We use cookies to improve functionality and personalize your content. To ensure compliance with the latest e-Privacy directive, we kindly ask for your consent to set these cookies. Learn more.
Significant Figures and Uncertainty
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 (accuracy, precision, mean and standard deviation) 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.
Significant figures communicate how precisely a value is known; uncertainty states the range within which the true value is expected to lie. This section covers the rules for both, and how to apply them to ICP-OES data.
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.
What Are the Rules for Significant Figures?
A significant figure is any digit that carries real information about a measurement. Five rules determine which digits count:
| Rule | What it means | Example | Sig figs |
|---|---|---|---|
| All non-zero digits are significant | 1–9 always count | 24.7 | 3 |
| Zeros between non-zero digits are significant | "Captive" zeros always count | 1.008 | 4 |
| Leading zeros are never significant | They are placeholders only | 0.00456 | 3 |
| Trailing zeros are significant only if there is a decimal point | 25.00 counts them; 2500 is ambiguous | 25.00 | 4 |
| Exact and counted numbers have infinite sig figs | Definitions and counts do not limit a result | 12 flasks; 1 in = 2.54 cm | ∞ |
Rules for calculations
- Multiplication and division: the result carries the same number of significant figures as the input with the fewest. 1.4589 × 1.2 = 1.75068, reported as 1.8.
- Addition and subtraction: the result carries the same number of decimal places as the input with the fewest. 5.789 + 105 = 110.789, reported as 111.
- Rounding: look at the first digit dropped. If it is 5 or greater, round up; if less than 5, leave the last significant figure unchanged. 0.0013567 to three significant figures is 0.00136.
- Carry extra digits through intermediate steps and round only once, at the end, to avoid compounding rounding error.
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.
Why Are Significant Figures Important?
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.
How to Report Significant Figures Correctly: Examples
Example 1: Measuring Length
If 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 Mass
If 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 Figures
If 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 ICP
A 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.
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 contains 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.
How Many Significant Figures Should You Report from an Instrument?
Report only the digits your instrument can resolve — those known with certainty, plus the first uncertain digit. An instrument reading 0.253 AU supports three significant figures; one reading 0.25 AU supports two.
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.
How Do Significant Figures Work in Multi-Step Calculations?
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.
For example, suppose you are conducting a series of calculations to determine the concentration of an element in a solution. You might start by measuring the mass of the sample with a balance that has an uncertainty of ± 0.01 grams, then measure the volume of the solution using a volumetric flask with an uncertainty of ± 0.1 mL. After multiplying or dividing these values, the final result should be rounded according to the least number of significant figures in the original measurements.
How to Calculate Uncertainty
Uncertainty is the range within which the true value of a measurement is expected to lie. It is expressed in two forms:
Absolute uncertainty — the uncertainty in the same units as the measurement. A 100 mL volumetric flask specified as 100.0 ± 0.1 mL has an absolute uncertainty of 0.1 mL.
Relative uncertainty — the absolute uncertainty divided by the measured value, usually expressed as a percentage.
Relative uncertainty = ( absolute uncertainty / measured value ) × 100
For the flask above: (0.1 / 100.0) × 100 = 0.1%.
Combining uncertainties
- When quantities are added or subtracted, add the absolute uncertainties in quadrature: u = √( u12 + u22 ).
- When quantities are multiplied or divided, add the relative uncertainties in quadrature: urel = √( u1,rel2 + u2,rel2 ).
Worked example: a 1000 mg/L stock standard certified at ± 0.3% is diluted 1:100 using a volumetric pipette (± 0.08%) and a flask (± 0.1%). The combined relative uncertainty of the diluted standard is √(0.3² + 0.08² + 0.1²) = 0.33%, giving 10.00 ± 0.03 mg/L.
What Is 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.
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.
What Is the Difference Between Type A and Type B Uncertainty?
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.
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. Many presentations on uncertainty are written in a language that is difficult for the beginner to grasp; this one is not.
For a second reference, see NIST Technical Note 1297: Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results.
Frequently Asked Questions
What are the rules for significant figures?
All non-zero digits are significant. Zeros between non-zero digits are significant. Leading zeros are never significant. Trailing zeros are significant only when a decimal point is present. Exact and counted numbers have unlimited significant figures.
Are zeros after the decimal point significant?
Yes. A trailing zero after a decimal point is significant because it communicates measured precision: 25.00 g has four significant figures and states that the mass is known to the nearest hundredth of a gram. Leading zeros — as in 0.00456 — are placeholders and are never significant.
How do you calculate uncertainty?
Absolute uncertainty is quoted in the units of the measurement (100.0 ± 0.1 mL). Relative uncertainty is the absolute uncertainty divided by the measured value, expressed as a percentage: (0.1 / 100.0) × 100 = 0.1%. When quantities are multiplied or divided, combine their relative uncertainties in quadrature.
What is the difference between Type A and Type B uncertainty?
Type A uncertainty is evaluated statistically from repeated measurements, typically as a standard deviation. Type B uncertainty is evaluated from other sources — calibration certificates, manufacturer specifications, or a reference material's certified value. A full uncertainty budget includes both.
How many significant figures should I report from an ICP-OES result?
Report only the digits the measurement supports: those known with certainty, plus the first uncertain digit. A trace result quoted as 0.00131 ppm Fe claims three significant figures, which is difficult to justify at trace levels. Quoting it as 0.00131 ± 0.00006 ppm makes the uncertainty explicit and the claim defensible.
Why are significant figures important?
They communicate how precisely a value is known. Reporting more digits than the measurement supports overstates confidence in the result; reporting fewer discards information you paid to acquire.