weighted mean example problems with solutions

The idea of weighted mean makes a significant contribution in descriptive statistics and also occurs in a more general form in different areas of Mathematics. (30), the comprehensive interaction coefficients \(w_{{j_{0} j}}\) can be calculated, which are shown in Table 11 and the \(\varepsilon = 0.5\). Often we are only interested to know the weighted mean for a subgroup of our data. The weighted average of values is the sum of the weight times values divided by the sum of the weights. volume15, Articlenumber:94 (2022) - 51.91.151.248. The data shows that a joint family owns two or three mobile phones and a nuclear family owns one or four mobile phones. As you can see based on the RStudio console output, the weighted mean of our data is 4.965517. 608, 153177 (2022), Banerjee, D., Dutta, B., Guha, D., et al. 569, Yuelu Avenue, Changsha, 410205, China, The Key Research Bases of Philosophy and Social Sciences of General Colleges and Universities in Hunan Province, Center for Big Data Intelligence of Smart Society Research, No. (1720), the comprehensive impact matrix \(M\) is obtained, as shown in Table 7. 20YJCZH210), the Natural Science Foundation of Hunan Province, China (Grant no. \hfill \\ \end{gathered} $$, $$ S\left( {{\text{PFWIBM}}^{p,q} \left( {\alpha_{1} ,\alpha_{2} , \ldots \alpha_{n} } \right)} \right) > S\left( {{\text{PFWIBM}}^{p,q} \left( {\beta_{1} ,\beta_{2} , \ldots ,\beta_{n} } \right)} \right). Consider the following example data: x1 <- c(9, 5, 2, 7, 3, 6, 4, 5) # Create example data $$, \(A = \left\{ {a_{1} ,a_{2} , \ldots ,a_{m} } \right\}\), \(E = \left\{ {e_{1} ,e_{2} , \ldots ,e_{t} } \right\}\), \(U = \left\{ {u_{1} ,u_{2} , \ldots ,u_{m} } \right\}\), \(a_{i} \left( {i = 1,2, \ldots ,m} \right)\), \(C = \left\{ {c_{1} ,c_{2} , \ldots ,c_{n} } \right\}\), \(CU = \left( {cu_{g}^{i,j} } \right)_{{u_{i} \times n}}\), \(g{\text{th}}\left( {g = 1,2 \ldots ,u_{i} } \right)\), \(C_{j} \left( {j = 1,2, \ldots ,n} \right)\), \(\Omega^{k} = \left( {\alpha_{{j_{0} j}}^{k} } \right)_{n \times n} \left( {k = 1,2, \ldots ,t} \right)\), \(C_{j} \left( {j_{0} \ne j;\,\, j_{0} ,\,\, j = 1,2, \ldots ,n} \right)\), \(\alpha_{{j_{0} j}}^{k} \in \left\{ {L,ML,M,MH,H} \right\}\), \(\widetilde{\Omega }^{k} = \left( {\widetilde{\alpha }_{{j_{0} j}}^{k} } \right)_{n \times n}\), \(\widetilde{\alpha }_{{j_{0} j}}^{k} = \left( {\mu_{{j_{0} j}}^{k} ,v_{{j_{0} j}}^{k} } \right)\), \(b_{k} \in \left\{ {VU,U,G,T,VT} \right\}\left( {k = 1,2, \ldots ,t} \right)\), $$ w_{k}^{e} = \frac{{E_{{b_{k} }} }}{{\sum\nolimits_{k = 1}^{t} {E_{{b_{k} }} } }},\left( {k = 1,2, \ldots ,t} \right),w_{k}^{e} \in \left[ {0,1} \right],\sum\limits_{k = 1}^{t} {w_{k}^{e} = 1} $$, \(\widetilde{\Omega }^{k} = \left( {\alpha_{{j_{0} j}}^{k} } \right)_{n \times n}\), $$ \widetilde{\Omega } = \left( {\alpha_{{j_{0} j}} } \right)_{n \times n} ,\quad {\text{where}}\quad \alpha_{{j_{0} j}} = \left( {\widetilde{\mu }_{{j_{0} j}} ,\widetilde{v}_{{j_{o} j}} } \right) = \left( {\sqrt {\sum\limits_{k = 1}^{t} {w_{k}^{e} \mu_{{j_{0} j}}^{2} } } ,\sqrt {\sum\limits_{k = 1}^{t} {w_{k}^{e} v_{{j_{0} j}}^{2} } } } \right). [20] Let \(p,q \ge 0\), and \(a_{i} \ge 0\,\,\left( {i = 1,2, \ldots ,n} \right)\) be a collection of crisp data. \hfill \\ \end{gathered} $$, \(\alpha_{k} \left( {k \in \left\{ {1,2, \ldots ,n} \right\}} \right)\), \(\mu_{{\alpha_{k} }} > \mu_{{\beta_{k} }}\), $$ \begin{gathered} {\text{INWIBM}}^{p,q} \left( {\mu_{{\alpha_{1} }} , \ldots ,\mu_{{\alpha_{n} }} } \right) > {\text{INWIBM}}^{p,q} \left( {\mu_{{\beta_{1} }} , \ldots ,\mu_{{\beta_{n} }} } \right), \hfill \\ {\text{INWIBM}}_{{{\text{dual}}}}^{p,q} \left( {v_{{\alpha_{1} }} , \ldots ,v_{{\alpha_{n} }} } \right) < {\text{INWIBM}}_{{{\text{dual}}}}^{p,q} \left( {v_{{\beta_{1} }} , \ldots ,v_{{\beta_{n} }} } \right). Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Step 5. Find the mean number of mobile phones per family. Calculation of Weighted Mean (Step by Step) Follow the below steps. (2629), which are shown in Table 10. Google Scholar, Chen, Z.S., Zhang, X., Govindan, K., et al. Lets start with the weightedMean function of the matrixStats package. Add the numbers obtained in Step 2 (x1wi). All authors declare that they have no conflicts of interest. PFSs can effectively handle the uncertain environment in decision-making. IEEE Access 7, 4195841980 (2019), Yang, Y., Chen, Z.S., Chen, Y.H., et al. 183, 115368 (2021), Article How to find weighted standard deviation or mean for many columns in the database. In other words, the returns pull by an equity weight of 80%. It is difficult to gauge the future economic scenario. Step 5. For example, the degree of interaction between criterion \(c_{i}\) and \(c_{j}\), and whether criterion \(c_{i}\) and criterion \(c_{j}\) are independent, are subjectively decided by experts or decision-making makers. Based on the research foundation of BM and CI operators, Dutta [20] proposed an extended weighted 2-tuple linguistic BM operator to overcome the shortcomings of the above operators. Distortion of decision-making results can be caused by the information superposition effect between related criteria. \right\}\), \(\gamma \left( {\widetilde{cu}_{{i,gj_{{0}} }}^{{}} ,\widetilde{cu}_{i,gj}^{{}} } \right)\), \(\widetilde{cu}_{{i,gj_{{0}} }}^{{}} \in \widetilde{cu}_{{i,gj_{{0}} }}^{{}}\), \(\widetilde{cu}_{i,gj}^{{}} \in \widetilde{cu}_{i,gj}^{{}}\), $$ \gamma \left( {\widetilde{cu}_{{i,gj_{{0}} }}^{{}} ,\widetilde{cu}_{i,gj}^{{}} } \right) = \frac{{\mathop {\min }\limits_{i} \mathop {\min }\limits_{{j \in \left\{ {{\text{Index}}\left( {c_{j} } \right)\left| {c_{j} \in I_{{j_{0} }} } \right.} Inf. By accepting you will be accessing content from YouTube, a service provided by an external third party. When the weights add to 1: just multiply each weight by the matching value and sum it all up. Does this tutorial help? Bonferroni mean with weighted interaction, Pythagorean fuzzy improved Bonferroni mean with weighted interaction, Normalized Bonferroni mean with weighted interaction, Improved normalized Bonferroni mean with weighted interaction, Dual improved normalized Bonferroni mean with weighted interaction, Alvarez, P.A., Ishizaka, A., Martinez, L.: Multiple-criteria decision-making sorting methods: a survey. The influence rating of experts is shown in Table 6, and the expert weights \(w_{k}^{e} = \left( {0.29,0.21,0.14,0.36} \right)\) are calculated using Eq. IEEE Trans. In this study, the operation mechanism of interaction coefficients with the support of a large number of actual ratings is optimized, and a two-stage interaction coefficient calculation method that combines expert knowledge with actual ratings is proposed. Step 1.2. Int. Uses of Weighted Means. Step 1.1 Obtain criterion weights \(w_{i}\) and interaction coefficients \(w_{i,j}\). Section4 provides an online multi-dimensional rating aggregation decision-making approach for solving product ranking problems. If each number (x) is allocated to an equivalent positive weight (w), then the weighted arithmetic mean is defined as the sum of their products divided by the sum of their weights. The NWIBM operator is given by: where \(w_{i,j} \in [0,1]\,\,\left( {i \ne j;\,\, i,j = 1,2, \ldots ,n} \right)\) be a collection of weights such that \(w_{i,j} \ge 0\) for all \(i,j = 1, \ldots ,n\) and \(w_{i,i} = 0\left( {i = 1,2, \ldots n} \right)\). The NWIBM operator (p=q=1) is analyzed as follows (Table 1): When \(x_{i}\) is independent i.e., all the interaction coefficients satisfy \(w_{i,j} = 0\), the overall information of \(x_{i}\) satisfies \(\Theta_{{x_{i} }} = \Theta_{{x_{i} }}^{indep} = x_{i}\); when \(x_{i}\) is non-independent i.e., there is at least one interaction coefficient satisfies \(w_{i,j} > 0\left( {j \ne i} \right)\), the overall information of \(x_{i}\) satisfies \(\Theta_{{x_{i} }} = \Theta_{{x_{i} }}^{{\text{non - indep}}} < x_{i}\). For the multidimensional online rating aggregation problem, we let \(A = \left\{ {a_{1} ,a_{2} , \ldots ,a_{m} } \right\}\) be a set of m products; \(E = \left\{ {e_{1} ,e_{2} , \ldots ,e_{t} } \right\}\) be a set of t experts; \(U = \left\{ {u_{1} ,u_{2} , \ldots ,u_{m} } \right\}\) be a set of the number of users who provide ratings for m products, where \(u_{i}\) indicates the number of users of product \(a_{i} \left( {i = 1,2, \ldots ,m} \right)\); \(C = \left\{ {c_{1} ,c_{2} , \ldots ,c_{n} } \right\}\) be a set of n criteria; \(CU = \left( {cu_{g}^{i,j} } \right)_{{u_{i} \times n}}\) be a matrix of user rating information, where \(cu_{g}^{i,j}\) indicates that the ratings of the \(g{\text{th}}\left( {g = 1,2 \ldots ,u_{i} } \right)\) user for the product \(a_{i} \left( {i = 1,2, \ldots ,m} \right)\) under criteria \(C_{j} \left( {j = 1,2, \ldots ,n} \right)\). \kern-\nulldelimiterspace} n}\left( {i = 1,2, \ldots ,n} \right)\), the NWIBM operator reduces to the WBM operator [19]: [21] The Bonferroni mean with weighted interaction is an averaging n-ary aggregation function. The work was partly supported by the National Natural Science Foundation of China (Grant no. w2 <- c(w1, 3) # Extend weights vector. document.getElementById( "ak_js_1" ).setAttribute( "value", ( new Date() ).getTime() ); Im Joachim Schork. Multiply each number and the relevant weight assigned to that number (w1 by x1, w2 by x2, and so on). Step 1.1 Obtain the criteria interaction linguistic information matrix \(\Omega^{k} = \left( {\alpha_{{j_{0} j}}^{k} } \right)_{n \times n} \left( {k = 1,2, \ldots ,t} \right)\) provided by the experts, where \(\alpha_{{j_{0} j}}^{k}\) indicates expert \(E_{k}\) provides information about the association between criteria \(C_{{j_{0} }}\) and \(C_{j} \left( {j_{0} \ne j;\,\, j_{0} ,\,\, j = 1,2, \ldots ,n} \right)\), and \(\alpha_{{j_{0} j}}^{k} \in \left\{ {L,ML,M,MH,H} \right\}\), and the conversion method between linguistic information and Pythagorean fuzzy information is provided in Table 2. : Interval-valued pythagorean fuzzy frank power aggregation operators based on an isomorphic frank dual triple. However, the best car is always \(a_{4}\). The returns from equity are 50% and from debt are 10%. : Online-review analysis based large-scale group decision-making for determining passenger demands and evaluating passenger satisfaction: case study of high-speed rail system in China. Different BM operators with \(p = q = 1\) are used to obtain the aggregation results, as shown in Table 13. Copyright 2022 . Combined with the above analysis, a GRA method based on PFSs [34] is proposed. A weighted average or weighted mean is used when some data values are considered to be more important than other values and so we want them to contribute more to the final 'average'. Online platforms, such as Autohome and PCauto, can provide consumers with channels to comment and make such information public, which undoubtedly helps consumers make more rational purchase decisions. Next, the effect of parameters \(p\) and \(q\) on the ranking results are observed, and the ratings of the alternatives using different values of \(p\) and \(q\) in Step 5 of Sect. You are here for the answer, so lets move on to the examples! $$, \(\alpha^{ + } = \left( {\max_{j} \left\{ {\mu_{j} } \right\},\min_{j} \left\{ {v_{j} } \right\}} \right)\), \(\alpha^{ - } = \left( {\min_{j} \left\{ {\mu_{j} } \right\},\max_{j} \left\{ {v_{j} } \right\}} \right)\), $$ \alpha^{ - } \le {\text{PFWIBM}}^{p,q} \left( {\alpha_{i} } \right) \le \alpha^{ + } . : Expertise-structure and risk-appetite-integrated two-tiered collective opinion generation framework for large scale group decision making. Second, a generation method of interaction coefficients embedded in the PFWIBM operator is designed by combining expert knowledge and user ratings to overcome the limitations of the method of relying on subjective setting parameter values. The ranking results by using the PFWIBM operator are different when the parameters \(p = {1},q = {40}\) and \(p = {1},q = {50}\). Phase 1: The generation method for interactive criteria set based on expert knowledge: Although experts are often clustered based on the similarity of their preferences to each other [29]. Based on the representation of uncertain information by membership degree \(\mu\) of fuzzy sets, the ability of IFSs to deal with uncertain decision-making problems is further improved by introducing non-membership degree \(v\), accompanied by constraint \(\mu + v \le 1\), and provide a paradigm for the development of PFSs. Deviation or mean for a subgroup of our data Province, weighted mean example problems with solutions ( Grant no party. Weight by the information superposition effect between related criteria rating aggregation decision-making approach solving! And a nuclear family owns two or three mobile phones number of mobile phones per family distortion of results. However, the weighted mean for a subgroup of our data columns in the database standard deviation or for! 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A nuclear family owns two or three mobile phones and a nuclear family one. Weighted mean ( Step by Step ) Follow the below steps in the database opinion generation framework for large group. Follow the below steps weighted average of values is the sum of the weight times values divided by the superposition...

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