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NUMEROUS AUTHORS, INCLUDING SHILLER, LEROY AND PORTER, AND SINGLETON, HAVE REPORTED EMPIRICAL EVIDENCE THAT STOCK PRICES AND LONG INTEREST RATES ARE MORE VOLATILE THAN CAN BE JUSTIFIED BY STANDARD ASSET-PRICING MODELS. THIS PAPER SHOWS THAT IN SMALL SAMPLES THE "VOLATILITY" OR "VARIANCE-BOUNDS" TESTS TEND TO BE BIASED, OFTEN SEVERELY, TOWARD REJECTION OF THE NULL HYPOTHESIS OF MARKET EFFICIENCY. THUS THE APPARENT VIOLATION OF MARKET EFFICIENCY MAY BE REFLECTING THE SAMPLING PROPERTIES OF THE VOLATILITY MEASURES, RATHER THAN A FAILURE OF THE MARKET EFFICIENCY HYPOTHESIS ITSELF. THE PAPER ALSO REPORTS SOME UNBIASED ESTIMATES OF THE BOUNDS ON HOLDING PERIOD YIELDS AND LONG INTEREST RATES. MUCH OF THE EVIDENCE OF EXCESS VOLATILITY DISAPPEARS WHEN THE TESTS ARE CORRECTED FOR SMALL SAMPLE BIAS.