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Suppose I have two poll questions that can be answered with a Yes or a No with the following results:

Poll 1

  • Yes: $200$
  • No: $100$

Poll 2

  • Yes: $2$
  • No: $1$

Both polls have a $66\%$-$33\%$ split, but getting them to a $50\%$-$50\%$ split is much harder with the first poll than with the second poll. There would have to be $100$ new "No"s for the first as opposed to $1$ new "No" for the second one, two orders of magnitude in difference. Is there a mathematical/statistical name for this "resistance to change"?

2 Answers 2

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Sometimes a result is said to be "robust" if it holds despite small changes in input.

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    @Elen'dia Starman: There is a lot of information on **robust statistics**. As usual, one can start with Wikipedia. Then look for books on robust statistics, or more generally non-parametric statistics. However, robust statistics, it seems to me, is not what the problem as described is about. Robust statistics (in part) tells you what to do when you do an experiment $4$ times and get the results $11.3$, $12.1$, $9.7$, and $444.4$. Do you average the results? Of course not, you throw the "outlier" $444.4$ away. Well, it is not always that obvious, hence robust statistics.2011-06-15
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Most of what you are trying to get at in the supplied example is captured by the notion of sample variance, or its square root, the sample standard deviation.

In both cases, the sample mean of the "yes" responses simplifies to $\frac{2}{3}$. But in the case of the poll in which your sample size was $300$, the estimate $2/3$ is a far more reliable indicator of the beliefs of the general population than the poll based on a sample size of $3$!

The notion of sample variance (or standard deviation) is the usual way of trying to capture numerically this notion of reliability. It is the mathematics behind the often heard phrase "the result is accurate to $\pm 3$ percent $19$ times out of $20$."

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    @user6312: Ah, thanks very much! :)2011-06-15