Statistics · Superchallenge problem

The heights of $30$ students are whole numbers of centimeters. A histogram shows: $120$ to $129$ cm: $4$ students; $130$ to $139$ cm: $7$ students; $140$ to $149$ cm: $10$ students; $150$ to $159$ cm: $6$ students; $160$ to $169$ cm: $3$ students. What is the largest possible value of the mean height minus the median height, in centimeters? Give your answer as a decimal.