Let’s Talk About Significant Figures

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In this post, I want to talk about the proper way to express test results on certificates of analysis. To explain this, I am going to talk about the concept of significant figures (or significant digits). What are significant figures?

According to Wikipedia:

“The significant figures of a number are digits that carry meaning contributing to its measurement resolution.”

Confused? Let me explain.

Here’s the problem. When you report a test result or an answer to a mathematical problem, the number of digits that you report implies information regarding the accuracy of the measurement.

Here is an example:

Let’s say that you want to calculate the sales tax on a price. The price is $5.25. The sales tax in Michigan is 6%. You would multiply 5.25 x 0.06 = 0.315. The sales tax is $.315 or 31 1/2 cents. The problem is that we don’t have half cents. The answer to this problem overstates the accuracy. The correct way to express the answer to this math problem is to round the answer to two digits. Typically, you would round .315 to .32 or 32 cents.

Here is another example:

Let’s say you weighed something on a scale that was accurate to a thousandth of a gram and the item weighed 5.236 g, and you reported the weight as 5.23600 g. Those extra zeros make it appear that the scale was much more accurate than it was. The scale was not accurate to 0.00001 g.

When reporting test results, we don’t want to imply accuracy beyond the capability of the measurements that we are taking. So we need to use rounding to remove the extra digits that do not belong. The question is how to determine the number of significant figures to retain in a calculation.

Significant Figure Rules

  1. When doing a mathematical calculation, the significant figures retained should be no more than the significant figures in the number with the least significant figures in the calculation. For example, plug 2.5 x 3.42 into your calculator and you will get an answer of 8.55. This answer should be rounded to 8.6. The reason is that 2.5 has two significant figures and 3.42 has three. Choose the number with the least amount of significant figures and that is how many your answer should have.

Another way to think of this is that a chain is no stronger than its weakest link. The accuracy of your result should reflect the least precise operation in your calculation.

  1. Direct measurements from things like thermometers or rulers have as many significant figures as the device can measure. Keep in mind that sometimes you can estimate a slightly more accurate measurement than the device can read. In this case you could add a significant figure. For example, if you were using a graduated cylinder or beaker that was divided into 1 ml graduations but you could estimate values between graduation marks. You might be able to report 9.6 ml for instance if you could see that the level was 6/10ths of the way between the 9 and 10 ml marks.
  2. Mathematical constants have infinite significant figures. An example of this would be if you had to convert a measurement into another unit during your calculation. For example, when I test for izod impact, my machine measures in in-lbs. I need to convert the reading from the machine into ft-lbs during the calculation. I do this by dividing the number by 12. The 12 in considered to be a mathematical constant and has infinite significant figures, thus it does not affect the significant figures reported.

Another way of determining how many significant figures to report is by looking at the accuracy of the test method that you are performing. The only way to really know the accuracy is to look at studies of inter-laboratory testing. Some people call these round-robin studies. Basically, you take a sample and send it out to a bunch of laboratories and have them all perform the test and send the results back to you. A statistical analysis of the test results will give you some idea of the accuracy of the test method. This accuracy is usually expressed as a standard deviation which is shown as the Greek letter Sigma σ or sometimes as just capital S.

The good news is that ASTM and ISO have performed inter-laboratory studies on many of their test methods and include the results at the end of the method. This data is usually in the section entitled “Precision”. If you take a look at the data, you can look at the standard deviations that they have calculated. This will give you a clue as to how accurate the test method is.

First, find the standard deviation that is appropriate to the test that you are performing. Take a look at the first significant digit in the standard deviation (ignore zeros at the beginning of the number). That is typically the last digit that you will want to report. For instance, if the standard deviation is 0.471, the 4 is the first significant digit and it is the first digit to the right of the decimal. I would typically report this result to one decimal place. Decimal places are different from significant figures though so I would report the whole number result plus one decimal place. If my whole number result is 1 digit, then I would report 2 significant figures, one to the left of the decimal and one to the right. For instance, Instead of reporting 7, I would report 7.1.

Sometimes, ASTM and ISO do this work for you and just tell you how many significant figures to report. If they do, it will typically be listed in the “Calculation” section of the test method.

Rounding

Once you know how many significant figures to keep, you have to round the test results to the proper number of significant figures. The last digit that you are going to keep is called the rounding digit. I think that we all know how to round numbers off.

  1. In the case of whole numbers, if the digit to the right of the rounding digit is 1-4, then the rounding digit stays the same and all digits to the right of the rounding digit become 0. If the digit to the right of the rounding digit is 5-9 then the rounding digit goes up 1 and the numbers to the right become 0. If the number to the right of the rounding digit is 0, then all digits to the right become 0. For example 344 rounded to the nearest ten would be 340. 346 rounded to the nearest 10 would be 350.
  2. In the case of decimal numbers (when you are rounding digits to the right of the decimal), the rules are the same except that you drop the digits to the right of the rounding digit. For example 21.56 becomes 21.6 if you are rounding to the nearest 1/10th.

Simple right? However, there is an alternative method that you may not have ever heard of. The problem with the method that I described above is that you are rounding down for 1,2,3 & 4 and up for 5,6,7,8 & 9. This is rounding down 4 times and up 5 times. If you were rounding lots of numbers to then average the numbers, this would have an effect on the average and thus is considered to be somewhat inaccurate. Here is the alternative method of rounding.

  1. If the digit to the right of the rounding digit is a five then you increase the rounding digit by 1 if it is odd and leave the rounding digit the same if it is even. The last digit kept will always be even by this method. For example 2.315 would become 2.32 rounded to the nearest 1/100th. However 2.325 would also become 2.32 because the rounding digit is even, it does not get changed.

The rationale is that when the digit to the right of the rounding digit is a 5, you will be rounding up about half the time and down about half the time instead of always rounding up. Over the long term, this would make your rounding more accurate.

Significant Figures for Common Test Methods

Based on the principles above, here are the significant figures that should be kept for some common test methods that we use for testing plastic materials.

  • Test Discription                        Test Method        Significant Figures to Report  
  • Izod Impact                            ASTM D256          2 (or 3) – see note A
  •                                                ISO 180                  2 – see note B
  • Tensile Strength                    ASTM D638         3 – see note B
  •                                                 ISO 527                3 – see note B
  • Tensile Elongation or Strain  ASTM D638       2
  •                                                 ISO 527               2 – see note B
  • Tensile Modulus                     ASTM D638        3
  •                                                 ISO 527                3 – see note B
  • Felexural Strength                 ASTM D790        3
  •                                                 ISO 178                 3 – see note B
  • Flexural Modulus                   ASTM D790         3
  •                                                ISO 178                  3 – see note B
  • Melt Flow Rate                       ASTM D1238        2 (or 3) – see note C
  •                                                ISO 1133               2 if >10, 3 if <10–see note B
  • Linear Mold Shrinkage        ASTM D955            3
  •                                                ISO 294                  3
  • Durometer Hardness           ASTM D2240        2 (or 3) – see note D
  •                                                ISO 868                  2 (or 3) – see note D
  • Ash Content                          ASTM D2584          3
  •                                               ISO 3451                  3
  • Density (or SG)                      ASTM D792           4
  •                                               ISO 1183                  4

Note A: For izod impact machines with a dial and pointer, 2 significant figures. If your machine has a digital readout that reads 3 significant figures, then 3 significant figures should be reported.

Note B: Specified in the test method.

Note C: The calculation for this test would recommend 3 significant figures however the inaccuracy associated with this test causes many lab managers to report 2 significant figures.

Note D: If you have a dial type tester, 2 significant figures. Most digital testers report 3 significant figures so 3 significant figures can be reported.

Conclusion

Believe it or not, I have just skimmed the surface on this subject. ASTM has published a standard practice on the use of significant digits which can be very helpful. The designation is E29. It has more information than you probably ever wanted to know. A Google search on significant figures or rounding rules will bring up many pages on the subject as well.

Rounding test results to an appropriate number of significant figures will provide enough information to determine compliance with specifications without providing information that is misleading about the accuracy of the test method. I have looked at a lot of material certificates of analysis over the years and I find that many if not most report more significant figures than I think are appropriate. You might consider looking at how you report results.

For more information:

Excellent page on significant figures and rounding from Chemteam

To purchase copy of ASTM E29

Wikipedia page on significant figures

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