Question
Which sect did Ramakrishna’s family belong to?
Answer
From the preface of the book "A short biography of Sri Ramakrishna":
This is in response to a great demand for a shorter biography of Sri Ramakrishna whose message is spreading far and wide with ever increasing rapidity. The matter is taken mostly from the Life of Sri Ramakrishna, published by us, though other available books have been consulted and the plan is quite different.
So, this book is a condensed form of his biography. From it's chapter 2:
Towards the middle of the eighteenth century there lived in the village of Derepore in the District of Hooghly, Bengal, a Brahmin family of which Manik Ram Chattopadhyaya was the head. He was a pious and kind-hearted man. With fifty acres of land at his disposal, he was able to meet the needs of his family as also to lend a helping hand to the poor and distressed of the village in times of calamity. In about 1775 he was blessed with a son who was named Khudiram. Two other sons and a daughter were also subsequently born to him. After the death of Manik Ram, the entire charge of the family devolved on his eldest son, Khudiram, who, trained in the family traditions of an orthodox house, was eminently fitted to attend to the manifold religious and secular duties of the household. Both Khudiram and his wife, Srimati Chandramani, were exceptionally devoted to their tutelary deity Sri Ramachandra, and soon earned the love, respect, and admiration of the villagers for their charity, truthfulness, and kindness.
Hence, as i have said in the comments, Sri Rama was the Kula Devata of the family that Sri Ramakrishna was born into. So, he was a Vaishnava by birth.
Anymore information, like which sub-sect of Vaishnavism did the family belong to, is not available, as far as i know. Probably they did not subscribe to any such sub-sects.
(Actually, even today, i hardly find any Bengali who says that "I belong to such and such sub-sect of this larger sect". If he is a Vaishnava, then he will simply say "I am a Vaishnava")
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