Large Language Models (LLMs) have significantly advanced natural language processing applications, yet their widespread use raises concerns regarding inherent biases that may reduce utility or harm for particular social groups. Despite the advancement in addressing LLM bias, existing research has two major limitations. First, existing LLM bias evaluation focuses on the U.S. cultural context, making it challenging to reveal stereotypical biases of LLMs toward other cultures, leading to unfair development and use of LLMs. Second, current bias evaluation often assumes models are familiar with the target social groups. When LLMs encounter words beyond their knowledge boundaries that are unfamiliar in their training data, they produce irrelevant results in the local context due to hallucinations and overconfidence, which are not necessarily indicative of inherent bias. This research addresses these limitations with a Local Integrated Bias Recognition and Assessment Framework (LIBRA) for measuring bias using datasets sourced from local corpora without crowdsourcing. Implementing this framework, we develop a dataset comprising over 360,000 test cases in the New Zealand context. Furthermore, we propose the Enhanced Idealized CAT Score (EiCAT), integrating the iCAT score with a beyond knowledge boundary score (bbs) and a distribution divergence-based bias measurement to tackle the challenge of LLMs encountering words beyond knowledge boundaries. Our results show that the BERT family, GPT-2, and Llama-3 models seldom understand local words in different contexts. While Llama-3 exhibits larger bias, it responds better to different cultural contexts. The code and dataset are available at: https://github.com/ipangbo/LIBRA.