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Sign test or Wilcoxon signed-rank test

Compares the centre of a skewed measurement in one group with a known reference value, using signs or ranks.

Clinical example

Whether the median waiting time in your clinic differs from a target of 30 days. Waiting times are strongly skewed and the sample is small.

When to use it

The outcome is a measurement, one value per patient.
You have one group and a known reference value to compare its centre with.
The data are clearly skewed and the sample is small.
It also works for paired data: apply it to the within-pair differences, with 0 as the reference. It then tests whether the median difference is zero.

Assumptions to check

Independent observations

One value per patient.

No normality needed

Both tests use ranks or signs, so skew and outliers matter much less. Report the median and IQR.

The Wilcoxon version needs symmetry

The signed-rank version assumes the distribution is symmetric around its median. The sign test does not, but it has less power.

What to report

Median and IQR
Median difference (IQR)

Common mistakes

Reporting a mean ± SD for a heavily skewed variable

Using the Wilcoxon signed-rank test for clearly skewed differences, because it assumes symmetry

Alternatives

If your situation is…	Use
The data are roughly normal, or the sample is large	One-sample t-test

In your software

# R
wilcox.test(d$waiting_days, mu = 30, conf.int = TRUE)

# Sign test
binom.test(sum(d$waiting_days > 30), sum(d$waiting_days != 30))

SPSS

NPTESTS
  /ONESAMPLE TEST (waiting_days) WILCOXON(TESTVALUE=30).

Sources

An interval for the median

The sign test and the Wilcoxon test give a p-value. If you also want a confidence interval for the median, a bootstrap confidence interval is one option.