The median test

THE MEDIAN TEST

A SIGN TEST FOR TWO INDEPENDENT SAMPLES

FUNCTION

• It will give information as to whether it is likely that two independent groups (not necessarily of the same size) have been drawn from populations with the same median.

• at least an ordinal scale

• t test for comparing the means of independent samples

• Null hypothesis: the two groups are from populations with the same median

• Alternate hypothesis: the median of one population is different from that of the other (two-tailed test) or the median of one population is higher than that of the other (one-tailed test)

METHOD

• Determine the combined median of the m + n scores.

• Split each group’s score at that combined median – those which exceed the

median and those which do not. Enter the resultant frequencies into a 2 x 2

table. Group

I IICombine

d

No. of scores above combined median

A B A + B

No. of scores below combined median

C D C + D

Total m n N

SP1.1

METHOD

• Find the probability of the observed values by either the Fisher exact test if

m + n ≤ 20 or its chi – square corrected for continuity if m + n > 20.

𝜒2 =𝑁 𝐴𝐷 − 𝐵𝐶 −

𝑁2

2

(𝐴 + 𝐵)(𝐶 + 𝐷)(𝐴 + 𝐶)(𝐵 + 𝐷)𝑝 =

𝐴 + 𝐵 ! 𝐶 + 𝐷 ! 𝐴 + 𝐶 ! 𝐵 + 𝐷 !

𝑁! 𝐴! 𝐵! 𝐶!𝐷!

SP1.1 SP1.2

SAMPLE PROBLEM 1

The following are observations for two independent samples:

Sample I: 10 10 10 12 15 17 17 19 20 22 25 26

Sample II: 6 7 8 8 12 16 19 19 22

The combined

median is 16.

Sample

I IICombine

d

No. of scores above combined median

No. of scores below combined median

7

5

3 10

116

12 9 21

M1 M2

χ² = 0.48125

SAMPLE PROBLEM 1

I. H₀: the two groups are from populations with the same median

H₁: the median of one population is different from that of the other

II. Statistical Test: Median TestIII. Level of significance is at .01 with df = 1

IV. Decision Rule: if χ² ≥ 6.64, reject H₀

if χ² < 6.64, accept H₀

V. Computation

SP1.1

VI. Decision: Since 0.48125 < 6.64, accept H₀ at .01 level

M2

SAMPLE PROBLEM 2

In a cross-cultural test of some behavior theory hypotheses adapted from psychoanalytic theory, Whiting and Child studied the relation between child-rearing practices and customs related to illness in various nonliterate cultures. One hypothesis of their study, derived from the notion of negative fixation, was their oral explanations of illness: Illness results from eating poison, from drinking certain liquids, and from verbal spells and incantations performed by others. Judgements of the typical oral socialization anxiety in any society were based on the rapidity of oral socialization, the severity of oral socialization, the frequency of punishment typical in oral socialization, and the severity of emotional conflict typically evidenced by the children during the period of oral socialization.

SAMPLE PROBLEM 2

Excerpts from ethnological reports of nonliterate cultures were used in the

collection of the data. By using only excerpts concerning customs relating to

illness, judges classified the societies into two groups – those with oral

explanations of illness present and those with oral explanations absent. Other

judges, using the excerpts concerned child-rearing practices rated each society

on the degree of oral socialization anxiety typical in its children. For the 39

societies for which judgements of the presence or absence of oral explanations

were possible, these ratings ranged from 6 to 17.

SAMPLE PROBLEM 2

Societies with oral

explanations absent

Societies with oral

explanations presentCombined

Societies above

median on oral

socialization anxiety

3 17 20

Societies below

median on oral

socialization anxiety

13 6 19

Total 16 23 39

M2

χ² = 9.39

SAMPLE PROBLEM 2

I. H₀: there is no difference between the median oral socialization anxiety in

societies which give oral explanations of illness and the median oral

socialization anxiety in societies which do not give oral explanations of illness

H₁: the median oral socialization anxiety in societies with oral explanations

present is higher than the median in societies with oral explanations absent

II. Statistical Test: Median Test

III. Level of significance is at .01 with df = 1

SP2.2

SAMPLE PROBLEM 2

IV. Decision Rule: if χ² ≥ 3.32, reject H₀

if χ² < 3.32, accept H₀

V. Computation

VI. Decision: Since 9.39 > 3.32, reject H₀ at .01 level

VII. Conclusion: The median oral socialization anxiety is higher in societies

with oral explanations of illness present than is the median in societies with

oral explanations absent.

REFERENCES

• Siegel, S. and Castellan, N. J. (1988). Nonparametric Statistics for Behavioral

Sciences. New York: McGraw Hill.

• Ferguson, G. A. and Takane, Y. (1989). Statistical Analysis in Psychology and

Education. United States: McGraw Hill.