Working from a "less than" table for 50 observations, a student used each cumulative entry as though it were the class frequency. His $\sum f_i$ came to 151. What is the correct diagnosis?

In a "less than" table each entry already contains all the earlier classes, so the entries are running totals, not frequencies. Subtracting each entry from the next recovers the true $f_i$, and the check is that $\sum f_i$ must equal the number of observations, 50. A total of 151 is the immediate warning sign. Changing what you divide by, or which limit you use as $x_i$, does not repair frequencies that were never frequencies.