Measurement Uncertainties

No measurement is perfect, all measurements have an associated "uncertainty."  Measurement uncertainty is a quantitative expression of the doubt that exists about the result of any measurement. Uncertainty represents the range of values within which the true value of the measurement is expected to lie. ​The numerical value of a ± uncertainty value tells you the range of the result. For example a result reported as 1.23g ± 0.05 means that the experimenter has some degree of confidence that the true value falls in between 1.18g and 1.28g.

Measurement uncertainty is crucial in science for several reasons:

• Reliability and Validity:  Stating the uncertainty of a measurement allows for evaluation of the reliability (precision) and validity (accuracy) of the results. 
• Comparability and Reproducibility:  Measurement uncertainty enables scientists to compare results from different experiments, laboratories, and times. It provides a standardized framework for determining if the differences between two measurements are significant or simply due to the inherent uncertainty of the measurement process. 
• Decision-Making:  In many fields, critical decisions are made based on measurement results. The uncertainty helps determine if a value is within a specified tolerance or if it has changed significantly. For example, knowing the uncertainty of a blood test result can help a doctor decide if a patient's condition has genuinely improved or if the change is within the margin of measurement variation.​

Measuring to the Correct Number of Digits
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Accurately recording measurements with the correct number of significant digits is crucial for representing the precision of your data.  Significant digits of a measurement include all the digits known with certainty, plus one estimated digit. The last digit recorded is always the estimated digit, indicating a degree of uncertainty based on the instrument's precision and your judgment.​

There should be no variation in the precision of raw data. In other words, the same number of digits past the decimal place should be used for all measurements made with the same tool.  Additionally, as it is the last digit that is uncertain, the uncertainty must be expressed to the same number of decimal places as those in the measurement.

• 34.0 cm3 ± 0.5 cm3
• 34.10 cm3 ± 0.05 cm3​​

Determining Measurement Uncertainty

For general purposes, the uncertainty of a measurement device is one half of the smallest measurement possible with the device.  To determine uncertainty:

ANALOG TOOLS

• Find the smallest increment of measurement on the measurement device
• Divide it by two
• Round to the first non-zero number so the uncertainty only has one significant digit.  An uncertainty value represents an estimated range of doubt. It's an approximation, not a perfectly known quantity. Including more than one significant digit in the uncertainty would suggest a level of precision in the uncertainty value that is not justified.

For example:

• Beaker A:  each line on the beaker indicates 10mL, so the increment of measurement is 10mL.  10mL divided by two is 5mL.  There is already only 1 significant digit, so the uncertainty of beaker A is ±5mL.
• Beaker B:  each line on the beaker indicates 25mL, so the increment of measurement is 25mL.  25mL divided by two is 12.5mL.  Uncertainty values can only have 1 significant digit, so the uncertainty needs to be rounded to include only one-non-zero number. 12.5 rounds to 10, so the uncertainty of beaker B  is ±10mL.

DIGITAL TOOLS
The simplest way to estimate the uncertainty of a digital measurement (such as electronic scale) is to report the smallest digit on the display as the uncertainty. For example, if a scale displays mass to the hundredths of a gram (0.01 g), the uncertainty can be estimated as ±0.01 g. 

The uncertainty of a stopwatch measurement is a bit more complex because it's heavily influenced by human factors. One must consider both the device's resolution and human reaction time.

• Instrument Resolution: This is the smallest time interval the stopwatch can display. For a stopwatch that shows time to the hundredths of a second (0.01 s), the uncertainty from the device itself is ±0.01 s.
• Human Reaction Time: This is a significant source of uncertainty. Reaction time is not constant and can vary between individuals and even for the same person. A typical estimated value for human reaction time is around ±0.2 seconds.

Recording and Reporting Measurement Values

The number of digits reported in the measurement should always have the same number of digits (decimal places) as the measurement uncertainty. So, when reporting measurement values, adjust the last reported digit of the measured value to match the decimal place of the rounded uncertainty.​​ It  would be confusing to suggest that there is confidence in the digits of the hundredths (or thousandths) place when there is uncertainty at the tenths place. 

• Wrong: 1.234 s ± 0.1 s   
• Correct: 1.2 s ± 0.1 s 

​Just as for units, in a data table, report the uncertainty in the column heading.  It is not necessary to keep rewriting if for every measurement in the table. 

For data derived from processing raw data (i.e., means), the level of precision should be consistent with that of the raw data.