Sheikh M. S. Al-Munajjid, a prominent Saudi Muslim lecturer and author, states:
One sheep is sufficient as a sacrificial animal for one man and his family and whoever he wants among the Muslims, based on the hadith of `A’ishah (may Allah be pleased with her), who said that the Prophet (peace and blessings be upon him) ordered that a ram with black legs, black belly and black (circles) round the eyes should be brought to him so that he could sacrifice it. He said to `A’ishah, “Give me the knife,” and she did that. He took it, then he took the ram, placed it on the ground and then slaughtered it (i.e., prepared to slaughter it), saying: “In the name of Allah, O Allah, accept (this sacrifice) on behalf of Muhammad and the family of Muhammad and the Ummah of Muhammad.” Then he sacrificed it (Muslim).
It was narrated that Abu Ayyub Al-Ansari (may Allah be pleased with him) said: At the time of the Prophet (peace and blessings be upon him), a man would sacrifice a sheep on behalf of himself and his household, and they would eat some and give some to others (Ibn Majah and At-Tirmidhi).
If a man sacrifices a single sheep or goat on behalf of himself and his family, that will suffice for everyone he intended of his family, whether living or deceased. If he did not intend anything specific, then it includes all those who are included in this word (family or household), either customarily or linguistically. Customarily it refers to all those whom he supports of wives, children, and relatives; linguistically it includes all those who are related to him of his own children and the descendents of his father, grandfather, and great-grandfather.
One-seventh of a camel or cow is equivalent to one sheep. If a man sacrifices one-seventh of a camel or cow on behalf of himself and his family, that is sufficient because the Prophet (peace and blessings be upon him) said that one-seventh of a camel or cow takes the place of a sheep in the case of the hadi (sacrifice offered during Hajj), so that applies also to the udhiyah because there is no difference between the udhiyahand the hadi in this regard.
If two or more people buy a sheep and sacrifice it, that is not sufficient, because no such thing has been narrated in the Qur’an and Sunnah. Similarly, if eight or more people share one camel or one cow, that is not sufficient (but it is permissible for seven people to share a camel or cow), because acts of worship are as prescribed in the Qur’an and Sunnah and are not subject to personal opinion; it is not permissible to go beyond the set limits with regard to how much is to be done or the way in which it is to be done. This does not have to do with including others in the reward because it was narrated that there is no limit to the number of people on whose behalf the sacrifice may be offered.
Excerpted, with slight modifications from, : www.islam-qa.com