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I have data arranged as following:

enter image description here

What I'd like to do is create a graph that looks something like the following (which I generated with graphics software):

enter image description here

So, for example, to generate the first line, cell B1 is the start time and cell B4 is the value at the start time. Cell B2 is the end time and cell B3 is the value at the end time. Between them is the line. Then repeat for each column.

Is this possible? And, just as important, is it possible without restructuring how the data series is structured in the spreadsheet?

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You can create such a graph, but i assume you can't with that cell structure. The main problem is IMHO that there's no common y axis for the different data series, as in this example:

graph

To get this result, you will have to modify the plot options of the data series to continue the lines even if values are missing:

plot options

So, i fear the only option is to modify the cell structure.

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