11.3.4. Representing Uncertainty on Graphs: Error Bars

Error bars are visual representations on a graph of the uncertainty (random error) associated with each data point.

Error bars provide a visual indication of the uncertainty in measurements. Each error bar extends from a data point either vertically (for y-values) or horizontally (for x-values) to represent the amount of uncertainty around that measurement. This helps viewers quickly understand the reliability of the data presented in the graph.

Imagine you are throwing darts at a dartboard. The bullseye represents the true value, but if your throws vary in precision, the radius around the bullseye where the darts land can be seen as the error bars. The larger the spread of darts, the less precise your throws are.

Each error bar is a line segment drawn through a data point, extending a distance equal to the absolute uncertainty above and below the point (for uncertainty in the y-variable) or to the left and right (for uncertainty in the x-variable).

To accurately represent error bars, you first need to calculate the absolute uncertainty for each measurement. For y-variables, this means drawing lines vertically from the data point up and down by the amount of uncertainty for that specific measurement. If the uncertainty is represented in the x-variable, similar horizontal lines are drawn. This visual representation allows us to analyze the spread of data points more effectively.

Think of each data point on the graph as a tree planted in a garden. The height of each tree represents its measurement. The error bars are like flags tied to each tree showing how tall the trees could actually be, indicating the possible range of heights due to measurement uncertainty.

They provide a visual indication of the precision of each individual measurement. A best-fit line should be drawn such that it passes within or at least through the majority of the error bars.

Error bars serve several important purposes. They indicate how confidently we can interpret data points, highlighting the potential range of errors in measurements. When drawing a best-fit line through the data, it should ideally intersect or remain within most of the error bars. If the line lies outside the error bars consistently, it may point to inaccuracies in the uncertainty estimates or indicate a systematic error.

Consider a basketball player shooting hoops. Each shot can be thought of as a data point with a certain level of uncertainty — represented by how often the ball lands within a certain area around the hoop (the error bars). If a coach is analyzing the player's shots, they'd want to see how many make it through the hoop compared to the area indicated by the error bars. If the shots fall outside this area, the coach might suspect the player has a misaligned shooting technique.

The spread of the error bars can be used to estimate the maximum and minimum possible gradients of a linear relationship, thereby providing an uncertainty for the calculated gradient itself.

Error bars also play a critical role in determining the slope (gradient) of a line on a graph. By examining the highest and lowest points of the error bars, you can identify the steepest (maximum gradient) and flattest (minimum gradient) slopes possible. This information gives insight into the uncertainty surrounding the gradient calculation, which is especially important for more complex analyses.

Imagine you are reporting how steep a hill is after measuring it with a level. If you take several measurements, but some vary due to instrument uncertainty (the error bars), you'd assess the steepest incline based on the highest measurement and the flattest incline from the lowest, helping you conclude the possible gradients of the hill.

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