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We study compression techniques for electroencephalograph (EEG) signals. A variety of lossless compression techniques, including compress, gzip, bzip, shorten, and several predictive coding methods, are investigated and compared. The methods range from simple dictionary based approaches to more sophisticated context modeling techniques. It is seen that compression ratios obtained by lossless compression are limited even with sophisticated context based bias cancellation and activity based conditional coding. Though lossy compression can yield significantly higher compression ratios while potentially preserving diagnostic accuracy, it is not usually employed due to legal concerns. Hence, we investigate a near lossless compression technique that gives quantitative bounds on the errors introduced during compression. It is observed that such a technique gives significantly higher compression ratios (up to 3-bit/sample saving with less than 1% error). Compression results are reported for EEG's recorded under various clinical conditions.