The following stem-and-leaf diagram shows the scores of students in a math test.

1. Find the position of the median score

There are $2+9+7+3=21$ scores.

Since $21$ is odd, the median is the $\frac{21+1}{2}=11$th score.

2. Identify the 11th score from the ordered stem-and-leaf diagram

The stem-and-leaf diagram is already in ascending order.

Scores in the 60s: $68,69$ give $2$ scores.

The 70s row has leaves $0,2,3,3,4,4,4,7,8$.

The 11th score overall is the $11-2=9$th score in the 70s row, which is $78$.

1. List or count the ordered scores and locate the quartile positions

There are $21$ scores. Since the median is the $11$th score, exclude the median when finding the quartiles.

Lower half: first $10$ scores, so $Q_1$ is the average of the $5$th and $6$th scores.

Upper half: last $10$ scores, so $Q_3$ is the average of the $16$th and $17$th scores.

2. Find $Q_1$, $Q_3$ and calculate the interquartile range

Ordered positions from the stem-and-leaf diagram give:

$5$th score $=73$, $6$th score $=73$, so
$Q_1=\frac{73+73}{2}=73.$

$16$th score $=86$, $17$th score $=87$, so
$Q_3=\frac{86+87}{2}=86.5.$

Interquartile range
$=Q_3-Q_1=86.5-73=13.5.$