Is there a lot of redundancy in written and spoken language? I read years ago in a pop science book that written and spoken language can be shown to have a high level of redundancy. The speculation was that this served to allow error correction because language is encountered in noisy environments and understanding has a high survival value. The figure I remember is that if 35% of the information remains an attentive listener or reader can still make out the message. This has become a cherished notion in my world view as an indicator that language and human endeavor contains complexity we are not consciously aware of.

I am looking for a knowledgeable confirmation of the idea that spoken or written language contains this redundancy. An impolite smack down due to unexamined confirmation bias is always welcome as well.
It seems that for written English, the figure is 50%.
From pages 27 to 28 of The making of cognitive science: Essays in honor of George A. Miller (Cambridge: Cambridge University Press, 1988)

Estimates of redundancy.  Shannon (1948, 1951) had himself estimated the redundancy of printed English to be about 50 percent.  He had used a technique in which a subject was given a passage of text and then required to guess the next letter until the correct response (i.e., that corresponding to the original text) was given.  Redundancy was calculated from the distribution of the numbers of guesses required.  Garner and Carson (1960) [...] also estimated the redundancy of printed English to be about 50 percent.  Newman and Gerstman (1952) [...] estimated redundancy to be 52 percent.
Uses of redundancy.  [...] Chapanis (1954) and Miller and Friedman (1957) both showed that when text was mutilated by deleting different percentages of letters, subjects were able to restore the missing letters with a high degree of accuracy.  Such restoration is possible because of redundancy, so these experiments showed that redundancy was useful to humans.
[...]
In summary, printed English is redundant, and thus constrained, both in letter sequences within words and in sequences of words themselves.  This redundancy is known to humans, who can use it to reconstruct mutilated text and to recognize and learn words and sequences of words that reflect varying degrees of this constraint.  [...]

From page 1086 of A new kind of science by Stephen Wolfram (Wolfram Media, Inc., 2002):

[...] English text typically remains intelligible until about half its characters have been deleted, indicating that it has a redundancy of around 0.5.  Most other languages have slightly higher redundancies, making documents in those languages slightly longer than their counterparts in English.