Uncertainty in measurements can be quantified by determining the range of possible values and expressing it in the form of ± value.

In essence, uncertainty serves as a means to quantify the doubt surrounding a measurement result. Whenever we conduct a measurement in a scientific experiment, there is invariably a degree of uncertainty attached to it. This uncertainty arises because all measurements are subject to some level of error, and recognizing this is crucial when analyzing results.

There are two primary types of uncertainties in measurements: random uncertainties and systematic uncertainties.

• Random uncertainties arise from unpredictable statistical variations. These can be minimized by averaging a large number of observations.

• Systematic uncertainties, in contrast, are predictable and often remain constant or proportional to the true value. Such uncertainties typically stem from flaws in the measurement instrument or the methodology used for observation.

To calculate the uncertainty in a measurement, it is essential first to identify the type of uncertainty involved. For random uncertainties, the standard deviation of your measurements can be calculated, which provides a measure of how dispersed your data is around the mean. The standard deviation can be determined using the following statistical formula:

Here, $\sigma$ represents the standard deviation, $N$ is the total number of observations, $x_i$ denotes each individual measurement, and $\bar{x}$ is the mean of those measurements.

For systematic uncertainties, you can estimate the uncertainty based on the precision of the measuring instrument. For instance, if you are using a ruler with millimeter markings, the uncertainty can be expressed as ± $0.5$ mm, as this value accounts for the need to estimate the final digit of the measurement.

Once you have calculated the uncertainty, it is typically expressed as a ± value. For example, if you measure the length of a table to be $2.00$ m with an uncertainty of $0.01$ m, you would report the measurement as:

This notation indicates that the true length of the table is likely to fall between $1.99$ m and $2.01$ m.

It is important to consider uncertainty in your measurements, as it provides a range of values that could represent the true value. This consideration is an integral part of scientific measurement and analysis.