The clearness index for 16 meteorological stations in Sudan has been expressed in terms of the fraction of bright sunshine duration, applying a variety of regression forms, namely linear, exponential, power and second-order polynomial. The significance and performance characteristics of the relations have been viewed using several statistical procedures. Although all of the four relationships are empirical and exhibit relative discrepancies, it has been found that they can reasonably be used to represent the underlying effect of relative duration of sunshine on clearness index for mean values. The estimates of the relationships are statistically significant at the 99.9% confidence level. For around 85% of the cases, the absolute error is fairly less than 4%. No estimate is found to fall beyond about +/-10% accuracy level. Unless otherwise investigated under extreme conditions, especially S/S-o = 0 and 1, the predictive value of the relations remains questionable for values of S/S-o outside the ranges covered herein. From the bulk of the data available, the important finding is that solar radiation is plentiful in this vast country and could be regarded as an essential, renewable source of energy for various applications.