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Evidence can support a claim without compelling every reasonable reader to accept it. Examine both the support and any alternative interpretation; disagreement does not automatically reduce the example to decoration.
Precise figures on a case you half remember are worse than an honest generalisation, because precision disarms the reader's guard. If you are unsure whether the mortality fell by a third or by half, write the direction of the change and leave the figure out until you can look it up.
Write the claim beside the paragraph, then read the case on its own. Ask what conclusion the details support and whether that is the conclusion you intended.
As you study, keep brief notes on useful cases: what was claimed, how it was justified and where to find the source. This gives you material to test against a title later.
Plan space for analysing your cases as well as describing them. If a case grows, check that you have not squeezed out the evaluation of a competing interpretation.
Choose the example with the least explained connection to its claim. Investigate the details, consider another reading, and decide whether to strengthen the explanation, narrow the claim or replace the example.