To answer your question, I obfuscated 1MB of data that consisted of a single 1 followed by all 0's, using your technique, and fed the results to ent:
Entropy = 0.000039 bits per byte.
Optimum compression would reduce the size
of this 1048576 byte file by 99 percent.
Chi square distribution for 1048576 samples is 267385856.00, and randomly
would exceed this value less than 0.01 percent of the times.
Arithmetic mean value of data bytes is 0.0000 (127.5 = random).
Monte Carlo value for Pi is 4.000000000 (error 27.32 percent).
Serial correlation coefficient is 0.499999 (totally uncorrelated = 0.0).
As you can see, in the worst case scenario, this method does not provide very good statistical randomness. In order to produce output with any real entropy, the input already needs to contain a good amount.
Suggestions
However, that's not to say there's no way to improve your method to provide decent statistical randomness. A few small changes can provide positive results.
First, consider using a nonce. Since we'll be iterating through the array anyways, we can include the iterator index into our calculations. This will ensure that successive similar values do not produce the same result. This will contribute to solving the problem of low entropy input not producing high entropy output.
Second, your technique as proposed suffers from low diffusion. Each byte is influenced exclusively by the byte that follows it. So a change in one location in the input influences the output only a small amount.
One way to fix this could be by combining via XOR all the bytes up front, then successively remove each byte, "encrypt it" using the XOR of the rest of the bytes as the "key", re-insert the "encrypted" byte back into the "key" using XOR, and repeat. The removal step is only required for the steps to be invertible.
Note that if you use the above technique, you will want to use modular addition for any further combination operations, otherwise you'll simply be stacking/moving bytes around.
A similar (but evolved) version of your technique that incorporates these recommendations is capable of producing ent results that provide good statistical randomness after a single application:
Entropy = 7.889023 bits per byte.
Optimum compression would reduce the size
of this 1048576 byte file by 1 percent.
Chi square distribution for 1048576 samples is 163954.21, and randomly
would exceed this value less than 0.01 percent of the times.
Arithmetic mean value of data bytes is 128.0863 (127.5 = random).
Monte Carlo value for Pi is 3.031574370 (error 3.50 percent).
Serial correlation coefficient is 0.008583 (totally uncorrelated = 0.0).
I think what you're effectively looking at/for is an unkeyed psuedorandom permutation. Interestingly, you can take an unkeyed psuedorandom permutation and make a secure cipher using the Even-Mansor construction. But key management and encryption is a whole 'nother level and not what you were attempting to accomplish judging by your description.