Processing Uncertainties

An uncertainty is an interval that indicates a range within which there is confident that the true value lies.  Uncertainties occur when measuring variables during data collection, and when processing the data during descriptive and inferential analysis. 

Uncertainties of Measurement

The uncertainty of a single data point stems from limitations of the measuring instrument and the experimental procedure.  Measurement uncertainty is expressed as the measured value plus-or-minus (+/-) the uncertainty.  It quantifies the inherent doubt in any measurement, due to the precision of the measuring instrument.  Click here to learn how to determine and represent uncertainties when making measurements. 

Uncertainties in Data Sets

A data set is a group of related numbers representing a single entity, such as the data collected when collecting multiple trials of the same measurement.  Uncertainties of data sets are a quantitative measure of the dispersion of data within the data set, indicating the range within which the true value is expected to lie. These uncertainties stem from various sources, including inherent randomness in measurements (random errors), flaws in the measurement tool or process (systematic errors) and sampling limitations. Understanding and quantifying uncertainty data sets crucial for making accurate, confident decisions and interpreting data correctly. Click here to learn how to determine and represent uncertainties of data sets, using calculations such as:

• range
• standard deviation
• standard error
• interquartile range

Uncertainties in Relationships/Trends between Variables

The correlation coefficient (r) and the coefficient of determination (R^2) are key tools for assessing the strength of a relationship between variables, which helps to quantify the uncertainty of that relationship. Calculations of r and R2 provide a measure of the goodness of fit of a linear model (best fit line), which is a direct indicator of the relationship's uncertainty.

• The correlation coefficient, represented by r, is a descriptive statistic with a value between -1 and +1 that measures the strength and direction of a linear relationship between two variables. Click here for more information about the correlation coefficient.
• ​The coefficient of determination, or R2, is an inferential statistics with a value between 0 and 1 (often expressed as a percentage) that represents the proportion of the variance in the dependent variable that is predictable from the independent variable(s). Click here for more information about the coefficient of determination.

Presenting Uncertainties in Graph

Presenting uncertainties in graphs provides a complete and honest representation of data.  A single data point is often the best estimate within a range of possible values. If this uncertainty is not shown, a graph can be misleading, making a finding appear more definitive than it truly is. By including uncertainty, a graph acknowledges the limitations of the data and promotes transparency in scientific and statistical communication.

Error bars are graphical lines on a graph that indicate the estimated variability of a reported value. They provide a visual clue about the potential range within which the true value might lie. There are several types of error bars, each suited for different types of data.  Click here for more information about the types of error bars used on graphs.

Uncertainties in Making Inferences from Sampled Data

When making inferences about a large population from a small sample, we don't have data on the whole population, so any conclusion is just an estimate.  Describing the uncertainty of inferential statistical tests indicates how confident one can be in a conclusion and prevents the making of overly strong claims.

Statistical hypothesis testing is a formal procedure for using sample data to make statistical inferences about a population, determining if a result is statistically significant or likely due to random chance.

• Begin with a null hypothesis (H0​), which is a statement of no effect or no difference, and an alternative hypothesis (Ha​), which asserts that there is a real effect, difference, or relationship in the population being studied
• Calculate a inferential statistic (like a t-score or a chi-square) and determine the p-value.
• The p-value is the probability of the inferential statistic result, assuming the null hypothesis is true. The standard of a p-value of 0.05 is a commonly used significance level, often denoted as α (alpha). This threshold is used to decide whether to reject the null hypothesis.
• If the p-value is small (typically less than 0.05), it suggests that the result is unlikely to have occurred by chance alone. Reject the null hypothesis in favor of the alternative hypothesis.
• Conversely, a large p-value means the observed data is likely to have occurs if the null hypothesis is true, so you fail to reject the null hypothesis.