I have always heard that
A change in just one bit of the original text shall change half of the bits of its hash.
I think this property is called the diffusion property.
However, I wonder why the diffusion property is so important in hashing functions.
I understand that for encryption algorithms this is a needed property. Without it, an attack based on language stats could be carried out.
Hashing functions however are one-way, which means that you can not get the original text from the hash. Why should hash functions exhibit the diffusion property then?
Given that you could use hashing functions for hiding information or for assuring the integrity of a document, I think it is a good idea to separate both aspects.
If we are using it for hiding information such as password this property in needed. Look at Lucas Kauffman's answer.
But if you are using hash functions to guarantee the integrity of a document. Is it still a needed property then?