The histogram below shows the heights, in cm, of male A-level students at a particular school - AQA - A-Level Maths: Mechanics - Question 12 - 2018 - Paper 3

To determine the median height, we first need to recognize that the median is the middle value in a dataset. In histograms, we find the median class interval based on the cumulative frequency.

1. Calculate the total frequency by summing the heights in each interval:

   • For the interval (155, 160): Height frequency = 1
   • For the interval (160, 170): Height frequency = 3
   • For the interval (170, 180): Height frequency = 5
   • For the interval (180, 190): Height frequency = 1

   Total Frequency = 1 + 3 + 5 + 1 = 10.

2. The median position is given by $\frac{n + 1}{2} = \frac{10 + 1}{2} = 5.5.$ This means the median falls between the 5th and 6th observations.

3. Calculate the cumulative frequency:

   • Interval (155, 160): Cumulative Frequency = 1
   • Interval (160, 170): Cumulative Frequency = 1 + 3 = 4
   • Interval (170, 180): Cumulative Frequency = 4 + 5 = 9
   • Interval (180, 190): Cumulative Frequency = 9 + 1 = 10.

4. The median class interval is the one where the cumulative frequency reaches or exceeds 5.5, which is the interval [170, 180].