A Penalty Goes a Long Way: Measuring Lexical Diversity in Synthetic Texts Under Prompt-Influenced Length Variations

Synthetic text generated by Large Language Models (LLMs) is increasingly used for further training and improvement of LLMs. Diversity is crucial for the effectiveness of synthetic data, and researchers rely on prompt engineering to improve diversity. However, the impact of prompt variations on response text length, and, more importantly, the consequential effect on lexical diversity measurements, remain underexplored. In this work, we propose Penalty-Adjusted Type-Token Ratio (PATTR), a diversity metric robust to length variations. We generate a large synthetic corpus of over 20M words using seven models from the LLaMA, OLMo, and Phi families, focusing on a creative writing task of video script generation, where diversity is crucial. We evaluate per-response lexical diversity using PATTR and compare it against existing metrics of Moving-Average TTR (MATTR) and Compression Ratio (CR). Our analysis highlights how text length variations introduce biases favoring shorter responses. Unlike existing metrics, PATTR explicitly considers the task-specific target response length ($L_T$) to effectively mitigate length biases. We further demonstrate the utility of PATTR in filtering the top-10/100/1,000 most lexically diverse responses, showing that it consistently outperforms MATTR and CR by yielding on par or better diversity with high adherence to $L_T$.