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What is the standard convention regarding the error bars of the independent quantity in a graph? In what situations should I show the $x$ error bars? In case both $x$ and $y$ uncertainties are comparable or neither can be disregarded, should I show both or only the greater one?

I know this matter is more related to aesthetics and convenience, but I wonder if there is more straight rule for that.

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If you are measuring y at some value x, and both quantities have uncertainty, then in principle you should show the uncertainties on both axes.

In some circumstances you might omit the x error bars. This would be the case if the y value depends on x such that $$\Delta y \gg |dy/dx| \Delta x,$$ where $dy/dx$ is your best estimate of the gradient of $y(x)$. In other words, where y changes by much less than its error bar over a change in x equal to the x error bar.

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