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Can I start labeling my $x$ and $y$ axes from non-zero values when drawing a graph? Or is there any convention to only label $x$-axis from non-zero value when using a false origin and not $y$-axis? Please clarify this.

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It is generally true that suppressed or offset zeros should only be used if necessary and the labeling must make the situation clear. And to get on one of my favorite hobby horses, everyone should read Tufte's The Visual Display of Quantitative Information at least once. And Probably "How to Lie With Statistics", too. –  dmckee Apr 30 '13 at 20:28
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up vote 3 down vote accepted

According to the blog Jacobs Physics: Resources for teachers and students of introductory physics:

It's perfectly acceptable, and sometimes desirable, not to begin an axis at zero.

And:

Students will attempt to demand a hard-and-fast rule about scaling graphs from the origin, but such a rule does not exist. The scaling of a graph depends on the circumstances of the data.

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