Recent advances in large language models (LLMs) have enabled automatic generation of chain-of-thought (CoT) reasoning, leading to strong performance on tasks such as math and code. However, when reasoning steps reflect social stereotypes (e.g., those related to gender, race or age), they can reinforce harmful associations and lead to misleading conclusions. We present the first systematic evaluation of social bias within LLM-generated reasoning, focusing on reasoning language models (e.g., DeepSeek-R1, OpenAI o1) that natively produce reasoning chains as part of their answers. Using the BBQ dataset, we analyze both prediction accuracy and reasoning bias across a broad spectrum of models, including instruction-tuned and CoT-augmented variants of DeepSeek-R1 (8B/32B), ChatGPT, and other open-source LLMs. We quantify how biased reasoning steps correlate with incorrect predictions and often lead to stereotype expression. To mitigate reasoning-induced bias, we propose Answer Distribution as Bias Proxy (ADBP), a lightweight mitigation method that detects bias by tracking how model predictions change across incremental reasoning steps. ADBP outperforms Stereotype-free Reasoning Pattern (SfRP) baseline in most cases, mitigating bias and improving the accuracy of LLM outputs. Evaluation and mitigation code is available at https://github.com/elviswxy/LLM_reasoning_bias.