The Friedman test investigates the score differences between
$k$ matched samples of size
$n$, the scores in the
$i$th sample being denoted by
(Thus the sample scores may be regarded as a twoway table with
$k$ rows and
$n$ columns.) The hypothesis under test,
${H}_{0}$, often called the null hypothesis, is that the samples come from the same population, and this is to be tested against the alternative hypothesis
${H}_{1}$ that they come from different populations.
The test is based on the observed distribution of score rankings between the matched observations in different samples.
The test proceeds as follows
(a) 
The scores in each column are ranked, ${r}_{ij}$ denoting the rank within column $j$ of the observation in row $i$. Average ranks are assigned to tied scores. 
(b) 
The ranks are summed over each row to give rank sums ${t}_{\mathit{i}}={\displaystyle \sum _{j=1}^{n}}{r}_{\mathit{i}j}$, for $\mathit{i}=1,2,\dots ,k$. 
(c) 
The Friedman test statistic $F$ is computed, where

G08AEF returns the value of
$F$, and also an approximation,
$p$, to the significance of this value. (
$F$ approximately follows a
${\chi}_{k1}^{2}$ distribution, so large values of
$F$ imply rejection of
${H}_{0}$).
${H}_{0}$ is rejected by a test of chosen size
$\alpha $ if
$p<\alpha $. The approximation
$p$ is acceptable unless
$k=4$ and
$n<5$, or
$k=3$ and
$n<10$, or
$k=2$ and
$n<20$; for
$k=3\text{ or}4$, tables should be consulted (e.g.,
Siegel (1956)); for
$k=2$ the Sign test (see
G08AAF) or Wilcoxon test (see
G08AGF) is in any case more appropriate.
 1: X(LDX,N) – REAL (KIND=nag_wp) arrayInput
On entry: ${\mathbf{X}}\left(\mathit{i},\mathit{j}\right)$ must be set to the value, ${x}_{\mathit{i}\mathit{j}}$, of observation $\mathit{j}$ in sample $\mathit{i}$, for $\mathit{i}=1,2,\dots ,k$ and $\mathit{j}=1,2,\dots ,n$.
 2: LDX – INTEGERInput
On entry: the first dimension of the array
X as declared in the (sub)program from which G08AEF is called.
Constraint:
${\mathbf{LDX}}\ge {\mathbf{K}}$.
 3: K – INTEGERInput
On entry: $k$, the number of samples.
Constraint:
${\mathbf{K}}\ge 2$.
 4: N – INTEGERInput
On entry: $n$, the size of each sample.
Constraint:
${\mathbf{N}}\ge 1$.
 5: W1(K) – REAL (KIND=nag_wp) arrayWorkspace
 6: W2(K) – REAL (KIND=nag_wp) arrayWorkspace
 7: FR – REAL (KIND=nag_wp)Output
On exit: the value of the Friedman test statistic, $F$.
 8: P – REAL (KIND=nag_wp)Output
On exit: the approximate significance, $p$, of the Friedman test statistic.
 9: IFAIL – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit:
${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
For estimates of the accuracy of the significance
$p$, see
G01ECF. The
${\chi}^{2}$ approximation is acceptable unless
$k=4$ and
$n<5$, or
$k=3$ and
$n<10$, or
$k=2$ and
$n<20$.
This example is taken from page 169 of
Siegel (1956). The data relates to training scores of three matched samples of
$18$ rats, trained under three different patterns of reinforcement.