Let \(S(x)\) denote the sum of the decimal digits of a positive integer \(x\). For a prime \(p\) and an exponent \(n \geq 1\), we study \(\widetilde{A}_p(n) = \frac{S(p^n)}{\lfloor n\log_{10}p\rfloor+1}\), the average value of the decimal digits of \(p^n\). We computed this statistic for the first fifty primes, from 2 through 229, and for exponents up to \(n = 8000\), giving 400,000 prime–exponent observations. Over this range, the prefix means move toward and remain close to \(9/2\), while the empirical prefix standard deviations decrease substantially as larger exponents are included. These observations motivate two conjectures: that the Cesàro mean of the normalized digit-sum statistic tends to \(9/2\), and that its prefix standard deviation tends to zero. The results are empirical and concern only this scalar statistic; they do not establish normality, independence, mixing, or uniform distribution of the decimal digits of prime powers.