A concatenated coding scheme, recently proposed by Mahdavifar et al., is considered. The scheme uses polar codes as inner codes and maximum distance separable codes, such as Reed-Solomon codes, as outer codes. It was shown by Mahdavifar et al. that the concatenated coding scheme has a significantly better asymptotic error decay rate compared to Arikan's polar codes. However, the scaling of the required blocklength with respect to the gap between the code rate and the channel symmetric capacity was not considered. Following the analysis of the scaling problem for Arikan's polar codes by Guruswami and Xia, it is shown that the scaling of blocklength in the concatenated scheme is still inverse polynomial with the gap to the symmetric capacity. It is also shown that improved bounds can be derived for the concatenated scheme, compared to plain polar codes, both for the asymptotic error decay and for the scaling of the blocklength with respect to the gap to the symmetric capacity. An improved result for the error burst length that can be corrected is also derived for the concatenated coding scheme.