Non Parametric Statistics

1. THE MEDIAN TEST
A SIGN TEST FOR TWO INDEPENDENT SAMPLES

2. 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)

3. 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 II
Combine
d
No. of scores above
combined median
A B A + B
No. of scores below
combined median
C D C + D
SP1.1

4. 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 −
𝑁
2
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

5. 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.
Sampl
e
I II
Combine
d
No. of scores above
combined median
No. of scores below
7
5
3 10
116
12 9 21
M1 M2
χ² = 0.48125

6. 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 Test
III. 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

7. 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.

8. 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.

9. SAMPLE PROBLEM 2
Societies with oral
explanations absent
Societies with oral
explanations present
Combined
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

10. 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

11. 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.

12. 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.