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I have a relatively large set of ratios (n=274). Technically, there is no upper bound, but obviously, the ratios cannot go below 0. Practically, the actual ratios are around 1 (0.8 - 1.2). I am trying to determine whether the median of the distribution is significantly different from 1. I thought about using a single sample t-test, but the ratios are not normally distributed, so I do not think that a t-test is valid? Would (log) normalizing help? Any suggestions would be appreciated!

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  • $\begingroup$ Sounds like a problem for the bootstrap! $\endgroup$ Oct 21 '15 at 5:37
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How about a sign test. Count the number of values over one and if 1 is the median, the number of values over 1 should follow the Binomial(n=274,p=0.5) distribution. So, for instance, if the number of values over 1 was K = 100:

Ho: median is one vs Ha: not

p-value = $P(K\le 100) + P(K\ge 175)$ I'll let you find those binomial probabilities from your favorite statistics package.

Or you can use (equivalently) a chi-square goodness of fit test with values (174, 100), if K were 100, and proportions 0.5, 0.5.

================ You are right not to do the t-test because it tests for differences in means, so it would only check medians if you knew that the distribution of ratios was symmetric.

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