DITHER

Dither creates the illusion of there being more tones. Without having a 51% gray pixel, an area of 51% gray can be simulated by randomly mixing 50% and 52% pixels.
A grayscale photo in a book is usually printed with just black ink on white paper. The illusion of gray tones works with just two values, but the more values the better the quality. High quality Ink-jet printers have black ánd gray inks, just two tones that together with dithering create the illusion of a large spectrum of gray tones (example at the bottom of this page).

A normal digital photo has 256 tones per channel, but when using dithering the amount of tones can be reduced to only 32 and still look pretty good.

As for compression; 256 tones is 8 bits (2^8), and 32 tones is 5 bits (2^5) per pixel, thus this loss of image quality doesn't really give much compression: 5/8 = 0.62
More interesting is the reduction of combinations.
At 256 tones, the amount of combinations of a 8*8 image block is 256^(8*8) =
134078079299425970995740249982058461274793658
205923933777235614437217640300735469768018742
981669034276900318581864860508537538828119465
69946433649006084096
But at 32 tones, the amount of combinations of a 8*8 image block is only 32^(8*8) =
213598703592091008239502170616955211460270452
235665276994704160782221972578064055002296208
6936576
The ratio is (32^(8*8)) / (256^(8*8)) =
0.00000000000000000000000000000000000000000000000000000000016

The gray spectrum on the far left has only 32 tones in it, doesn't look smooth at all.


The right gray spectrum, has only 32 gray tones also, but now it seems smooth because it is dithered.
32 tones
 32 tones


COMPARISON

Full grayscale: 256 tones of gray
(246 here, to be exact)
Only 32 tones of gray
Can you spot the difference?!

16 tones of gray...
8 tones of gray
Can you NOW spot the difference!!