Median Test

AKA: Mood's Median Test.
Context:
- It assumes that the following conditions are met:
  - Observations, measurements are independent both within and between samples.
  - Population and sampling distributions have the same shape.
- It tests the following hypotheses:
  - Null hypothesis ([math]\displaystyle{ H_0 }[/math]): the population medians all are equal.
  - Alternative hypothesis: ([math]\displaystyle{ H_A }[/math]): the population medians are not all equal.
- Test Statistic: It is based on a chi-squared test statistic.
  - It rejects the null hypothesis based on a pre-established significance level or evaluation of a critical region.
- …
Counter-Example(s)
- Pearson's Chi-Squared Test.
- ANOVA Algorithm.
See: Frequency Distribution, Statistics, Pearson's Chi-Squared Test, Nonparametric Test, Null Hypothesis, Median, Statistical Population, Sampling (Statistics).

References

(ITL-SED, 2017) ⇒ Retrieved 2017-01-08 from NIST (National Intitute of Standards and Technology, US) website http://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/meditest.htm
- The median test is a special case of the chi-square test for independence. Given k samples with [math]\displaystyle{ n_1, n_2, \cdots, n_k }[/math] observations, compute the grand median of all [math]\displaystyle{ n_1 + n_2 + \cdots + n_k }[/math] observations. Then construct a [math]\displaystyle{ 2\times k }[/math] contingency table where row one contains the number of observations above the grand median for each of the [math]\displaystyle{ k }[/math] samples and row two contains the number of observations below or equal to the grand median for each of the </math> k samples. The chi-square test for independence can then be applied to this table. More specifically

[math]\displaystyle{ H_0: }[/math] All k populations have the same median.
[math]\displaystyle{ H_a: }[/math] All least two of the populations have different medians

Test Statistic: [math]\displaystyle{ \frac{N^2}{ab}\sum^k_{i=1}\frac{(O_{1i}−n_ia/N)^2}{n_i} }[/math]

where

[math]\displaystyle{ a }[/math] the number of observations greater than the median for all samples
[math]\displaystyle{ b }[/math] the number of observations less than or equal to the median for all samples
[math]\displaystyle{ N }[/math] the total number of observations
[math]\displaystyle{ O_{1i} }[/math] the number of observations greater than the median for sample i

Significance Level: [math]\displaystyle{ \alpha }[/math]

Critical Region: [math]\displaystyle{ T\gt \chi^2_{1−\alpha;k−1} }[/math]

where [math]\displaystyle{ \chi^2 }[/math] is the percent point function of the chi-square distribution and k-1 is the degrees of freedom

Conclusion: Reject the independence hypothesis if the value of the test statistic is greater than the chi-square value.

(Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/Median_test Retrieved:2016-12-17.
- In statistics, Mood's median test is a special case of Pearson's chi-squared test. It is a nonparametric test that tests the null hypothesis that the medians of the populations from which two or more samples are drawn are identical. The data in each sample are assigned to two groups, one consisting of data whose values are higher than the median value in the two groups combined, and the other consisting of data whose values are at the median or below. A Pearson's chi-squared test is then used to determine whether the observed frequencies in each sample differ from expected frequencies derived from a distribution combining the two groups.