The list below shows the time (in minutes) taken by 12 students to solve a maths problem - Junior Cycle Mathematics - Question 5 - 2018

To find the inter-quartile range (IQR), we first need to calculate the first quartile (Q1) and the third quartile (Q3).

The data in order is: 3, 5, 6, 7, 9, 10, 12, 13, 14, 14, 15.

• Q1 is the median of the first half of the data:

  • First half: 3, 5, 6, 7, 9 → Q1 is 6.

• Q3 is the median of the second half of the data:

  • Second half: 10, 12, 13, 14, 14, 15 → Q3 is 14.

Now, we calculate the IQR:

$\text{IQR} = Q3 - Q1 = 14 - 6 = 8$

So, the inter-quartile range of the data is 8 minutes.

The range is 12 minutes, and ( \frac{1}{4} ) of the range is:

$\frac{1}{4} \times 12 = 3$

Since the inter-quartile range (IQR = 8) does not equal 3, this statement does not match any histogram.

Calculating ( \frac{1}{2} ) of the range:

$\frac{1}{2} \times 12 = 6$

Again, the IQR (8) does not equal 6, so this statement also does not match any histogram.

Now, calculating ( \frac{3}{4} ) of the range:

$\frac{3}{4} \times 12 = 9$

Since the IQR (8) does not equal 9, this also does not match.

Histogram B shows the data distribution where most values are concentrated towards the middle. The quartiles are positioned in such a way that the middle 50% of the data is represented, demonstrating that the IQR is shorter relative to the total range. Hence, Histogram B presents a balanced distribution in the middle with minimal extremities.