Question

Ezra counts the number of people who were already on the bus when he gets on each morning. His data for the last 15 days are shown below. what is the interquartile range of the data?

8,9,9,10,12,12,13,14,15,15,15,16,18,19,20

Answer 1

To find the interquartile range (IQR) of the data, we need to follow these steps:

1. **Order the data**: The data is already ordered:
\(8, 9, 9, 10, 12, 12, 13, 14, 15, 15, 15, 16, 18, 19, 20\)

2. **Find the first quartile (\(Q_1\))** and the third quartile (\(Q_3\)):
- The first quartile (\(Q_1\)) is the median of the first half of the data (the lower half).
- The third quartile (\(Q_3\)) is the median of the second half of the data (the upper half).

3. **Split the data into two halves**:
- Lower half: \(8, 9, 9, 10, 12, 12, 13\) (7 data points)
- Upper half: \(14, 15, 15, 15, 16, 18, 19, 20\) (8 data points)

4. **Calculate \(Q_1\)**:
- The median (middle value) of the lower half is the 4th value:
- \(Q_1 = 10\)

5. **Calculate \(Q_3\)**:
- The median of the upper half: Since there are 8 points, the median will be the average of the 4th and 5th values:
- Values for the upper half are \(15, 15, 15, 16\) (the 4th value is 15, and the 5th value is 16).
- \(Q_3 = \frac{15 + 16}{2} = \frac{31}{2} = 15.5\)

6. **Calculate the interquartile range (IQR)**:
- \(IQR = Q_3 - Q_1 = 15.5 - 10 = 5.5\)

Thus, the interquartile range of the data is **5.5**.