A researcher collects data on the number of hours spent studying per week by a group of students. The frequencies are as follows: values $1$ through $10$ appear $5$ times each; values $11$ through $20$ appear $10$ times each; values $21$ through $30$ appear $15$ times each; and values $31$ through $40$ appear $8$ times each. Let $\mu$ be the mean, $M$ be the median, and $d$ be the median of the modes of this data set. Which of the following statements is true?