XAT 2024 — QA & DI Question 20

Basics of AverageEasy

Passage / Data

Read the following scenario and answer the THREE questions that follow.

The upper hinge of a dataset is the median of all the values to the right of the median of the dataset in an ascending arrangement, while the lower hinge of the dataset is the median of all the values to the left of the median of the dataset in the same arrangement. For example, consider the dataset 4, 3, 2, 6, 4, 2, 7. When arranged in the ascending order, it becomes 2, 2, 3, 4, 4, 6, 7. The median is 4 (the bold value), and hence the upper hinge is the median of 4, 6, 7, i.e., 6. Similarly, the lower hinge is 2.

A student has surveyed thirteen of her teachers, and recorded their work experience (in integer years). Two of the values recorded by the student got smudged, and she cannot recall those values. All she remembers is that those two values were unequal, so let us write them as A and B, where A < B. The remaining eleven values, as recorded, are: 5, 6, 7, 8, 12, 16, 19, 21, 21, 27, 29. Moreover, the student also remembers the following summary measures, calculated based on all the thirteen values:

Minimum: 2
Lower Hinge: 6.5
Median: 12
Upper Hinge: 21
Maximum: 29

Which of the following is a possible value of B?

Answer & solution

A
2
B
6
8
D
13
E
29

Solution

The known observations are: 5, 6, 7, 8, 12, 16, 19, 21, 21, 27, 29

Median of all 13 observations is 12, hence there must be 6 values greather than or equal to 12 and 6 values less than or equal to 12.

We see that for known observations, we already have 6 values (16, 19, 21, 21, 27 and 29) which are greather than 12. Hence, both the unknown values i.e., A and B must be less than 12.

Also, the smallest of these observations is 2 (given) and we know A < B, hence A = 2.

The lowest 6 observations are 2, 5, 6, 7, 8 and B.
The lower hinge is 6.5. Hence, B should take a value such than 6 and 7 are the middle two numbers when arranged in ascending order.
This is possible when B is greater than or equal to 7 and less than or equal to 12.

∴ Possible values of B are 7, 8, 9, 10, 11 or 12.

Hence, option (c).