A data set consists of several different positive numbers. Which one of these changes leaves the MEAN of the set exactly as it was?

If the set has n numbers with mean m, its total is nm; adding one more number equal to m makes the total (n + 1)m over n + 1 numbers, so the mean is still m. The new value carries exactly its own share and asks nothing of the others. Each of the other changes moves the mean: adding 5 to every number lifts it by 5, doubling every number doubles it, and doubling the largest number alone adds that number's value to the total and so raises the mean by (largest ÷ n). Note that the added value does change the SPREAD — a value sitting at the mean pulls the standard deviation down.