Wind forecasting is an important consideration in integrating large amounts of wind power into the electricity grid. The wind power forecast error distribution assumed can have a large impact on the confidence intervals produced in wind power forecasting. In this work we examine the shape of the persistence model error distribution for ten different wind plants in the Electric Reliability Council of Texas (ERCOT) system over multiple timescales. Comparisons are made between the experimental distribution shape and that of the normal distribution. The shape of the distribution is found to change significantly with the length of the forecasting timescale. The Cauchy distribution is proposed as a model distribution for the forecast errors and model parameters are fitted. Finally, the differences in confidence intervals obtained using the Cauchy distribution and the normal distribution are compared.