Journal of Business and entrepreneurial
July - September Vol. 7 - 3 - 2023
http://journalbusinesses.com/index.php/revista
e-ISSN: 2576-0971
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Receipt: 03 July 2022
Approval: 09 April 2023
Page 27-32
Quality impact, with the use of more than three
sigmas in the statistical control of processes by
variables
Afectación de la calidad, con el uso de más de tres sigmas
en el control estadístico de procesos por variables
José Luis Hidalgo Torres
*
ABSTRACT
The main objective of this study is to show how the
use of more than ± 3σ in the formulas to determine
the limits of the variables in the statistical process
control charts affects the quality of the manufactured
products and, consequently, the final consumer. In
the research process, the induction and description
methods were used to determine the required values
of the parameters, constants and others used, which
allow showing the proposed objective. The results in
this study indicate that the use of more than ± 3σ in
the statistical process control formulas by variables
can have a negative impact on the quality of the final
product. This is because the use of wider limits may
allow more variability in the process to go
undetected, which may result in more defective or
out-of-specification products. Consequently, this can
negatively affect customer satisfaction and company
reputation. Therefore, it is important to take these
findings into account when setting control limits on
statistical process control charts.
Keywords: Show, limits, quality, methods, impact,
variability.
RESUMEN
El objetivo principal de este estudio es mostrar cómo
el uso de más de ± 3σ en las fórmulas para
determinar los límites de las variables en las gráficas
* MsC Universidad César Vallejo, Lima, Perú. jlhidalgot@hotmail.com
https://orcid.org/0000-0002-4671-3116
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28
de control estadístico de procesos afecta la calidad
de los productos fabricados y, en consecuencia, al
consumidor final. En el proceso de investigación se
utilizaron los métodos de inducción y de descripción,
para ir determinando los valores requeridos de los
parámetros, constantes y otros utilizados, que
permiten mostrar el objetivo propuesto. Los
resultados en este estudio indican que el uso de más
de ± 3σ en las fórmulas de control estadístico de
procesos por variables puede tener un impacto
negativo en la calidad del producto final. Esto se debe
a que el uso de límites más amplios puede permitir
que más variabilidad en el proceso pase
desapercibida, lo que puede resultar en una mayor
cantidad de productos defectuosos o fuera de
especificación. En consecuencia, esto puede afectar
negativamente la satisfacción del cliente y la
reputación de la empresa. Por lo tanto, es
importante tener en cuenta estos hallazgos al
establecer los límites de control en las gráficas de
control estadístico de procesos.
Palabras clave: Mostrar, límites, calidad, métodos,
impacto, variabilidad.
INTRODUCTION
Customer satisfaction and a company's reputation depend to a large extent on the
quality of the products it offers. To ensure this quality, companies use a variety of tools
and techniques, including statistical process control. This tool allows companies to
monitor and improve the quality of their products through the use of statistical process
control charts. However, the effectiveness of these charts depends largely on the choice
of control limits. In this study, we examine how the use of more than ± 3σ in the formulas
for determining the limits of variables in statistical process control charts affects the
quality of the manufactured products and, consequently, the final consumer. Our findings
indicate that the use of more than ± 3σ in statistical process control formulas by variables
can have a negative impact on the quality of the final product. This is because the use of
wider limits may allow more variability in the process to go undetected, which may result
in more defective or out-of-specification products.
This can have a negative impact on customer satisfaction and company reputation.
Therefore, our findings are important for companies seeking to improve the quality of
their products through the use of statistical process control. This study provides valuable
e-ISSN: 2576-0971. July - September Vol. 7 - 3 - 2023 . http://journalbusinesses.com/index.php/revista
29
information to help companies make informed decisions when setting control limits on
statistical process control charts.
MATERIALS AND METHODS
Increments of sigmas, in the formulas for the calculation of the limits for the control
charts by variables, the following research methods have been used: This method was
used to go calculating, interpreting and concluding the results from the resolution of
specific cases of examples proposed in several books of statistics that, in some of their
topics, are related to the analysis of the statistical control of processes by variables.
Descriptive method: It was used to describe the processes and procedures applied to
solve the problems and that allowed obtaining the results that will be shown later.
Increments of sigmas, in the formulas for the calculation of the limits for the control
charts by variables, the following research methods have been used:
Inductive method: This method was used to calculate, interpret and conclude the
results from the resolution of specific cases of examples proposed in several statistics
books that, in some of their topics, are related to the analysis of the statistical control
of processes by variables. Descriptive method: It was used to describe the processes
and procedures applied to solve the problems and that allowed obtaining the results that
will be shown later.
RESULTS
Before starting to present the results obtained, it should be pointed out that the
problems that were used were not solved according to the requirements requested in
each one of them, but rather by applying the corresponding formulas for the calculation
of parameters that are strictly related to the control charts by variables; that is, the data
provided by them were used to be used for the aforementioned purposes of the
research.
In order to show the results of the research, several stages have been established that
will indicate the process and procedures applied:
MENTION OF THE PROBLEMS TO APPLY THE STATISTICAL
CONTROL BY VARIABLES.
Taken from (Gutierrez_Pulido, 2009) It is desired that the strength of an article be at
least 300 psi. To verify that such a quality characteristic is met, small periodic inspections
are made and the data are recorded on an X - R chart. The subgroup size used is three
items, which are taken consecutively every two hours. The data for the last 30
subgroups are shown below. Answer:
a) Given that the mean of averages is 320.73, does the process meet the lower
specification (EI = 300)? Explain.
b) Calculate the limits of the X - R chart and interpret them.
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c) Obtain the charts and interpret them (out points, trends, cycles, high variability,
etc.).
d) Give a preliminary estimate of the instability index, St. Louis.
e) Does the process show a reasonable stability or state of statistical control?
f) Make an analysis of the capacity of the process:
i) Estimate the standard deviation of the process.
ii) Calculate the real limits of the process and interpret them.
iii) Obtain a histogram for the individual data.
iv) Calculate the Cpi index and interpret it.
v) Using Table 5.2 (Chapter 5), estimate the percentage of product that does not
meet the lower specification.
vi) Is the process capable of meeting specifications?
g) If you proceeded properly, in the previous paragraph you will find that the process
capability is bad, but how do you explain this if none of the data in table 7.5
is less than 310.0? Please argue your answer.
h) To which aspect would you recommend focusing improvement efforts: capacity
or stability? Argument.
Taken from (Kelmansky, 2009) Let us continue with volumetric capacity data
(section 21.1.1.1.1.) for a process with mean 47 dm3 and standard deviation of 0.666
dm3. We use this time a control chart X, with n=5. That is, we average the volumetric
capacity of 5 carafes every hour, for 16 hours.
Taken from (Ruiz & Rojas, 2006) The gauge of the sinkers is a key characteristic for
their good performance. The following table shows measurements of 20 samples of size
5. Construct the control charts X - R, X - S and X - S*.
Taken from (Gutiérrez_Pulido, 2009) In the manufacture of optical discs a
machine metallizes the disc. To ensure uniformity of the metal on the disc, the density
should be 1.93, with a tolerance of ±0.12. Table 7.7 shows the data obtained for an
initial study with subgroup size of 5.
a) Calculate the control limits for the X-R charts and interpret them.
b) Plot the X-R chart and interpret it.
c) Does the process have an acceptable stability? Argument.
d) Conduct a capacity study for this purpose:
i) Estimate the standard deviation of the process.
ii) Calculate the real limits of the process and interpret them.
iii) Obtain a histogram for the individual data, insert specifications and interpret in
detail.
iv) Calculate the capacity indexes and interpret them.
v) Using Table 5.2 (Chapter 5), estimate the percentage of product that does not
meet specifications.
vi) Is the process capable of meeting specifications?
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e) On which aspect would you recommend focusing improvement efforts: capacity
or stability?
The results of this study have important implications for companies seeking to improve
the quality of their products through the use of statistical process control. Our findings
indicate that the use of more than ± 3σ in the formulas for determining the limits of
variables in statistical process control charts can have a negative impact on the quality
of the final product. This is because the use of wider limits may allow more variability in
the process to go undetected, which can result in more defective or out-of-specification
products.
The findings have important implications for companies seeking to improve the quality
of their products through the use of statistical process control. They suggest that it is
important to take care in setting control limits on statistical process control charts and
to consider the impact that the use of wider limits may have on final product quality.
However, it is also important to keep in mind the limitations of this study and the need
for future research to further explore these issues.
In summary, this study provides valuable information on how the use of more than ± 3σ
in the formulas for determining the limits of variables in statistical process control charts
can affect the quality of the final product. The findings suggest that it is important to take
care when setting control limits and to consider the impact that the use of wider limits
may have on final product quality.
Future research could further explore these issues and provide additional information
to help companies make informed decisions on how to improve the quality of their
products through the use of statistical process control by variables.
CONCLUSIONS
Only by assuming a ± 1σ, it is possible to determine the existence of F.C.E. points, with
more sigmas the existence of points that are F.C.E. is almost null; points that represent
all the elements of each one of the samples, which can be that all or some of the elements
of the sample are F.C.E., causing that the whole sample is F.C.E.
Effects of increasing sigmas in the use of control charts by variables:
By calculating the limits varying from ± 1σ, ..., ± 6σ, it is determined and visualized
graphically that the values of the limits expand towards the lateral extremes in the
manner of an arithmetic progression, with respect to ± 1σ.
The expansion of the limits causes more elements to be accepted as valid.
Regarding the process capability Cp; it has that with more ± 1σ and in the best of the
cases with ± 2σ it is had that the processes are ≥ 1,33; parameter averagely accepted
worldwide with purposes that the results of the manufacturing processes, deliver quality
products for the consumer. With ≥ ± 2σ the process capability indexes show a total
deficiency.
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