Friedman's Test is a nonparametric alternative to twoway analysis of variance. The hypothesis is:
H_{0} the means of all the samples are equal
H_{1} the mean of at least one of the samples is different
Consider the example where three treatments are evaluated on four different patients:
Therapy

Andrew

Belinda

Chris

Dave

Relaxed 
110

140

100

130

Normal 
115

150

105

135

High Intensity 
117

155

100

135

The test involves:
 impose ranks on each of the columns. If values are equal, average the ranks they would have got if they were slightly different:
Therapy

Andrew

Belinda

Chris

Dave

Relaxed 
1

1

1.5

1

Normal 
2

2

3

2.5

High Intensity 
3

3

1.5

2.5

 calculate the F_{r} statistic using the formula:
Where:
I 
Number of samples (treatments) 
J 
Number of blocks 
R_{i} 
sum of the ranks in row 'i' 
This gives a value for F_{r} of 4.65
The value of F_{r} has an approximately chisquare distribution with I  1 degrees of freedom.
This is a distribution free alternative to ANOVA. It will compare several samples and test the hypothesis:
H_{0} the means of all the samples are equal
H_{1} the mean of at least one of the samples is different
The test involves:
 sort the combined results into size order and allocate ranks
 form a table that contains the ranks, instead of the values
 calculate the test statistic 'K using the formula:
Where:
R_{i} sum of the ranks in row 'i'
J number of values in row 'i'
N the total number of values
I the number of rows
'K' has an approximately c^{2} distribution with degrees of freedom I  1
The ANOVA method relies on the assumption that the variance is the same for all the samples. Levene's test is:
H_{0} the variance is the same for all the samples
H_{1} the variance of at least one of the samples is different
The test uses the statistic 'W' where:
Where:
N 
total number of values 
Ni 
number of values in row 'i' 
k 
number of levels 

median of level 'i' 

the average of all the z_{ij} values 

the average of the z_{ij} values in row 'i' 
This is the nonparametric version of the two sample ttest; it compares the means of two samples. It tests the hypothesis:
H_{0} the means are equal
H_{1} the mean of sample 'm' is [less than/greater than/not equal to] the mean of sample 'n'
It is similar in concept to the Wilcoxon Signed Rank Test and is also known as the Wilcoxon Rank Sum Test. The test involves:
 sort the combined results into size order and allocate ranks
 find the sum 'w' of the ranks of the sample with the smallest number of values (if they both have the same number of values, select either). This is one of the critical values.
For small data sets the critical value is found from published tables. Because the sum of the ranks must be an integer, it is not usually possible to find the exact critical value:
m is the number of values in the sample with the least number of values
n is the number of values in the other sample
m_{1} is associated with the sample containing the smallest number of values
See Mood's Median Test
The Mood's Median Test is a nonparametric equivalent of ANOVA. It is an alternative to the KruskalWallace test. The hypothesis is:
H_{0} the medians of all the samples are equal
H_{1} the median of at least one of the samples is different
The test involves:
 find the median of the combined data set
 find the number of values in each sample greater than the median and form a contingency table:

A

B

C

Total

Greater than the median





Less than or equal to the median





Total





 find the expected value for each cell:
 find the chisquare value from:
Most types of hypothesis tests require that the population conforms to a particular distribution, usually the normal distribution. Where this is not the case a nonparametric test can be used.
Nonparametric tests make no assumptions about the type of distribution (although they may require symmetry, or some other property). The disadvantage is that they are not as efficient; for a given data set the nonparametric test will give a higher pvalue.
Parametric Hypothesis Test 

A Parametric test make assumptions about the underlying distribution of the population from which the sample is being drawn, and which is being investigated. This is typically that the population conforms to a normal distribution.
Alternative name for Mood's Median Test
See the Mann Whitney Tests
Wilcoxon Signed Rank Test 

This is the nonparametric equivalent of the one sample ttest. It tests the hypothesis:
H_{0} the mean equals zero
H_{1} the mean [is less than/greater than/not equal to] zero
The test involves:
 sort the values into order of their absolute magnitude (ignoring the signs) and allocate ranks
 calculate the sum the ranks of the data values are positive (S_{+})
An example would be:
Rank 
1

2

3

4

5

6

Data Value 
1.0

2.0

2.5

3.0

3.0

3.5

If the mean is nonzero the positive values will be clustered at one end or other of the ordered data set and S_{+} and S_{} will be very different.
For the example S_{+} = 3 + 4 + 5 + 6 = 18
For small data sets values the critical value is found from published tables. Because the sum of the ranks must be an integer, it is not usually possible to find the exact critical value:
For larger data sets the zstatistic can be found from:
This gives the upper tail test (the mean is less than zero). For the lower tail test, use the sum S+ and for the two tailed test use both, remembering to use the table value twice.
The pvalue can be calculated using normal distribution tables.
