Use this one-sided test to determine whether the difference between the population medians of sample 1 and sample 2 is less than 0, and to get an upper bound. This one-sided test has greater power, but it cannot detect when the median of the first sample is greater than the median of the second sample.
For example, an analyst uses this one-sided test to determine whether the median strength of one type of steel is statistically less than the median of a second type. This one-sided test has greater power to detect whether the strength of the first type of steel is less than the second type of steel, but it cannot detect whether the first type is greater than the second type.
Use this two-sided test to determine whether the population medians differ and to get a two-sided confidence interval. This two-sided test can detect differences that are less than or greater than 0, but it has less power than a one-sided test.
For example, a transportation analyst tests whether the lengths of time that two brands of paint persist on a road differ. Because any difference in the length of time is important, the analyst uses this two-sided test to determine whether the time that one paint persists is greater than or less than the time that the other paint persists.
Use this one-sided test to determine whether the population median of the first sample is greater than the population median of the second sample, and to get a lower bound. This one-sided test has greater power, but it cannot detect when the median of the first sample is less than the median of the second sample.
For example, a technician uses this one-sided test to determine whether the median difference between the speeds of two filling machines is greater than 0 seconds per box. This one-sided test has greater power to detect whether the difference in speed is greater than 0, but it cannot detect whether the difference is less than 0.