The **midrank** is a tied observation’s average rank. An observation’s *rank* is its position in an order list, from smallest to largest. A midrank is one way to deal with tied ranks in nonparametric tests that based on ranks, like the Wilcoxon rank-sum test. You can extend a test to allow for ties by “patching” the data and assigning midranks instead of ranks (Stark, 2021).

For example, let’s say your ordered observations were {1.3, 1.7, 1.7, 2.5}. These would be ranked {1, 2, 2, 4}. However, you take the average of where the two tied ranks should be (i.e. ranked 2 and 3} and report the ranks with midranks instead: {1, 2.5, 2.5, 4}.

## Finding the Midrank

The midrank depends on the number of tied ranks (*l*) and number of items that are lower ranked (*k*). It can be found with the formula (Scholz, 2013):

For example, the ordered observations given above {1.3, 1.7, 1.7, 2.5} have a mid-rank of 2.5:

## References

Agresti, A. (2003). Categorical Data Analysis. Wiley.

Scholz, F. (2013). Stat 425 Introduction to Nonparametric Statistics Rank Tests for Comparing Two Treatments. Retrieved February 14, 2021 from: http://faculty.washington.edu/fscholz/SpringStat425_2013/Stat425Ch1.pdf

Stark, P. (2021). SticiGui. Retrieved February 14, 2021 from: https://www.stat.berkeley.edu/~stark/Teach/S240/Notes/ch4.htm

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