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What does "appropriate controls and replication" mean here? In the following sentence "Experiments must have been conducted rigorously, with appropriate controls and replication", what does "appropriate controls and replication" mean?

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This is a very scientific sentence that many native English speakers may not fully understand. When conducting experiments, you need to control variables - which could be anything. For example, if you are testing out a new medication on people to see if there are side effects, you may want one of your "controls" to be that the people testing it cannot be on any other medication. There are an infinite amount of variables depending on which experiment you are referring to. Replication means to do the experiment more than once to make sure that your results are valid. I hope this helps

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