If it's the same code identically for each case but with a different dataset based on the index, that's all you need -- you don't need a switch construct at all -- just use the index. You don't even need the temporary "i" index variable, but it may be handy just to shorten the typing...
...
i = app.Specimen.Value
cla(app.UIAxes);
plot(app.UIAxes, app.data.spec(i).si(:,10), app.data.spec(i).si(:,14));
As long as the two fixed column numbers are fixed, that should be all you need to do...though I'd recommend to also make those variables with some more meaningful identification names instead of burying magic constants in the code itself. Besides the local documentation, makes fixing them if there's a need to change the data storage simpler if they're always referred to and defined in one place. They could be likely candidates to be globals app variables defined in startup code if needed more than one place.
