Significant Figures And Uncertainty - Inorganic Ventures

ICP Operations Guide: Part 15 By Paul Gaines, Ph.D.

Introduction to Significant Figures and Uncertainty

There are certain basic concepts in analytical chemistry that are helpful to the analyst when treating analytical data. The last section (part 14) addressed accuracy, precision, mean and deviation as related to chemical measurements in the general field of analytical chemistry. This section will address significant figures and uncertainty.

What are Significant Figures?

When working with analytical data it is important to be certain that you are using and reporting the correct number of significant figures. The number of significant figures is dependent upon the uncertainty of the measurement or process of establishing a given reported value. In a given number, the figures reported, i.e. significant figures, are those digits that are certain and the first uncertain digit. It is confusing to the reader to see data or values reported without the uncertainty reported with that value.

Why Is Uncertainty Important in Analytical Chemistry?

Significant figures are essential for ensuring both accuracy and precision in measurements as they communicate the level of uncertainty in a given value. They represent the digits in a measurement that are known with certainty, plus one digit that is estimated. The number of significant figures in a measurement reflects how accurately it represents the true value, with more significant figures indicating greater accuracy. 

They also convey the precision of the measurement tool used, as a tool with finer markings will result in more significant figures. In calculations, the number of significant figures in the result must correspond to the least precise measurement involved, preventing overestimation of certainty. Overall, significant figures are crucial for maintaining reliable and consistent scientific data.

How to Report Significant Figures Correctly: Examples 

Example 1: Measuring LengthIf you measure the length of a pencil with a ruler that is marked in millimeters, and the length is 12.3 cm, the measurement has three significant figures (1, 2, and 3). This indicates that the measurement is accurate to the nearest tenth of a centimeter.

Example 2: Measuring MassIf you weigh an object and the scale reads 25.00 grams, the measurement has four significant figures (2, 5, 0, and 0). The two zeros show that the mass is measured precisely to the nearest hundredth of a gram.

Example 3: Calculations with Significant FiguresIf you multiply 3.2 (which has two significant figures) by 2.45 (which has three significant figures), the result should be rounded to two significant figures, as the least precise number is 3.2. The result would be 7.8, not 7.84.

Example 4: Measurements on ICPA sample is measured using ICP-OES and reported to contain 0.00131 ppm of Fe. This value implies with certainty that the sample contains 0.0013 ppm Fe and that there is uncertainty in the last digit (the 1). 

However, we know how difficult it is to make trace measurements to 3 significant figures and may be more than a little suspicious. If the value is reported as 0.00131 ± 0.00006 ppm Fe this indicates that there was an estimation of the uncertainty. 

Rounding is an essential process when working with significant figures to ensure that the precision of the result matches the precision of the data. When rounding, you look at the digit in the place value immediately after the last significant figure. If this digit is 5 or greater, you round the last significant figure up by one; if it's less than 5, you leave the last significant figure unchanged.

For example, if the result of a calculation is 0.0013567 and you need to report it to three significant figures, you round it to 0.00136. The digit after the third significant figure is a 6, so the 5 is rounded up to 6, making the final value 0.00136. This ensures the number reflects the appropriate level of precision, avoiding overstatement of the measurement's accuracy.

A statement of how the uncertainty was determined would add much more value to the data in allowing the user to make judgments as to the validity of the data reported with respect to the number of significant figures reported.

Examples of Significant Figures in Measurements

You purchase a standard solution that is certified to contain 10,000 ± 3 ppm boron prepared by weight using a 5-place analytical balance. This number contains 5 significant figures. However, the atomic weight of boron is 10.811 ± 0.01. It is, therefore, difficult to believe the data reported in consideration of this fact alone.

For more information about uncertainties related to atomic weights, please read the article linked below. This resource goes far beyond what we discuss here, but offers a wealth of information relating to the determination of uncertainties related to atomic weight measurements with regard to isotopic abundance variances. https://www.degruyter.com/document/doi/10.1515/pac-2016-0302/html

The number 0.000013 ± .000002 contains two significant figures. The zeros to the left of the number are never significant. Scientific notation makes life easier for the reader and reporting the number as 1.3 x 10-5 ± 0.2 x 10-5 is preferred in some circles.

A number reported as 10,300 is considered to have five significant figures. Reporting it as 1.03 x 104 implies only three significant figures, meaning an uncertainty of ± 100. Reporting an uncertainty of 0.05 x 104 does not leave the impression that the uncertainty is ± 0.01 x 104, i.e., ± 100.

A number reported as 10,300 ± 50 containing four significant figures. If the number is reported as 10,300 ± 53, the number of significant figures is still 4 and the number reported this way is acceptable, but the 3 in the 53 is not significant.

Significant figures in Laboratory instruments 

In laboratory instruments, significant figures vary based on the precision of the equipment and the measurement being taken. For example, a spectrophotometer measures the absorbance of light by a sample at a specific wavelength, often used to determine the concentration of a substance in solution. If the instrument is capable of reporting absorbance to three decimal places, like 0.253 AU (absorbance units), it indicates a measurement with three significant figures. 

The spectrophotometer 

The precision of the spectrophotometer dictates how many decimal places are meaningful. If the device only reports absorbance to two decimal places (e.g., 0.25), then the result is limited to two significant figures, and any additional digits beyond this would be uncertain. Each instrument has inherent limitations in terms of precision, which directly impacts the number of significant figures in the data reported. 

By understanding the instrument's specifications and the level of precision it offers, scientists ensure that measurements are reported with appropriate accuracy and reliability, preventing overstatement of the results.

Mathematical calculations

Mathematical calculations require a good understanding of significant figures. In multiplication and division, the number with the least number of significant figures determines the number of significant figures in the result. With addition and subtraction, it is the least number of figures to the left or right of the decimal point that determines the number of significant figures.

Examples

• The number 1.4589 (five significant figures) is multiplied by 1.2 (two significant figures). The product, which is equal to 1.75068, would be reported as 1. 8 (two significant figures).

• The number 1.4589 (five significant figures) is divided by 1.2 (two significant figures). The dividend, which is equal to 1.21575, would be reported as 1.2 (two significant figures).

• The addition of 5.789 (four significant figures) to 105 (three significant figures) would be reported as 111.

The importance of understanding how significant figures affect precision when performing multiple-step calculations in scientific research:

Understanding how significant figures affect precision is critical when performing multiple-step calculations in scientific research because it ensures that the results are not overstated in terms of their accuracy.

In multi-step calculations, the precision of the final result is determined by the least precise measurement involved, and this principle must be applied consistently throughout each step of the calculation.

Example 

For example, suppose you are conducting a series of calculations to determine the concentration of an element in a solution. In that case, you might start by measuring the mass of the sample with a balance that has an uncertainty of ± 0.01 grams (reported with two significant figures), then measure the volume of the solution using a volumetric flask with an uncertainty of ± 0.1 mL (reported with three significant figures). 

After multiplying or dividing these values, the final result should be rounded according to the least number of significant figures in the original measurements. In this case, even though the volume measurement has three significant figures, the mass measurement only has two significant figures, so the final result should be rounded to two significant figures.

Failure to properly apply significant figures in multiple-step calculations can lead to results that imply a higher level of precision than is justified by the measurements, which can cause misleading conclusions or inaccurate predictions. For instance, if intermediate steps in a calculation are carried out without accounting for significant figures and the result is overly precise, the final value may be unrealistically precise, giving the false impression that the measurement can be trusted to a higher degree than is scientifically valid. 

Thus, understanding and applying the rules for significant figures at each stage of a calculation helps maintain the integrity of scientific findings and ensures that conclusions are based on realistic, reliable data. This is especially important for reproducibility and for the validity of experimental results.

Uncertainty in ICP-OES Measurements

The International Vocabulary of Basic and General Terms in Metrology (VIM) defines uncertainty as:

"A parameter associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand."

NOTE 1: The parameter may be, for example, a standard deviation (or a given multiple of it), or the width of a confidence interval.

NOTE 2: Uncertainty of measurement comprises, in general, many components. Some of these components may be evaluated from the statistical distribution of the results or series of measurements and can be characterized by standard deviations. 

Understanding Uncertainty in ICP-OES Measurements

The other components, which also can be characterized by standard deviations, are evaluated from assumed probability distributions based on experience or other information The ISO Guide refers to these different cases as Type A and Type B estimations respectively.

In an ICP-OES analysis, uncertainty plays a significant role in determining the reliability of the measurement results. For example, if the ICP-OES system reports a concentration of 2.0 ppm (parts per million) for an element in a sample with an associated uncertainty of ± 0.1 ppm, it means the true concentration of the element could range from 1.9 ppm to 2.1 ppm. This uncertainty reflects various factors that influence the measurement.

Types of Uncertainty in ICP-OES: Type A vs Type B

ICP-OES measurements often involve both Type A and Type B uncertainties. Type A uncertainty is derived from statistical analysis of repeated measurements, such as variations in repeated readings due to random fluctuations in the instrument or sample. Type B uncertainty arises from non-statistical sources, such as instrument calibration, environmental conditions (like temperature and humidity), or the quality of reagents and sample preparation. For instance, instrument drift during the analysis or slight inconsistencies in the calibration standard can contribute to Type B uncertainty.

Thus, uncertainty in ICP-OES measurements combines these different factors, highlighting that the reported concentration is an estimate with a range, rather than an exact value. Understanding and quantifying uncertainty is essential for interpreting the accuracy and precision of the results.

There are numerous publications concerning uncertainty calculations. I am concerned that many presentations on the topic are written in a language that may be difficult for the beginner to easily grasp. However, there is a clear and complete guide that I highly recommend. https://www.eurachem.org/images/stories/Guides/pdf/QUAM2012_P1.pdf 

https://eurachem.org/index.php/publications/guides/quam 

Practical Tips for Students and Beginners

Grasping the Concepts of Significant Figures and Uncertainty

When you're starting out in science, understanding significant figures and uncertainty is key to making sure your measurements and calculations are accurate and reliable. Here’s some advice to help you get started:

1. What are Significant Figures?

Significant figures are the digits in a measurement that are important for expressing its accuracy. These include all the non-zero digits, any zeros between them, and any trailing zeros that are after a decimal point. For example, in the number 0.00456, there are three significant figures: 4, 5, and 6. The zeros at the beginning are just placeholders, so they don't count.

2. Why Do Significant Figures Matter?

Significant figures help show how precise a measurement is. For example, if your balance shows a weight of 12.45 grams, it means the weight is known precisely to the hundredths place. If it just showed 12 grams, it would only be accurate to the nearest whole gram. The more significant figures, the more precise the measurement.

3. Dealing with Uncertainty

Uncertainty refers to how much we expect a measurement to vary. In science, there's always some degree of uncertainty because no instrument is perfect. For example, a thermometer might show 25.0°C, but the actual temperature could be a little higher or lower. This uncertainty is usually given as a range. If you see a concentration of 2.0 ppm with a ± 0.1 uncertainty, it means the concentration is somewhere between 1.9 ppm and 2.1 ppm.

4. How to Round Numbers Correctly

When performing calculations with measurements that have significant figures, you need to make sure the final result is rounded to the correct number of significant figures. Here's a simple rule: if the next digit is 5 or more, round up; if it’s less than 5, leave the last significant figure unchanged. For example, if you get 4.567 and need to round it to two significant figures, the result will be 4.6.

5. Practice Is Key

The best way to understand significant figures and uncertainty is to practice. Start by looking at measurements and figuring out how many significant figures they have. Then, practice rounding numbers and doing calculations, making sure you pay attention to the rules for significant figures. Over time, it will become second nature!

Remember, significant figures and uncertainty are tools to help you better interpret data, not just rules to follow. They help ensure your results are both accurate and realistic, which is essential in scientific research.

Recommended Reading

Whether you're a beginner or an experienced student of the subject, I strongly encourage you to read Quantifying Uncertainty in Analytical Measurement, published by Eurachem.

Of the numerous volumes of publications on this topic I have seen over the years, this one stands out above all others. It is quite thorough, written in an understandable manner and it includes several good examples.

Further Reading

• Part 16: Traceability

• ICP Operations Guide: Table of Contents

• More Guides and Papers