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What if you need to test a claim about a dataset packed with extreme outliers and heavy skew where a standard t-test completely fails?

When the Bell Curve Breaks

A standard t-test assumes your data comes from a symmetric bell curve, but real-world data like app load times or house prices is often heavily skewed. When distribution assumptions fail, we switch to a non-parametric testβ€”a statistical method that makes no assumptions about the shape or parameters of the underlying population distribution.
How can we test a claim about the population median without knowing the distribution's shape?

Turning Data into Signs

By definition, the population median (denoted Ξ·\eta) divides a continuous distribution into two equal halves, meaning any single random observation has exactly a 50%50\% chance of landing above Ξ·\eta and a 50%50\% chance of landing below Ξ·\eta. In 1710, Scottish physician John Arbuthnot used this exact logic to analyze London birth records, creating the world's first recorded non-parametric hypothesis test.
Under the null hypothesis H0:Ξ·=Ξ·0H_0: \eta = \eta_0, each observation xix_i is replaced by a positive sign (++) if xi>Ξ·0x_i > \eta_0 or a negative sign (βˆ’-) if xi<Ξ·0x_i < \eta_0. Because each data point independently has a probability p=0.5p = 0.5 of being positive, the total count of positive signs SS follows a binomial distribution: S∼B(n,0.5)S \sim B(n, 0.5).
What does a complete exam-style calculation look like from start to finish?

Worked Example: Streaming Latency

A tech company claims its median video buffering time is Ξ·0=20\eta_0 = 20 milliseconds. A consumer group suspects the median latency is actually higher, recording 1010 independent tests: 22,19,28,20,31,25,27,18,35,2422, 19, 28, 20, 31, 25, 27, 18, 35, 24. Let's test this at the Ξ±=0.05\alpha = 0.05 significance level.