Right, but for the wrong reasons...

From The Daily Mail:

Children are less likely to walk to school as their parents’ income increases, a study revealed today.

Richer parts of the country such as Barnet, Surrey and Bath all had low levels of walking, while relatively deprived areas such as Hull, Barking and Luton have high levels. The town which has the highest rate of school runs by car is well-heeled Windsor, where half its children get lifts to lessons.

But the figures, released by the Department of Transport, also dispel the myth of so-called Chelsea tractors ferrying young Henrys and Henriettas to private schools in London. Well-off parts of the capital like Kensington, Westminster and Fulham have some of the lowest car use in the country – with less than one in 10 children getting to lessons by vehicle.

The article drivels pleasantly on in the same manner for several column inches, but then they inadvertently give the real explanation:

In fact London boroughs, with the exception of more leafy outlying ones like Barnet, have some of the highest rates of pupils going by foot to their classes.

Yes, there is a correlation between parents' income and whether kids get taken to school by car: that is not because wealthier people own more cars (they probably do, but only slightly more), it is because wealthier people tend to live in more sparsely populated outer urban regions.

Conversely, kids in the hyper-rich but densely populated parts of west/central London walk to school because it is far quicker than driving. And kids in the poorer but equally densely populated parts of London like Barking also walk to school.

Further, wealthier parents are more likely to send their kids to a private school, and as there are far fewer private schools than there are state schools, by definition, the average distance from home to private school will be much further than from home to nearest state school.