This paper presents both theoretical and empirical evidence about a probability distribution which describes the behavior of share price changes. Osborne's Brownian motion theory of share price changes is modified to account for the changing variance of the share market. This produces a scaled t-distribution which is an excellent fit to series of share price indices. This distribution is the only known simple distribution to fit changes in share prices. It provides a far better fit to the data than the stable Paretian, compound process, and normal distributions.