Accuracy, precision and significant figures: reading the numbers on a COA

A certificate of analysis (COA) is full of numbers: a purity of 99.12%, a molecular weight of 1419.6, a water content of 4.3%. Each one looks exact. But every measurement has some spread, and a number is only as good as the test behind it. This guide explains three ideas that help you read those numbers: accuracy, precision and significant figures. It also covers how to tell whether two results really differ.

Terms are defined in the glossary.

Accuracy and precision: the difference

The two words are often used as if they mean the same thing. In measurement they don't.

• Accuracy is how close a result is to the true value.
• Precision is how close repeated results are to each other.

A dartboard shows the difference. Darts bunched tightly in one corner are precise but not accurate. Darts spread evenly around the bullseye are accurate on average but not precise. Darts bunched on the bullseye are both.

A method can be very precise and still wrong. If a balance is out of calibration by 2 mg, it will give the same wrong answer every time. That steady offset is called bias. The international vocabulary of measurement calls closeness to the true value on average trueness, and keeps "accuracy" for a single result, which depends on both trueness and precision (BIPM, VIM).

Labs control bias by calibrating instruments against reference standards. They control precision by keeping the method the same each time and by checking how much repeated results vary.

How labs measure precision

Precision is measured by testing the same sample more than once. The ICH guideline on validating analytical methods describes three levels (ICH Q2(R2)):

The spread grows from top to bottom. Two results from different labs are expected to differ more than two injections on the same instrument.

Standard deviation and RSD

The spread is usually expressed as a standard deviation: roughly, how far a typical result sits from the average. Here is a worked example. Five injections of one sample give these HPLC purities:

98.9%, 99.1%, 99.0%, 99.2%, 98.8%

1. The average (mean) is 99.0%.
2. Each result's distance from the mean is −0.1, +0.1, 0, +0.2 and −0.2.
3. Square them and add them up: 0.01 + 0.01 + 0 + 0.04 + 0.04 = 0.10.
4. Divide by one less than the number of results: 0.10 ÷ 4 = 0.025.
5. Take the square root: the standard deviation is about 0.16 percentage points.

The relative standard deviation (RSD) is the standard deviation as a percentage of the mean. Here it is 0.16 ÷ 99.0 × 100, about 0.16%. RSD makes it easy to compare spreads between results of different sizes.

Significant figures

Significant figures are the digits in a number that carry meaning. Writing a result to the right number of significant figures tells the reader how precise it is.

Counting them

• Non-zero digits always count. 99.12 has four significant figures.
• Zeros between other digits count. 1,405 has four.
• Leading zeros never count. They only place the decimal point. 0.050 has two significant figures: the 5 and the final 0.
• Trailing zeros after a decimal point count. 4.30 has three. Writing the zero says the second decimal place was measured.
• Trailing zeros in a whole number are unclear. 1,400 could have two, three or four. Scientific notation removes the doubt: 1.40 × 10³ has three.

Rounding

To round to a set number of significant figures, keep that many digits from the first non-zero digit, then look at the next digit. If it is 5 or more, round up.

• 1419.53 to four significant figures is 1420.
• 0.04567 to two significant figures is 0.046.
• 98.96 to three significant figures is 99.0. The final zero stays, because it is significant.

In calculations

A calculated result can't be more precise than the measurements that went into it.

• Multiplying or dividing: keep the fewest significant figures of any input. 2.5 × 3.14159 is 7.9, not 7.853975.
• Adding or subtracting: keep the fewest decimal places of any input. 12.1 + 0.035 is 12.1, not 12.135.

Round only at the end. Rounding at each step adds small errors that can build up. For more on precision when making up solutions, see molarity, mg/mL and dilutions.

Reading the numbers on a COA

Purity

A purity of 99.12% is written to four significant figures. That suggests the hundredths place means something. Whether it does depends on the method's precision. As the HPLC purity guide explains, the gradient, wavelength and integration settings can each move the result by a fraction of a percent.

So, are 99.12% and 98.87% different? Only if the gap of 0.25 percentage points is clearly bigger than the method's spread. If the repeatability standard deviation is about 0.16, as in the example above, a gap of 0.25 could easily come from normal variation. And if the two figures come from different labs or different methods, they aren't directly comparable at all. Reproducibility between labs is wider than the spread within one lab.

The useful questions are whether a result meets its specification and whether it came from a stated method. A ranking of lots by the second decimal place usually tells you little.

Results against a specification

A COA often shows a specification, such as "≥98.0%", next to the result. Labs round the result to the same number of decimal places as the limit before comparing. This is the convention in the pharmacopoeias. Under it, a measured 97.96% rounds to 98.0% and meets a ≥98.0% limit, but would fail a ≥98.00% limit. The way a limit is written matters as much as the result.

Molecular weight

A mass spectrometry result is compared with the value expected from the sequence. For example, "found 1419.6 Da, expected 1419.5 Da". The two won't match exactly. The question is whether the gap is within what that instrument can resolve. A COA should state the expected value, the found value and, ideally, the tolerance. See mass spectrometry and identity confirmation for what counts as a match.

Water content and other small figures

Small figures, such as a water content of 4.3% by Karl Fischer titration, carry the same caution. A result of 4.3% versus 4.6% on two lots may be within the method's spread.

Measurement uncertainty

Some results are given with a range, for example 99.0 ± 0.3%. The ± figure is the measurement uncertainty: the range the lab expects the true value to lie in. Most often it is an "expanded uncertainty" with a coverage factor of 2. That means about 95% confidence that the true value is inside the range (BIPM, GUM). The Eurachem guide sets out how analytical labs work it out (Eurachem, QUAM).

Uncertainty makes comparisons easier. If two results' ranges overlap a lot, there is no good evidence they differ. Labs accredited to ISO/IEC 17025 must evaluate uncertainty for their tests, though they don't always print it on a report. Many manufacturer COAs don't state it. If you need it, ask the lab that did the test. Peptide purity testing in the UK covers accredited labs.

A short checklist

When reading a number on a COA, check:

• the method and the lab that produced it,
• how many significant figures it is given to, and whether the method supports that many,
• the specification it is compared against, and how many decimal places the limit has,
• any stated uncertainty or replicate results, and
• whether two figures you are comparing came from the same method and lab.

Related guides

For how a purity figure is produced, see HPLC purity explained. For a full walk-through of a lot document, see how to check a certificate of analysis. Every lot has its own COA, published on the product page and on that lot's own page. See Quality.