A new efficiency criterion is proposed for the Markowitz portfolio selection approach. It is shown that the use of standard deviation as a measure of risk in the original Markowitz analysis and elsewhere in economic theory is sometimes unreasonable. An Investment with a relatively high standard deviation (\sigma) will be relatively safe if its expected value (E) is sufficiently high. For the net result may be a high expected floor, E - K\sigma, beneath the future value of the investment (where K is some constant). The revised efficient set is shown to eliminate paradoxical cases arising from the Markowitz calculation. It also simplifies the task left to the investor. For it yields a smaller efficient set (which is a subset of the Markowitz efficient set) and therefore reduces the range of alternatives from among which the investor must still select his portfolio. The proposed criterion may also be somewhat more easily understood by the nonprofessional.