Question 639·Hard·Math

Each bar height is a frequency, so a rating contributes its value times its height to the total. The graph shows $1$, $2$, $2$, $k$, and $1$ students at ratings $1$ through $5$. Since total = mean times number of students, write:

The $4k$ counts all the students who gave a $4$; the $k$ on the right counts those same students in the group size.

Type the equation in Desmos, replacing $=$ with $\sim$ so Desmos fits the unknown frequency $k$. Under PARAMETERS, it shows $k=10$. So ten students gave a rating of $4$.

Type $1+2+2+k+1$ on the next Desmos line. It shows $16$, so the bars represent 16 students, not five students for the five rating labels.

Type $1+2+2$ to count ratings below $4$. Desmos shows $5$. Those students fill positions $1$ through $5$; the ten ratings of $4$ fill positions $6$ through $15$, since $5+10=15$.

The median is the middle of the ratings in order. With $16$ students, the middle positions are $16/2=8$ and $9$. Both fall among the ratings of $4$ in positions $6$ through $15$, so their average is $4$. Count the students each bar represents, not just the rating labels. The median rating is $4$. Choice C.