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Significant Figures and Margin of Error—Or Why the Fourth Decimal Place in Your Potency Reading is Probably Meaningless
The concept of significant figures tells us, given the amount of error in a measurement, which numbers in a reading are meaningful and which are not. This is vitally important when determining whether low level values are real, for comparing readings over time, and for comparing results across laboratories and instruments. In a recent trend, cannabis laboratory customers are demanding more decimal places in the readings they are paying for, and third party laboratories have unfortunately obliged. I will show why those extra decimal places may be meaningless, and how to use significant figures to discern the true meaning of your cannabis analysis measurements.
Figure 1: A plot of two quantities versus each other, the vertical blue lines are the error bars in each data point.
In the first installment of this column (1), I stated that two of its purposes are to introduce the basic concepts of cannabis analysis to people new to the field, and to discuss current issues in the cannabis analysis universe. Since that time, I have done not enough of the former, and perhaps too much of the latter. But this is because there have been a lot of things to discuss given the state of cannabis analysis, the constant change in regulations, and the newness of the field. It is now time to rectify things and discuss one of the most basic topics in chemistry: significant figures, or significant digits as some call it. It was my intention to cover this topic earlier, but of course other issues got in the way.
What has prompted this column is two observations. I work for a company that makes cannabis and hemp analyzers (2). Recently I have had customers asking me for more “data points” in the readings our instruments make. Additionally, I have seen potency readings on certificates of analysis (COAs) from third party cannabis analysis laboratories suddenly go from having two decimal places to having four decimal places. This is concerning because cannabis businesses are coming to conclusions and making business decisions based on those decimal places, when it turns out that those extra decimal places are probably meaningless and flawed business decisions may be made as a result. Hence the need for this column on significant figures.
Introduction to Significant Figures
My first brush with significant figures was in a Chemistry 101 course my freshman year of college. I am old enough that handheld calculators were a novelty when I took that course, and I had a trusty Texas Instruments TI-30 calculator, for which I paid the princely sum of $30. It reported numbers to nine decimal places and by golly I just knew that had to be correct. Thus, I recorded my answers to my assignment with that many digits in it. That assignment came back from my professor with red ink all over it and a very poor grade. My mistake had been to ignore significant figures.
At its base, significant figures take into account that scientific measurements have error associated with them. We talked about accuracy, precision, systematic error, and random error in a previous column (3). Briefly, precision is a measure of reproducibility, accuracy is a measure of how far away you are from the true value, and random error is error from a combination of sources whose sign and value vary randomly. When a reading has a known margin of error, the value is reported as a number with a ± next to it. For example, when a pollster reports the percentage of people that support candidate X for president, they will report it for example as 43 ±3%. This means that 43% of the people polled support candidate X, and the error is ±3%. What the margin of error, or “error bar” as some of us scientists call it, means is that the true value of the reading, the true number of people who support candidate X, is somewhere between 40% and 46%. For example, sometimes accuracies are quoted at a “95% confidence limit.” This means that we are 95% confident that the true value falls within the error bar for that measurement. Thus, a total tetrahydrocannabinol (THC) reading of 0.5% with a 95% confidence level of ±0.05% means we are 95% confident that the true value lies between 0.45% and 0.55%.
The use of error bars is illustrated in Figure 1. (See upper right for Figure 1, click to enlarge.)
The figure is a plot of two quantities versus each other, the blue vertical lines represent the error bars in each individual measurement. The dotted line denotes the fact that the error bars for X = 5 and X = 10 overlap. This is important because it means the true value for either measurement falls in a common range, and that the two measurements are not statistically different from each other. For example, if two hemp samples are measured with total THC values of 0.3 ± 0.1 wt.% and 0.35 ± 0.1 wt.%, the two readings would not be significantly different from each other since their error bars overlap. In this case the true value for both readings falls between the lowest and highest values for the two error bars, which means the true value falls between 0.2% and 0.45%, making it difficult to tell whether this sample is above or below the 0.3% total THC federal limit for hemp (4).
In addition to when comparing values to each other, margin of error and significant figures come into play when reporting values. Let’s say for the sake of argument that a COA for a hemp biomass sample states that the weight percent total THC measurement is 0.3142 ±0.05%. The third and fourth decimal places in these readings correspond to values of 0.004 and 0.0002. Note that these two values are less than the margin of error! The third and fourth decimal places here are said to be insignificant and should not be reported. This is the mistake I made in my freshman chemistry assignment. The purpose of the margin of error is to tell us how much confidence we have in our measurements. In this case where the third and fourth decimal place values are less than the margin of error, we have no confidence in these values and hence they are meaningless. Here the reading should be reported as 0.31 ±0.05%.
As another example, I have sitting on my desk a COA for a hemp biomass sample from a reputable, state-licensed, International Organization for Standardization (ISO)-certified third party cannabis testing laboratory. Column 1 reports the weight percent of cannabidiolic acid (CBDA) in a hemp biomass sample as 14.5828 and column 2 reports the margin of error as ±0.05%. What’s wrong with this picture? As above, the error is in the second decimal place, the third and fourth decimal places are meaningless, and the value should be reported as 14.58 ±0.05%. The COA, by printing the margin of error in column 2, proves that the number of decimal places printed in column 1 is incorrect. Come on people, we can do better than this! Any laboratory director who has taken a freshman chemistry course should know about significant figures so there are no excuses here. Just because your customers ask for more decimal places does not mean you have to report them, particularly because the extra decimal places are insignificant and useless for making business decisions.
A Practical Implication of Significant Figures
Growing hemp in the US was legalized with the passage of the 2018 Farm Bill (5). The bill states, in its current interpretation as promulgated by the United States Department of Agriculture (USDA) in its Interim Final Rule issued on October 29, 2019 (4), that legal hemp may not contain more than 0.3% by dry weight total THC. After the Farm Bill was passed there was great confusion about how to enforce that number. Some states read it literally and any value above 0.3000 was considered illegal. Other states took significant figures and margin of error into account and said that hemp up to 0.4% total THC was OK. This inconsistent enforcement is part of the hemp testing insanity I have written about in previous columns (6,7).
Here is the crux of the matter. If a hemp sample tests at 0.31% total THC, but the error is ±0.05%, this means that the true value of the reading is anywhere from 0.26% to 0.36%, straddling the 0.3% legal limit. Is this hemp crop legal or illegal? Is it fair to destroy a hemp farmer’s livelihood based on judgement of the accuracy in the hundredths decimal place?
Fortunately, the USDA’s Interim Final Rule has finally given us some guidance on this (4). The rule defines an “acceptable hemp THC level” which takes into account significant figures and margin of error. According to the rule (4), to calculate the acceptable hemp THC level as measured on a given instrument, take 0.3 wt.% total THC and add the margin of error to it. Thus, if the total THC margin of error on an instrument is ±0.05%, the acceptable hemp THC level as measured on that instrument is 0.35%. This means that any hemp sample reading from this instrument at or below 0.35% is legal, whereas any reading over this reading is illegal.
One of my complaints as to how the rule is written is that it rewards inaccurate readings. The rule did not establish any minimum acceptable margin of error. Thus, if faced with the choice of using an instrument with an accuracy of ±0.1% total THC versus ±0.05% total THC, one might be tempted to use the former because it gives a higher probability of passing your sample, particularly if it is a little hot.
I discussed the trend of cannabis laboratories reporting measured values to more and more decimal places. Significant figures determine how to report measured values taking into account the margin of error. We discussed how to properly compare and report values with the correct number of significant figures. This is important because cannabis businesses and regulators need the best quality data possible to make business and law enforcement decisions.
- B.C. Smith, Cannabis Science and Technology 1(3), 10–12 (2018).
- B.C. Smith, Cannabis Science and Technology 1(4), 12–16 (2018).
- 115th United States Congress, Senate Bill S.2667, Hemp Farming Act of 2018.
- B.C. Smith, Cannabis Science and Technology 2(5), 10–13 (2019).
- B.C. Smith, Cannabis Science and Technology 3(3), 12–15 (2020).
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