Espirales. Revista multidisciplinaria de investigación científica, Vol. 6, No. 42
October - December - 2022. e-ISSN 2550-6862. pp 1-20
DOI https://doi.org/10.31876/er.v6i43.827
Kruskal-Wallis, Friedman and Mood nonparametric tests
applied to business decision making
Pruebas no paramétricas de Kruskal-Wallis, Friedman y Mood aplicadas a la
toma de decisiones empresariales
Carlos Ernesto Flores Tapia*
Karla Lissette Flores Cevallos*
Received: July 13, 2022
Approved: September 11, 2022
* Commercial Engineer, Master in International
Negotiations and Doctor (PhD.) in Social and Legal
Sciences. Research Professor - Pontificia Universidad
Católica del Ecuador. http://orcid.org/0000-0002-
1892-6309
cflores@pucesa.edu.ecflorestapiacarlos@yahoo.com
* D. candidate in Business Management and
Economic Environment -University of Cadiz, Spain-.
Researcher. Executive Director - Fundación Los
Andes, Ecuador. http://orcid.org/0000-0003-0851-
5319 karla.floresceva@alum.uca.es
Abstract
The post COVID-19 crisis scenario demands continuous
improvement in the quality of the products and services offered,
supported by administrative and statistical tools that optimize
business decision-making. In this research, three non-parametric
statistical tests -Kruskal-Wallis, Friedman and Mood's median-
are applied to three case studies in order to test the hypotheses
posed, respectively: whether the distributions of the populations
of the units sold by a distribution company are equal for three
types of inputs, whether the stress levels differ or not for three
types of work modalities in an automobile commercialization
company and whether the medians of production yields in an
agricultural company are equal or not for four varieties of fruit.
The solution is sought using standard procedures for each of the
nonparametric tests.
Keyword:
Business, Statistics, Statistical analysis.
Cite this:
Flores, C., Flores, K. (2022).
Kruskal-Wallis, Friedman and
Mood nonparametric tests applied
to business decision making,
Espirales. Revista Multidisciplinaria
de investigación científica, 6(42), 1-
20
Kruskal-Wallis, Friedman and Mood nonparametric tests applied to business decision making
Espirales. Revista multidisciplinaria de investigación científica, Vol. 6, No. 42
October - December - 2022. e-ISSN 2550-6862. pp 1-20
2
Introduction
The scenarios surrounding industries in the globalized world of the 21st century, where
technological and scientific advances are constantly evolving, make the demand for
quality of products and services increasingly higher. Nowadays, one of the key factors
for the success of an industry is to make use of all its knowledge and learning capacity,
as well as its experience. Statistical inference through hypothesis testing in the business
sector is one of the elements that can contribute the most to learning and improvement
of products and processes; in this sense, the application of statistics is presented as an
effective tool to understand and optimize the supply of goods and services in the
business sector . (Koontz et al., 2017).
The statistics used for the verification of business hypotheses can be of parametric and
nonparametric type; the first ones start from the assumption in which the samples come
from populations with normal distribution, the second ones, are useful if the distribution
from which the sample comes from is not specified or it has been proven that they do
not meet the assumption of normality (Anderson et al., 2016; Flores-Ruiz et al., 2017).
In this research we proceed with the study of three cases referred to companies in the
province of Tungurahua-Ecuador, in which the application of non-parametric methods
using the Kruskal-Wallis, Friedman and Mood's median test is required. In the first case,
the administration of the company D&D Distribuciones (2021)In the first case, the
management of the company, dedicated to the commercialization of gifts in the
province of Tungurahua-Ecuador, is interested in determining whether or not there are
differences in sales between the three types of imported bases or supports used for the
decoration of these gifts - cardboard boxes, wooden bases and glass bases - in order
to forecast the volume of orders needed from suppliers in the international market. The
Resumen
El escenario post crisis del COVID-19 exige el mejoramiento continuo
de la calidad de los productos y servicios ofertados sustentado en
herramientas administrativas y estadísticas que optimicen la toma de
decisiones empresariales. En tal virtud, en la presente investigación
se aplican tres pruebas estadísticas no paramétricas -Kruskal-Wallis,
Friedman y mediana de Mood- a tres estudios de caso con el objetivo
de probar las hipótesis planteadas, respectivamente: si las
distribuciones de las poblaciones de las unidades vendidas por una
empresa distribuidora son iguales para tres tipos de insumos, si los
niveles de estrés difieren o no para tres tipos de modalidades de
trabajo en una comercializadora automotriz y si las medianas de
rendimiento de producción en una empresa agrícola son iguales o no
para cuatro variedades de frutas. La solución se busca mediante
procedimientos estándar para cada una de las pruebas no
paramétricas.
Palabras clave:
Empresa, Estadística, Análisis estadístico
Flores Tapia Carlos Ernesto, Flores Cevallos Karla Lissette
Espirales. Revista multidisciplinaria de investigación científica, Vol. 6, No. 42
October - December - 2022. e-ISSN 2550-6862. pp 1-20
3
Kruskal-Wallis test is applied because we are dealing with independent samples,
without necessarily assuming that the data have normal distributions.
In the second case study, the human talent management of the company, whose line of
business is the commercialization of spare parts and accessories for vehicles, needs to
verify Comercial Automotriz Romero (2021)whose line of business is the
commercialization of spare parts and accessories for vehicles, needs to verify, through
Friedman's test, whether three types of work modalities implemented during the first
six months of 2021 -5 days teleworking, 5 days on-site and 5 days of mixed workday: 3
teleworking and 2 days on-site-, in the context of the health crisis Covid-19, differ or not
in the stress levels of employees.
In the third case illustrated in the present research, the agricultural-commercial company
Bolhi Market (2021) verifies whether or not the yields of four varieties of fruit measured
in Kg/Ha, differ from each other; for which it uses Mood's median test because it has a
categorical variable and a continuous response variable, it is not known with certainty
whether the data of all the groups have similar distributions and the observations are
independent of each other. In the three cases studied, the solution is sought both by
conventional procedures for each of the non-parametric tests contemplated in this
research and by means of Minitab software.
Nonparametric methods are statistics used to test hypotheses in which the population
distribution does not follow the normal curve or other specific shape, which is why they
are also known as free distribution tests. Among the main nonparametric methods for
the analysis of ordered data and Spearman's rank correlation coefficient are the sign
test, Wilcoxon, Mann-Whitney, Kruskal-Wallis, Mood's Median and Friedman; while
among the goodness-of-fit tests are the Kolmogorov-Smirnov and chi-square tests,
mainly (Levin et al., 2014).
In nonparametric methods, the data generally respond to nominal and ordinal variables,
rather than interval or ratio variables, or there are few data. In addition, it usually
happens that the data do not meet the requirements of normality, level of measurement
and homogeneity required for the application of parametric tests -Z, t student, F or
ANOVA-, consequently, nonparametric methods are appropriate as alternatives to
parametric tests. However, their main disadvantage is the loss of sharpness in the
estimation of intervals in exchange for the possibility of using less information and much
faster and less laborious calculations. (Sailema, 2019).
In addition, if the distributions of the groups include outliers and there is one categorical
variable and one continuous response variable, Mood's median is used; if there is only
one categorical factor and non-normal data from three or more populations and it does
not include outliers, Kruskal-Wallis is used; and if there is a randomized block design to
test medians, Friedman's test is applied, this test being a non-parametric alternative to
the block design of experiments model and to an ANOVA with two factors. (Flores-
Tapia & Flores-Cevallos, 2021; Lind et al., 2021).
Kruskal-Wallis, Friedman and Mood nonparametric tests applied to business decision making
Espirales. Revista multidisciplinaria de investigación científica, Vol. 6, No. 42
October - December - 2022. e-ISSN 2550-6862. pp 1-20
4
The present research prioritizes the study in three business cases by applying the
Kruskal-Wallis test, the Friedman median test and the Mood median test.
Kruskal-Wallis test
It is a nonparametric test for ordinal sample data from three or more independent
populations classified by ranks. It does not require assumptions regarding the shape of
the populations and is also called Kruskal-Wallis one-way analysis of variance by ranks.
This test determines whether the medians of two or more groups differ, for which the
data must contain a categorical variable, a continuous response variable and
distributions with a similar shape in all groups. It can be considered a generalization of
the Wilcoxon, U and Mann-Whitney rank sum tests and the Kruskal-Wallis test is
analogous to the parametric one-way ANOVA test. Also, the ordering of information by
ranks rather than plus and minus signs wastes less data than the sign test. (González et
al., 2021; Triola, 2018).
Independent samples are randomly selected from k populations that differ only by
location and are not necessarily assumed to come from normal distributions. It is
important that each sample has at least five observations that are ranked from lowest to
highest as if they were a single group (Flores-Tapia & Flores-Cevallos, 2017b). The
Kruskal-Wallis statistic with k-1 degrees of freedom is denoted by the letter H and is
given by equation 1, as follows:
𝐻 =
!"
#$%#&!'
#
%
(
!
'
"
#
!
+%
%
(
"
'
"
#
"
+ ⋯+%
%
(
#
'
"
#)
!
'
− 3(𝑛 + 1)
(1)
Where:
∑R1, ∑R2, ..., ∑Rk: sums of the ranks of samples 1, 2, ..., k, respectively.
n1, n2, ..., nk: sample sizes 1,2, ..., k, respectively.
n: combined number of observations of all samples.
The Kruskal-Wallis test is right-tailed and seeks to test the following hypotheses:
H0: the samples come from populations with the same distribution.
H1: the samples come from populations with different distributions.
Friedman test
The Friedman test is the nonparametric alternative to the one-way ANOVA test with two
factors when samples are dependent and also to a block design of experiments model.
It is an extension of the Wilcoxon signed-rank test for more than two groups, based on
sum of ranks. Assuming certain simplifications, it can be considered as a comparison
between the medians of several groups, being appropriate when the data have a natural
order, that is, when they have to be ordered and also paired to make sense of them.
The concept is similar to the Wilcoxon rank sum for two groups, but this time assigning
the ranks within each row, since there are more than two measures for each individual
and then applying the rank sum for each column. (Freund et al., 2010).
Flores Tapia Carlos Ernesto, Flores Cevallos Karla Lissette
Espirales. Revista multidisciplinaria de investigación científica, Vol. 6, No. 42
October - December - 2022. e-ISSN 2550-6862. pp 1-20
5
The Friedman test allows determining whether the median of the treatment effect differs
or not in a randomized block design, for which it is required that the response factor be
continuous or ordinal, the treatment of categorical type and to have a categorical block
variable, the blocks being a group of experimental runs in constant conditions. Put
another way, this test is designed to verify that the probability distributions of k
treatments are identical or that at least two of the distributions differ in location (Guzman
& Monteoliva, 1984; Minitab, 2021).
The test statistic is given according to Equation 2, as follows:
𝐹𝐹
*
=
!"
+)%) &!'
%
∑
𝑅
,
")
,-!
− 3𝑏(𝑘 + 1)
(2)
Where:
b: blocks used in the experiment.
k: treatments.
(Ri)2: sum of the ranks of the observations corresponding to the squared treatment.
Also, as considered with the Kruskal-Wallis statistic, the null distribution of the Friedman
statistic is calculated with a λ2 distribution with k-1 degrees of freedom, provided that
the number of blocks used is large, noting also that the approximation is adequate if
the number of treatments or the number of blocks is greater than 5 (Levin et al., 2014).
The hypotheses for the Friedman test are as follows:
H0: all treatment effects are zero.
H1: not all treatment effects are zero.
Mood median test
Mood's median test determines the dispersion of the populations with respect to the
mean of the ranges of the two populations as if they were a single group, that is, it
calculates whether the medians of two or more groups are different and the range of
values that are likely to contain the difference between the medians of the populations.
It is applied when a categorical variable and a continuous type response variable are
available and it is not known with certainty whether the data from all groups have
similarly shaped distributions. In addition, it is taken into account that the sample size
has to be smaller than 15 or 20 observations and the observations must be independent
of each other and the data are measured at least on an ordinal scale. (Guatemal, 2019).
Mood's median test is less sensitive to outliers than the Kruskal-Wallis test; however,
the power is lower when the data come from distributions such as normal and can be
considered as a special case of Ji-squared (Ramos et al., 2019).
The median Mood statistic is shown in Equation 3.
𝑀 =%
4
𝑟
,
−%
.&!
"
6
"
(3)
Where:
N: n + m; where n is the size of sample 1 which is equal to or smaller than sample 2 or
m.
ri: is the rank of the i-th observation x in the ordering of the combined x and y values.
Kruskal-Wallis, Friedman and Mood nonparametric tests applied to business decision making
Espirales. Revista multidisciplinaria de investigación científica, Vol. 6, No. 42
October - December - 2022. e-ISSN 2550-6862. pp 1-20
6
(N+1) /2: average of the ranges assigned to the x and y values combined and ordered
from highest to lowest.
4
𝑟
,
−%
.&!
"
6
"
: deviation of each of the observations with respect to the mean range.
The above value in parentheses is similar to ∑ (Xi - !X)2 in the calculation of the standard
deviation; such that if the values of y are further apart than those of x (s 1 <s 2) the M
statistic tends to be small, whereas if those of x are much more dispersed than those of
y (s 1 >s 2) the statistic tends to a larger value (Guzman & Monteoliva, 1984).. Therefore,
the objective of Mood's median test is to locate M that is sufficiently large or small to
reject H0. The hypotheses posed for Mood's test are as follows:
H0: the medians of the population are all equal.
H1: population medians are not all the same.
On the other hand, for hypothesis testing of the Kruskal-Wallis, Friedman and Mood
Median tests, software tools are available to help analysts in the implementation of
computer models, such as Minitab, SPSS, R, Excel, Stata and Gretel software, mainly
(Gujarati & Porter, 2009; IBM, 2021; Microsoft, 2021; Minitab, 2021)..
Materials and methods
The study contemplated the application of non-parametric techniques -a
hypothesis test that does not require the population distribution to follow a
normal distribution- such as the Kruskal-Wallis, Friedman and Mood's Median
tests that fit the statistical methodology, particularly the hypothesis testing
(Flores-Tapia et al., 2017; Hernández-Sampieri et al., 2014; Levin et al., 2014;
Triola, 2018).The six stages or phases that were followed were foreseen:
• Definition of the problem of interest and collection of relevant data.
• Formulation of a mathematical model that represents the problem.
• Development of a computer-based procedure to derive a solution to the
problem from the model.
• Model testing and improvement according to needs.
• Preparation for the application of the model prescribed by the
administration.
• Implementation.
The scope of the study contemplated the application of the three nonparametric
tests contemplated, one to each of the three case studies proposed in the
research following three phases, namely: definition of the problem, application
of the mathematical models and development of the computer-based
procedure, and presentation of results (Flores-Tapia & Flores-Cevallos, 2017a;
Lind et al., 2021).. That is, once the problem was defined, a five-step hypothesis
test was used to solve the question posed by management, that is:
Step 1: Formulation of the null and alternative hypotheses.
Step 2: selection of the significance level.
Flores Tapia Carlos Ernesto, Flores Cevallos Karla Lissette
Espirales. Revista multidisciplinaria de investigación científica, Vol. 6, No. 42
October - December - 2022. e-ISSN 2550-6862. pp 1-20
7
Step 3: decision on the test statistician
Step 4: Formulation of the decision rule
Step 5: calculation of the statistic and making the decision regarding the null
hypothesis - not to reject the null hypothesis or to reject the null hypothesis and
accept the alternative. The procedure for calculating each of the statistics
considered in this research is detailed below.
Kruskal-Wallis test
In the case of the Kruskal-Wallis test, all samples were combined into a single
sample, sorted in ascending order and a rank was assigned to each sample value
-if there was a tie in the rank, the average of the ranks involved was assigned to
each observation. Then, for each sample, the sum of the ranks and the sample
size were determined. Subsequently, the hypothesis testing steps were followed,
that is, the null and alternative hypotheses were formulated, the significance
level was set, the Kruskal-Wallis statistic was identified by verifying compliance
with the positions for this test, the critical value was established by consulting
the Chi-square table with k-1 degrees of freedom, and the decision rule was
formulated. Finally, the Kruskal-Wallis statistic was calculated and a decision was
made regarding the null hypothesis, applying the decision rule, that is, if the
calculated Kruskal-Wallis value -H- is less than the critical value, the null
hypothesis is not rejected, otherwise, the null hypothesis is rejected and the
alternative hypothesis is accepted.
Friedman test
Since it is a statistic equivalent in ranks to the sum of squares for treatments -
SST- for the randomized block design, once the data of a randomized block
design were obtained, within each block the observed values of the responses
of each of the k treatments were classified from 1 to k blocks, from the smallest
to the largest, and each of the ranks was summed. In case of a tie between the
ranks involved, the average was assigned to each one, as long as the ties
occurred within the same block. Subsequently, the null and alternative
hypotheses were determined, the significance level was established, the
Friedman test was selected, verifying compliance with the assumptions for its
application, the decision rule was formulated indicating that the region of
rejection of the null hypothesis is given by Fr > λ2α with (k-1) degrees of
freedom. The critical value, as in the Kruskal-Wallis test, was consulted in the
table corresponding to the chi-squared distribution. Finally, the Friedman
statistic was calculated and the decision rule was applied with respect to the null
hypothesis.
Kruskal-Wallis, Friedman and Mood nonparametric tests applied to business decision making
Espirales. Revista multidisciplinaria de investigación científica, Vol. 6, No. 42
October - December - 2022. e-ISSN 2550-6862. pp 1-20
8
Mood median test
Once the data for each of the samples had been sorted, the median and the size
of each sample and of all the samples were calculated. Then the null and
alternative hypotheses were formulated, the level of significance and the critical
values of Mood's median test were determined by consulting the values in the
corresponding chi-square table, after verifying compliance with the assumptions
that allow its application. Subsequently, the sample medians were transformed
into a λ2 value, obtaining the test statistic, for which a table was constructed
with the observed frequencies, the observations in each sample above and
below or equal to the overall median of all samples were counted, and the Mood
median statistic was calculated with Equation 4.
𝜆
!
"
= ∑$
($
!
%$
"
)
#
$
"
(4)
Where:
∑: summation
Fo: observed frequency
Fe: expected frequency
The final step consisted of comparing the critical value with the calculated value
of the Mood median and a decision was made whether or not to reject the null
hypothesis by applying the decision rule.
Results
Following the methodology indicated above, the application is developed and
the results of the Kruskal-Wallis, Friedman and Mood Median tests considered
for the case studies, which are the object of this research, are shown.
Kruskal-Wallis test
Before proceeding with the application of the Kruskal-Wallis test and the
presentation of the results, the statement of the case study and the conditions
of the study are established as follows:
The company D&D Distribuciones located in Ambato-Ecuador is dedicated to
the production of gifts for special occasions. The management is interested in
knowing what type of bases - pedestals or supports - customers prefer for gifts
- cardboard boxes, wooden bases and glass bases - because this type of material
is imported and it is necessary to place the order with enough time in advance
and in the quantities that best suit the company. Specifically, we want to test
whether or not there is a difference in sales depending on the three types of
bases or supports used for gifts and thus make the best decisions regarding the
import order.
Flores Tapia Carlos Ernesto, Flores Cevallos Karla Lissette
Espirales. Revista multidisciplinaria de investigación científica, Vol. 6, No. 42
October - December - 2022. e-ISSN 2550-6862. pp 1-20
9
Following the methodology proposed in this research and considering for this
test a significance level of 5% and taking into account the units sold during the
months of February to June 2021, the null and alternative hypotheses are stated
as follows:
Ho: the distributions of the populations of units sold are equal for the three types
of gift bases.
H1: Not all population distributions are equal.
We proceed with the classification in ranges of each sample value with the
respective sum for each of the groups or types of gift bases (Table 1),
establishing the decision rule for the significance level of 0.05; that is, if the
calculated value of H is less than the critical value, the null hypothesis is not
rejected. Then, the critical value is located in the Chi-square table for 2 degrees
of freedom (k-1), being the value 5.991 and, once the assumptions for the
application of the Kruskal-Wallis statistic are verified, we proceed with the
calculation of H, obtaining a value of 8.06 and as the calculated value H is greater
than the critical value, the null hypothesis is rejected and the alternative is
accepted. That is, there is sufficient evidence to conclude that there is a
difference between sales in the three types of gift bases or supports, which
implies that the managers of the company D&D Distribuciones need to place
the import order with different quantities for each of the three types of gift bases.
Table 1.
Ranking and summation of ranks for each of the gift base types
-D&D Distributions
Months
2021
R3 Ranges
Cardboard
Boxes
(Units)
R2 Ranges
Wooden
bases
(Units)
R1 Ranks
Glass
bases
(Units)
February
230
March
April
110
5
May
June
1
Summation
Source:
Own elaboration based on D&D Distribuciones company data.
However, if the calculations are made using Minitab software with the Kruskal-Wallis
non-parametric statistics option, the results are as follows:
Kruskal-Wallis, Friedman and Mood nonparametric tests applied to business decision making
Espirales. Revista multidisciplinaria de investigación científica, Vol. 6, No. 42
October - December - 2022. e-ISSN 2550-6862. pp 1-20
10
Descriptive statistics
Type of base or
support
N
Median
Classification of averages
Z-value
Glass Base
5
3.4
-2.82
Wooden Base
5
9.8
1.10
Cardboard
Boxes
5
10.8
1.71
General
8.0
Test
Null hypothesis
H₀: All medians are equal.
Alternative
hypothesis
H₁: At least one median is different.
GL
H-value
p-value
8.06
0.018
The sample median estimates for the three groups are 50,150 and 200 and since the p-
value 0.018 is less than the significance level determined for this test of 0.05;
consequently, the null hypothesis is rejected.
Friedman test
Before proceeding with the application of the Friedman test and the presentation of the
results, the statement of the case study and the conditions of the same are established,
in the following terms:
The human talent management of the company Comercial Automotriz Romero
compares the stress level of three work modalities during the first six months of the year
2021 in the context of the Covid-19 health crisis: 5 days teleworking, 5 days face-to-
face and mixed workday - 3 days teleworking and 2 days face-to-face. The management
wants to observe whether stress levels differ or are the same for the three types of work
modalities and, consequently, conducts a randomized block experiment for 6
collaborators, recording the response on a scale of 1 to 3.
Once the data were obtained, the observed values were classified into ranges in each
of the treatments and the sum was calculated (Table 2).
Table 2.
Classification and summation of ranks for each of the work modalities -
Comercial Automotriz Romero-.
Collaborators
R3
Ranges
Telework
R2
Ranges
On-site
R1 Ranks
Mixed
day
1
1.0
1.21
3.0
1.56
2.0
1.48
1.5
1.63
3.0
2.01
1.5
1.63
1.0
1.42
2.0
1.70
3.0
2.06
1.0
1.16
2.5
1.27
2.5
1.27
Flores Tapia Carlos Ernesto, Flores Cevallos Karla Lissette
Espirales. Revista multidisciplinaria de investigación científica, Vol. 6, No. 42
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11
5
2.0
2.43
3.0
2.64
1.0
1.98
1.0
1.94
3.0
2.81
2.0
2.44
Source:
Own elaboration based on data from Comercial Automotriz Romero.
Then the standard statistical procedure is followed to test the null and alternative
hypotheses with a chosen significance level of 5%. As the number of blocks exceeds 5
(six in this case) the Friedman test is used, considering as a decision rule: if the calculated
value a of F
r
is less than the critical value, the null hypothesis is not rejected. The null
and alternative hypotheses are stated as follows:
Ho: all treatment effects are zero. That is, stress levels do not differ for the three types
of work modalities, they are the same.
H1: not all treatment effects are zero. That is, stress levels differ for the three types of
work modalities.
Subsequently, the critical value is located in the chi-squared table for 2 degrees of
freedom and α = 0.05, determining this value to be 5.991. Finally, the calculated value
F
r
= 6.75 is compared with the critical value, proceeding to reject the null hypothesis
and accept the alternative; that is, the stress levels of at least two of the three modalities
have probability distributions that differ in location.
The same result is also obtained using Minitab software with the Friedman test option.
Descriptive statistics
Work
modality
N
Median
Sum of classifications
Mixed
1.68167
12.0
On-site
1.87333
16.5
Telework
1.52500
7.5
General
1.69333
Test
Null hypothesis
H₀: All treatment effects are zero.
Alternative
hypothesis
H₁: Not all treatment effects are zero.
Method
GL
Chi-square
p-value
Not adjusted for ties
6.75
0.034
Tight for ties
7.36
0.025
The p-value for the stress level data is less than the significance level of 0.05, therefore,
human talent management rejects the null hypothesis and accepts the alternative.
Kruskal-Wallis, Friedman and Mood nonparametric tests applied to business decision making
Espirales. Revista multidisciplinaria de investigación científica, Vol. 6, No. 42
October - December - 2022. e-ISSN 2550-6862. pp 1-20
12
Mood median test
Before proceeding with the application of the Mood median test and the presentation
of the results, the statement of the case study and the conditions of the same are
established in the following terms:
The agricultural-commercial company Bolhi Market is conducting an experiment with
four fruit varieties to determine if there are significant differences in yield. The four
varieties were randomly assigned to 28 experimental plots located in the same locality.
The yield in Kg/Ha was recorded for each of the 28 plots, obtaining the following results
from Table 3.
Table 3.
Yield by type of fruit variety - Bolhi Market commercial agricultural company-.
Variety
Yield in Kg/Ha
F1
45
N1= 8
F2
99
N2= 7
F3
42
N3= 6
F4
45
N4= 7
N=28
Source:
Own elaboration based on data from Bolhi Market, a commercial agricultural
company.
The next step consists of ordering the data of each sample and after calculating the
median, whose result is 34, the hypotheses are formulated as follows:
Ho: M1=M2=M3=M4: the median yields Kg/Ha are equal among the four varieties.
H1: at least one pair of medians is different among the four varieties.
With a significance level of 5% the critical value for 2 degrees of freedom (k-1) consulting
the chi-squared table is 7.815. Then, the assumptions for the Mood median statistic are
verified and Table x is constructed, obtaining a value of 2.15 corresponding to the Mood
statistic.
Table 4.
Calculation of the median statistic of Mood -commercial agricultural company
Bolhi Market-.
fo
faith
fo - fe
(fo-fe) ^2
((fo-fe) ^2) /fe
3.71
-1.71
2.94
0.79
4.29
1.71
2.94
0.69
3.25
0.75
0.56
0.17
3.75
-0.75
0.56
0.15
2.79
0.21
0.05
0.02
3.21
-0.21
0.05
0.01
3.25
0.75
0.56
0.17
3.75
-0.75
0.56
0.15
λ
2.15
Source:
Own elaboration
Flores Tapia Carlos Ernesto, Flores Cevallos Karla Lissette
Espirales. Revista multidisciplinaria de investigación científica, Vol. 6, No. 42
October - December - 2022. e-ISSN 2550-6862. pp 1-20
13
Finally, the critical value is compared with the calculated value of the median Mood, in
this case as the calculated value is lower than the critical value, Ho is not rejected, the
median yield Kg/Ha is equal in all fruit varieties.
Using Minitab software, with the non-parametric option, Mood's median test, the same
result is obtained as with the previous procedure, as follows:
Descriptive statistics
C
Median
Overall median
of N <=
Overall median of N >
Q3 - Q1
95%
median
CI
F1
21.25
(14.8069,
38.4505)
f2
45
55.25
(21.9356,
79.3515)
f3
42
35.00
(18, 65)
f4
37.00
(15.6667,
57.4667)
General
Test
Null hypothesis
H₀: Population medians are all the same.
Alternative
hypothesis
H₁: Population medians are not all the same.
GL
Chi-square
p-value
2.21
0.530
The p-value of 0.530 is greater than the significance level of 0.05; consequently,
the null hypothesis is not rejected. Therefore, the median yield Kg/Ha is the
same for all fruit varieties. The Ji-square value is 2.21, practically the same (2.15)
obtained by the previous statistical procedure without software support - the
slight decimal variation is due to rounding of figures.
The results obtained for the first case study, D&D Distribuciones, show that not
all the distributions of the population are equal; that is, there is a difference in
sales depending on the three types of bases or supports used for gifts,
establishing that with this information the company's management is in a
position to make the best decisions regarding the management of the import
order of bases for gifts. In the second case study, it is concluded that at least
one of the three types of work modality implemented by the company Comercial
Automotriz Romero in times of the Covid-19 health crisis, during the first six
Kruskal-Wallis, Friedman and Mood nonparametric tests applied to business decision making
Espirales. Revista multidisciplinaria de investigación científica, Vol. 6, No. 42
October - December - 2022. e-ISSN 2550-6862. pp 1-20
14
months of the year 2021, has a different effect. In the third case study, the results
of the experiment conducted by the agricultural-commercial company Bolhi
Market determine that the differences between the median yields Kg/Ha of the
four fruit varieties are not statistically significant.
Now, after reviewing some studies related to the application of the three
parametric tests contemplated in this study, the following stands out Obando et
al. (2017)who use the Friedman test to demonstrate the existence of significant
favorable differences of the physical activity program in the intervention group
in the incidence of stress, determining the decrease of stress in the
administrative staff once compared the average ranges of the initial and final test
performed. Also, Acha (2018)The Kruskal-Wallis test is applied to a set of yam
production data to prove that five plots in which the crop is distributed have
identical distributions, previously the fulfillment of the assumptions to proceed
with this test was verified.
For their part, Flores Tapia et al. (2017) investigate the effect of stem size, the
most important cultivable rose varieties and bud condition on the total sales
volume in the floricultural company "High Connection Flowers" in Latacunga
canton. They use tools of quantitative methods for decision making, that is, the
design of experiments by means of complete factorial models, requiring for their
application, first, the testing of the hypothesis of equality of variances and the
Anderson-Darling normality test. Another author, Benavides et al. (2020)
evaluated the feeding behavior of seven Holstein dairy cows in agro-ecosystems
with alder and pasture, acacia and pasture and only pasture. The test lasts six
months and takes into account the time of pasture consumption, rumination and
rest. It is shown that there is a higher pasture consumption in combination with
alder than in the other cases. While, Mandicak et al. (2021). the Kruskal-Wallis
test was used to analyze whether information and communication technology
has an impact on sustainable chain and cost management in various construction
projects in Slovakia, and the hypothesis was positively confirmed. The research
samples consist mainly of foreign companies operating in Slovakia.
However, the studies referred to above do not use a methodological procedure
applying the three types of non-parametric tests prioritized in this research -
Kruskal-Wallis, Friedman and Mood Median- in two business cases in the same
article and referring to the search for answers to business concerns, as is done
in the present research; The usefulness of parametric tests for obtaining
information for business decision-making when the distribution of the data from
which the sample comes from is not specified or it has been proven that they do
not comply with the normality assumption, leaving the way open for future
Flores Tapia Carlos Ernesto, Flores Cevallos Karla Lissette
Espirales. Revista multidisciplinaria de investigación científica, Vol. 6, No. 42
October - December - 2022. e-ISSN 2550-6862. pp 1-20
15
business research in which this type of alternative tests to parametric tests such
as Z, ANOVA or t student, which are applied when the data are based on the
assumption that the samples come from populations with normal distribution,
are applied.
On the other hand, the results of the study are consistent with the theory
explained by. Flores-Tapia & Flores-Cevallos (2017b); Levin et al. (2014); Lind
(2012); Triola (2018)., among others.
Conclusions
The article verifies that non-parametric tests such as Kruskal-Wallis, Friedman
and Mood's median can be used to obtain hypothesis testing results that allow
companies to make the best decisions. In this sense, throughout the article, the
objectives of this case study research have been achieved, that is, each of the
concerns of the management of the companies under study have been answered
based on the data and results obtained in each case, that is: it has been
determined that there is no difference in sales between the three types of gift
bases marketed by the company D&D Distribuciones, it has been observed that
the stress levels of at least two of the three work modalities implemented by the
company Comercial Automotriz Romero have probability distributions that differ
in their effect and it has been verified that the yields of four varieties of fruit
grown by the company Bolhi Market, measured in Kg/Ha, do not differ from each
other.
On the other hand, it should be noted that the application of the goodness-of-
fit tests considered in this study using conventional statistical procedures and
with the support of computer tools, such as Minitab and other specialized
programs, speed up processing times and save significant costs for organizations
that need timely information for technical decision-making, confirming their
usefulness, especially if internal conditions and the business environment
become increasingly complex, as in the case of the crisis generated by COVID-
19.
Kruskal-Wallis, Friedman and Mood nonparametric tests applied to business decision making
Espirales. Revista multidisciplinaria de investigación científica, Vol. 6, No. 42
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16
..........................................................................................................
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