiplication and Division. 56.937/0.46 = 130.29782609

Tidiness at the end of a calculation.

So you have carried out a calculation that requires a series of seven or eight mathematical operations and at the end, after punching everything into your calculator, you see the result "14.87569810512...". The question you should ask yourself is how many digits to include when reporting your final answer.

It is at this point that you must refer back to the quality of the data you were given (i.e., how many significant digits are included with the given data). We illustrate this here with one final example.

Three scientists determine the mass of the same sample of FeCl3. Scientist A works in a field laboratory and carries a portable balance for determining the sample mass, the balance can determine masses to the nearest +/- 0.1 g. Scientist B has a better, but still somewhat crude balance, which reports the mass to the nearest +/- 0.01 g. Scientist C has a balance, like the analytical balances you will find in chemistry laboratories at WU, that can determine sample masses to the nearest +/- 0.0001 g. If each scientist wants to indicate the total number of moles of FeCl3 in the sample, how will each do this in a way that reflects the precision of the instrumentation they are using? The three scientists all use the atomic masses suggested by IUPAC (International Union of Pure and Applied Chemistry), which are included in the table below.

 
Scientist A
Scientist B
Scientist C
given data







  • sample mass:
    19.0 g
  • Fe atomic mass:
    55.847 g/mol
  • Cl atomic mass:
    35.4527 g/mol
  • sample mass:
    18.99 g
  • Fe atomic mass:
    55.847 g/mol
  • Cl atomic mass:
    35.4527 g/mol
  • sample mass:
    18.9925 g
  • Fe atomic mass:
    55.847 g/mol
  • Cl atomic mass:
    35.4527 g/mol
reported
moles FeCl3
  • 0.117 mol FeCl3
  • 0.1171 mol FeCl3
  • 0.11709 mol FeCl3
Why?
The balance used for the mass determination limits the result to 3 significant digits. The quality of the instrumentation is better, than that used by Scientist A, but the result is still limited to only 4 significant digits. Why not 6 significant digits in the reported result? This time the answer is limited by the uncertainty in the atomic mass of Fe, which is known to 5 significant digits!

This brings up an interesting question. Why is the atomic mass of chlorine known to 6 significant figures, while that of iron is only known to 5 significant figures? Click here for an explanation.

More Examples


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