As businesses strive to penetrate the competitive market landscape, the need for cutting-edge operational excellence strategies is paramount. Six Sigma, a data-driven methodology for eliminating defects in any process, has emerged as a powerful tool for business performance improvement. Selecting the right Six Sigma consultant, however, can be a daunting task. In this context, it is essential to engage in insightful dialogue with prospective consultants, asking the right questions that will guide you towards making an informed decision.

Understanding the relevance of Six Sigma in the current business landscape requires a holistic grasp of Lean Six Sigma, an advanced process improvement methodology that amalgamates the efficiency and speed of Lean with the effectiveness and robustness of Six Sigma. Lean Six Sigma emphasizes the elimination of waste, thereby providing a robust framework for improving business performance. The role of a Six Sigma consultant, therefore, is to guide an organization through the implementation of this methodology, ensuring process improvement and minimizing variability in business operations.

Firstly, gauge the prospective consultant's understanding of the Six Sigma methodology. Inquire about their approach to Design for Six Sigma (DFSS), a business-process management method related to traditional Six Sigma. DFSS aims at the design or redesign of different processes used in business, manufacturing, and transactional operations. Understanding the consultant’s approach to DFSS and their strategy for integrating it into your business processes can provide deep insights into their expertise and experience.

Delve deeper into their track record by asking about the projects they've overseen or implemented. Ask for tangible success stories or case studies where they've played a pivotal role. Furthermore, ask them about the specific Six Sigma training they've received or facilitated, and the level of certification they possess. Six Sigma is divided into various belt categories from White Belt (Beginner Level) to Master Black Belt (Expert Level). The higher the belt level, the more comprehensive the consultant's understanding and application of Six Sigma is likely to be.

A good consultant would also be able to strategically integrate Six Sigma with other business performance improvement methodologies such as Agile or Scrum. Therefore, it is pertinent to ask about their experience with, and approach to, integrating these methodologies.

Next, it's important to discuss how the consultant plans to extend Six Sigma principles to integral sectors of your business such as HR, Finance, or Supply Chain Management. This multidimensional application of the methodology would ensure the consultant’s ability to provide a holistic process improvement framework for your organization.

It's equally important to understand their approach towards statistical analysis, a cornerstone of Six Sigma. Ask questions that delve into their understanding of statistical tests, hypothesis testing, and data-driven decision making. Here, an understanding of statistical concepts like Normal Distribution, Standard Deviation, and Control Charts becomes crucial.

Lastly, a fruitful engagement with a Six Sigma consultant transcends the theoretical and statistical aspects of Six Sigma. It is also about the consultant's ability to align with your business culture, values and work alongside your team effectively. Therefore, it would be wise to ask about their previous experiences in engaging with the team at different levels of hierarchy and their ability to drive change within the organization.

In conclusion, choosing the right Six Sigma consultant requires careful consideration, a comprehensive understanding of Six Sigma principles, and a clear vision for your business. The aforementioned questions provide a robust framework for making this strategic decision. Remember, the right Six Sigma consultant can be a game-changer in your quest for operational excellence and business performance improvement.