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The use of a multi-variable chart is beneficial for analyzing variations and relationships between multiple variables in a process. It serves to highlight how different factors interact and influence each other, which is particularly useful for understanding within-sample variation, sample-to-sample variation within batches, and batch-to-batch variation.

When assessing within-sample variation, multi-variable charts can help visualize how different measurements within the same sample behave, which is essential for quality control processes. Similarly, when looking at sample-to-sample variation within batches, these charts effectively represent variations across samples, allowing for better monitoring of consistency and quality in production.

Batch-to-batch variation is another context where multi-variable charts excel, as they provide insights into how different batches compare to one another, essential for maintaining quality standards across production runs.

However, the analysis of weighted average-to-standard weights does not align with the primary use of multi-variable charts. This measurement typically involves statistical analysis that focuses on specific weighted averages rather than how multiple variables interrelate. Therefore, this context is less amenable to the insights that multi-variable charts are designed to provide. This distinction clarifies why this option stands out as an exception to the utility of multi-variable charts.