The Representativeness Index (RI) is a measure of how much the observed sample represents its population. It is a min-max normalized index and a distribution-free measure. There has been little empirical validation that a balanced sample is associated with a representative sample. Here, we investigate the relationship between the RI and the balanced sample by simulating sampling from uniform distribution and normal distribution. We quantify the quality of the balanced sample based on its mathematical definition, comparing the sample mean with the population mean for a variable of interest. Then, we propose a length-biased correction to a random sampling distribution to improve the RI. The length-biased distribution, the limiting distribution of spread in renewal process, weights the density function at each x by its length x. Lastly, we demonstrate how much the proposed length-biased correction to a simple random sample improves the RI using the 2010 Sample Redesign Primary Sampling Units for the American Housing Survey, Current Population Survey, and Survey of Income and Program Participation.