According to the International Center for Home Education Research (ICHER), a nonpartisan, scholarly repository of all things homeschooling, there are no studies to date that demonstrate one way or another about homeschooling academic prowess.
I have had the chance to observe and interact with many homeschooled and public-schooled children. Of the children I have encountered, the homeschoolers generally tend to be better behaved, especially when out in public. Regarding academics, though, I’d have to say the children I have met range from below- to above-average in comparison to their public-schooled counterparts (the below-average children have learning disabilities).
With respect to my own children, my oldest two, who were public-schooled for half of their education, are above-average in reading and mathematical ability and average to above-average in writing ability, depending on the type of writing required. My one child who has been homeschooled from the beginning is several grades ahead of her same-age public-schooled counterparts.
I have to agree with this statement from the article that parental involvement makes a huge difference:
Christians agree that educational success is normally tied to parental involvement. So whether one homeschools in a formal sense or not, encouragement and nurture from the parents are important.