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Yitzhaki (1996) showed that the OLS estimator of the slope coefficient in a simple regression is a weighted average of the slopes delineated by adjacent observations. The weights depend only on the distribution of the independent variable. In this paper I demonstrate that equal weights can only be obtained if and only if the independent variable is normally distributed. This characteristic is used to develop a new test for normality which is distribution free and not sensitive to outliers. Thedoi:10.2139/ssrn.2111936 fatcat:5z2qaeye6bfflhhqridnh5f27y