Uluru is not the centre of Australia, Alice Springs is not the centre of Australia, Finke is not the centre of Australia, even Central Mount Stuart isn't the centre of Australia. Nope, despite what anyone might have told you, none of them are the centre of Australia. All these places are in Central Australia, this is correct. I am talking about the centre of gravity of Australia as per The Australian Concise Oxford Dictionary:
"the point at which the weight of a body may be considered to act."
On 22 April 1860 John McDouall Stuart, the one the Stuart Highway is named after, reached 133º30'S 22ºE, the point he had calculated to be the geographic centre of Australia. The following day, Stuart and his men ascended a nearby (4 km NNE) mountain he christened Central Mount Sturt, planted a note declaring the mountain to be the centre of Australia, and erected a British flag over it. After the return of his expedition, the Australian government renamed the mountain Central Mount Stuart, after the first white explorer to go there rather than after said explorer's hero. This note now resides in the State Library of South Australia.
After further calculations with previously unavailable data, the centre of Australia was "moved" in 1988, and a new plaque erected by the Royal Geographical Society of Australia. The new coordinates were (and still are) 25º36'36.4"S 134º21'17.3"E, a point very near (22 km W) Finke. This point is known as the Lambert Centre, after Dr. Bruce P. Lambert, a significant national surveyor. Finke, or Apatula in the local lingo, is an Aboriginal community alongside the Finke River in the NT. Finke is home to the starting line of the Finke Desert Race.
The centre of Australia, like the centre of the Earth, is hot. Unlike the centre of the Earth, it is red not white. Actually the centre of Australia is pretty much the like the rest of Central Australia bar a dirty great plaque the government decided it would be a good idea to stick there. The place is frequented by camels, drop bears, kangaroos, king browns, scorpions, tropical centipedes, kangaroo rats, and various other interesting animals. Temperatures are extreme. As I said, it's hot. But it's not always hot. 45º is only during the day, -15º is perfectly normal for at night.
I highly recommend going to visit the centre of Australia if you're ever looking for something to do.
Further reading: http://www.ga.gov.au/education/geoscience-basics/dimensions/centre-of-australia-states-territories.html
 Of course if your plane doesn't make it, or you get eaten by drop bears or something, I totally didn't tell you to come here.