# Uncertainty-Based Trimmed Coefficient of Variation with Application

Authors: Ali Hussein Al-Marshadi; Muhammad Aslam; Alharbey Abdullah
Journal: Journal of Mathematics
 (2021)

---

## Abstract

In this paper, the neutrosophic trimmed average, neutrosophic trimmed standard deviation, and neutrosophic trimmed coefficient of variation (NTCV) are introduced. The application of the proposed neutrosophic trimmed descriptive statistics is given with the help of measurement data. The comparisons of the proposed NTCV are compared with the existing neutrosophic coefficient of variation (NCV). From the comparisons, it is concluded that the proposed NTCV is more efficient than NCV in terms of consistency and measures of indeterminacy. Based on the study, it is recommended to apply the proposed NTCV in the industry when there is a need to make decisions on the basis of measurement data.


---

## Body

## 1. Introduction

The statistical methods and techniques are playing an important role in decision-making in all fields of social sciences, medical sciences, and industries. Among them, the average and coefficient of variation (CV) have been widely used in decision-making in the presence of more than one characteristic. The average is used to select the variable of interest which is better on average, and CV is applied to check the consistency of that characteristic. For example, industrials are interested to make the decision about the product on the basis of measurements recorded by different operators. To make the decision, a more consistent operator is selected using the CV. It is important to note that better on the average does not mean more consistent than the others. In other words, the CV tells about the variation in the data. Less the variation means better the data for the decision-making. A CV of less than 10% is considered very good and larger than 30% is not acceptable. The variation in the data can be reduced by omitting the outliers from the data. The outlier of the data can be removed from the data using the idea of the trimmed mean. In this method, a preselected percentage of the values are removed from the starting and ending of the ordered data. The use of the trimmed average is helpful to reduce the variation by removing extreme observations from the data. Wu and Zuo [1] proposed trimmed measures using the scale deviation method. Alkhazaleh and Razali [2] worked on estimation using the trimmed average. Yusof et al. [3] discussed various trimmed methods. Wang et al. [4] introduced the mean approach in medical science. Lugosi and Mendelson [5] introduced heavy-tailed distribution. More information about the application of trimmed measures can be seen in [6–10].Uncertainty is defined as the lack of sureness about measurement, parameters, and observations. For example, measuring the water level, measuring rock joint roughness, and measuring the lifetime of a virus is done under uncertain environment. According to [11], “different sources of uncertainty may affect the quality of measurement results: environment, measurement setup, measuring instrument, appraiser, measuring object, measuring procedure, physical constants, the definition of the characteristic, software, and calculations.” In case, when uncertainty is presented in the data, the fuzzy logic can be applied for the analysis of the data. The trimmed average under fuzzy logic can be applied to remove the extreme observations from the fuzzy data. The authors of [12–15] discussed the applications of trimmed average using fuzzy logic. More applications can be seen in [16, 17].Fuzzy logic is based on membership and nonmembership values. Neutrosophic logic is a general form of logic that deals with three measures, namely, the measure of truth (membership), the measure of falsehood (nonmembership), and the measure of indeterminacy. The fuzzy logic is a special case of neutrosophic logic, see [18]. The information about the measure of indeterminacy can be obtained from the neutrosophic logic. The neutrosophic logic has been applied in a variety of fields, see [19–21]. Using the idea of neutrosophic logic, neutrosophic statistics which is the extension of classical statistics was introduced by [22]. The methods to analyze the neutrosophic data were discussed in [23, 24]. Aslam [25] introduced the neutrosophic coefficient of variation. Aslam and Bantan [26] introduced a measurement system using neutrosophic statistics. More information on dealing neutrosophic numbers can be seen in [27–29].As mentioned earlier, the trimmed average is a useful technique to reduce the variation by removing the extreme observations from the data. In this method, a small percentage of values are removed to minimize the variation in the data. The trimmed average helps to remove the outliers from the data before calculating the traditional average. The coefficient of variation under neutrosophic statistics is known as the neutrosophic coefficient of variation (NCV) and the coefficient of variation using the trimmed average under neutrosophic statistics is known as the neutrosophic trimmed standard deviation (NTSD). The coefficient of variation using trimmed average under classical is called the trimmed coefficient of variation. Aslam [25] introduced NCV. By exploring the literature and best of our knowledge, there is no work on neutrosophic trimmed average, neutrosophic trimmed standard deviation (NTSD), and NTCV. In this paper, the introduction of average, standard deviation, and coefficient of variation using the neutrosophic statistics will be given. In addition, we will give the application of the proposed NTCV using the measurement data from the industry. It is expected that the proposed NTCV will be helpful to increase the consistency as compared to NCV. Furthermore, the proposed NTCV will be helpful to minimize the measure of indeterminacy.


## 2. Methodology

LetINϵIL,IU be an indeterminacy interval associated with neutrosophic random numberXNi=XLi+XUiINi=1,2,3,…,nN of size nNϵnL,nU, where XL, nL and XU, nU are the lower and upper values, respectively. The basic operations such as multiplication, division, and inverse of these neutrosophic numbers can be seen in [23, 24]. Suppose a data analyst has a neutrosophic sample nNϵnL,nU and he is interested to find α% neutrosophic trimmed average (NTA). Suppose that XL and XU denote the lower and upper values of an indeterminate interval of measurement parts. The trimmed observation is denoted bykN=nNα, where α is the percentage of values trimmed from the data. Suppose that RN=nN−nNα shows the difference between the total observation and trimmed observations. The following process can be adopted to calculate α% neutrosophic trimmed average.Step 1: arrange XL and XU observations in the ascending orderStep 2: trim kNϵkL,kU observations at both ends of arranged data, where kN=nNαStep 3: compute NTA of remaining observations, RN=nN−nNα, RNϵRL,RUThe neutrosophic trimmed average, sayT¯L of values XL, is calculated as(1)T¯L=1RL∑i=kL+1nL−kLXLi,where index of summation runs from the lower value of kN to the lower value of RN.The neutrosophic trimmed average, sayT¯U of values XU, is calculated as(2)T¯U=1RU∑i=kU+1nU−kUXUi,where index of summation runs from the upper value of kN to the upper value of RN.The neutrosophic trimmed average, sayX¯Ni, using equations (1) and (2), is calculated by(3)X¯Ni=1RL∑i=kL+1nL−kLXLi+1RU∑i=kU+1nU−kUXUiIX¯N;IX¯NϵIX¯L,IX¯U.The neutrosophic trimmed sum of the square of observations fromX¯Ni is calculated by(4)∑i=kU+1nU−kUXNi−X¯Ni2=∑i=kU+1nU−kUminXLi+XUiILX¯L+X¯UIL,XLi+XUiILX¯L+X¯UIU,XLi+XUiIUX¯L+X¯UIL,XLi+XUiIUX¯L+X¯UIUmaxXLi+XUiILX¯L+X¯UIL,XLi+XUiILX¯L+X¯UIU,XLi+XUiIUX¯L+X¯UIL,XLi+XUiIUX¯L+X¯UIU,INϵIL,IU,where X¯L=1/nL∑i=1nLXLi and X¯U=1/nU∑i=1nUXUiThe neutrosophic trimmed standard deviation (NTSD), saysNT, is given by(5)sNT=1RN∑i=kN+1nN−kNXNi−X¯Ni2.The neutrosophic trimmed coefficient of variation (NTCV) tells about the consistency and is computed by(6)CVNT=sNTX¯Ni×100.


## 3. Application Using Measurement Data

Now, we present the case study from the automotive industry in Kachiran Company in Asia, see [30], for more details. The company is a manufacturing housing clutch used as automobile parts. To make a better decision about the performance of these parts, the company needs the measurements of these parts. The decision about the performance depends on the consistency of the operators. The operators working in the company have the instruction to measure the length of the parts. The measurements of these parts cannot be recorded completely; therefore, the measurement observations are neutrosophic. The measurements in mm by three operators are shown in Table 1.Table 1
The real example data.
Part no.Operators1231[62.14, 62.26][62.09, 62.21][62.09, 62.21]2[62.13, 62.25][62.13, 62.25][62.13, 62.25]3[62.05, 62.17][62.05, 62.17][62.04, 62.16]4[62.11, 62.23][62.11, 62.23][62.11, 62.23]5[62.19, 62.31][62.19, 62.31][62.19, 62.31]6[62.06, 62.18][62.06, 62.18][62.06, 62.18]7[62.07, 62.19][62.08, 62.20][62.07, 62.19]8[62.14, 62.26][62.14, 62.26][62.14, 62.26]9[62.24, 62.36][62.24, 62.36][62.23, 62.35]10[62.22, 62.34][62.22, 62.34][62.22, 62.34]From Table1, it can be seen that the use of classical statistics may mislead the managers in decision-making. Therefore, the consistency of the operators in measuring will be discussed with the help of the proposed methods. Let α=1% and nNϵ10,10. The application of the proposed method to find NTCV for operator 1 is stated as follows (Tables 2 and 3).Step 1:
arrangeXL and XU observations of operator 1 in the ascending order as shown in Table 2.Table 2
Observations of operator 1.
XL62.0562.0662.0762.1162.1362.1462.1462.1962.2262.24XU62.1762.1862.1962.2362.2562.2662.2662.3162.3462.36Step 2:
trimkN=1 observations at both ends of arranged data, wherekN=10×0.1. The remaining data are given in Table 3.Table 3
Trimmed observations of operator 1.
XL62.0662.0762.1162.1362.1462.1462.1962.22XU62.1862.1962.2362.2562.2662.2662.3162.34Step 3:
compute NTA of remaining observations,RN=10−2=8.The neutrosophic trimmed average of valuesXL is calculated as(7)T¯L=18∑i=28XLi=62.12.The neutrosophic trimmed average of valuesXU is calculated as(8)T¯U=18∑i=28XUi=62.24.The neutrosophic trimmed average is defined by(9)X¯Ni=62.12+62.24IN;INϵ0,0.0019.The neutrosophic trimmed sum of the square is calculated by(10)∑i=28XNi−X¯Ni2=0.0264,0.1059.The neutrosophic trimmed standard deviation (NTSD), saysNT, is given by(11)sNT=18∑i=28XNi−X¯Ni2=0.0575,0.1150.The neutrosophic trimmed coefficient of variation (NTCV) tells about the consistency and computed by(12)CVNT=0.0575,0.115062.12,62.24×100=0.0924,0.1851.The values of NTCV for other operators can be calculated in the same way as for operator 1. The neutrosophic descriptive statistics for three operators are shown in Table4. From the first column of Table 4, it can be seen that, on average in measurement, operator 2 is better than other operators. We also note that the indeterminacy interval of operator 3 is smaller than other operators. Therefore, operator 3 is more consistent in measuring the length of housing clutch parts. Based on this study, it is concluded that the management can make the decision about the product on the basis of measurement recorded by operator 3.Table 4
Neutrosophic descriptive statistics.
OperatorsX¯NisNTCVNTRange162.12,62.240.0575,0.11500.0924,0.18510.0927262.12,62.260.0514,0.10780.0826, 0.17310.0905362.12,62.240.0526,0.10530.0845,0.16950.085


## 4. Comparative Study

Aslam [25] introduced the neutrosophic coefficient of variation (NCV) under the neutrosophic statistics. In this section, we will discuss the advantages of the proposed NTCV with NCV. Note here that the proposed NTCV reduces to the existing NCV when no observation is trimmed from the data (α=0%). To show the efficiency of the proposed NTCV over NCV, we will consider the same descriptive neutrosophic statistics of measurement data are presented in the last section. The NCV and NTCV for three operators are shown in Table 5. From column four of Table 5, it can be noted that the values of NTCV from the proposed method are smaller than the existing NCV which indicates that the proposed NTCV is more consistent in measurement as compared to NCV. For example, for the measurement data given by operator 3, the indeterminate interval is from 0.0826 to 0.1695. On the contrary, this interval from the existing NCV is from 0.1029 to 0.2045. From Table 5, it can also be noted that the use of the proposed method increases the efficiency of the values of the coefficient of variation. From this study, it is concluded that the proposed NTCV is smaller than the existing NCV. We conclude that the proposed method is helpful to increase the consistency of measurement. The neutrosophic forms of NCV and NTCV along with the measures of indeterminacy are placed in Table 6. The first values in neutrosophic form denote the determined values under classical statistics and the second part is indeterminate parts. For example, in neutrosophic form 0.0826 + 0.1731 IN,INϵ0,0.5228, the value 0.0826 presents the value of the coefficient of variation (CV) for classical statistics. The value 0.1731IN,INϵ0,0.5228, is the indeterminate part with the measure of indeterminacy0.1731−0.0826/0.1731=0.5228. We note that the measure of indeterminacy from the existing method is given by [25] is 0.82. From this study, it is concluded that the proposed method is helpful to minimize the measure of indeterminacy. We also compared the results of the proposed study with interval statistics. The interval statistics used intervals in order to capture the data inside the intervals. Therefore, the interval statistics tells the values of NTCV from 0.0826 to 0.173 without giving any information about the measure of indeterminacy. Therefore, it is concluded that the proposed NTCV is more efficient in measure of indeterminacy than the existing CV proposed by [25] and interval statistics.Table 5
The comparison in NCV and NTCV.
OperatorsNCVStatusNTCVStatus1[0.1003, 0.2011]Good0.0924,0.1851Very good2[0.1013, 0.5628]Not acceptable0.0826, 0.1731Very good3[0.1020, 0.2045]Good0.0845,0.1695Very goodTable 6
The comparison in NCV and NTCV.
OperatorsNeutrosophic form of NCVNeutrosophic form of NTCV10.1003 + 0.2011IN,INϵ0,0.50120.0924+0.1851IN,INϵ0,0.520.1013 + 0.5628IN,INϵ0,0.820.0826 + 0.1731IN,INϵ0,0.522830.1020 + 0.2045IN,INϵ0,0.50120.0845+0.1695IN,INϵ0,0.5


## 5. Concluding Remarks

In this paper, the neutrosophic trimmed average, neutrosophic trimmed standard deviation, and neutrosophic trimmed coefficient of variation (NTCV) were introduced. The application of the proposed neutrosophic trimmed descriptive statistics was given with the help of measurement data. The comparisons of the proposed NTCV are compared with the existing neutrosophic coefficient of variation (NCV). From this study, it is can be seen that the proposed NTCV is more efficient than NCV in terms of measures of indeterminacy. In addition, it can be seen that the proposed NTCV reduces the variation in the measurement data. The proposed NTCV can be applied for the decision-making in the industry when the data are obtained from the measurement having the neutrosophy. The other trimmed statistical methods under neutrosophic statistics can be considered as future research.



---

*Source: 5511904-2021-10-04.xml*