Study case: calculating frequency density from a histogram class In this GCSE Histograms Frequency Density Problem Tutor, I use one study case to explain why histogram bars are not drawn using frequency directly when class widths are unequal. The problem is: a class interval from 20 to 30 has frequency 15. Find its frequency density and explain the bar height. The answer is frequency density 1.5, because frequency density = frequency divided by class width. I like this example because it is short but it teaches the key idea behind histograms. Before solving, I ask the student to identify the class width. The interval from 20 to 30 has width 10. The frequency is 15. In a histogram, the area of the bar represents frequency, not simply the height. This is why frequency density is used. If students understand that area equals frequency, histogram questions become much easier. If they only memorise a formula, they often become confused when the intervals are unequal. Method 1 is the formula method. Frequency density = frequency / class width. Substituting the values gives 15 / 10 = 1.5. Therefore the bar height should be 1.5. I explain that the vertical axis of a histogram is frequency density, so the bar should rise to 1.5, not 15. This is a common GCSE mistake: students plot the frequency as the height and ignore class width. Method 2 is the area method. If the class width is 10 and the frequency density is 1.5, then the area of the bar is width times height = 10 times 1.5 = 15. That matches the frequency. I like this method because it proves the answer and shows why the formula works. It also helps students reverse the problem if the histogram gives the height and asks for the frequency. Method 3 is a comparison method. I ask the student to imagine another class interval with width 5 and frequency 15. If the frequency is the same but the width is smaller, the bar must be taller because the same area is squeezed into a narrower width. Its frequency density would be 15 / 5 = 3. This comparison helps students understand why unequal intervals need frequency density. It is not just a technical rule; it is a way of keeping area proportional to frequency. After the three methods, I compare them with the student. The formula method is quickest. The area method gives the strongest understanding. The comparison method helps prevent the mistake of treating a histogram like a bar chart. I remind students that a histogram is different from a bar chart because the width of each bar matters. When intervals are unequal, height alone does not represent frequency. Common mistakes include using frequency as the bar height, calculating class width incorrectly, forgetting that 20 to 30 has width 10, and reading the vertical axis without checking whether it says frequency or frequency density. I use these mistakes as teaching clues. If a student plots height 15 instead of 1.5, I return to the area rule. If they get the class width wrong, we practise subtracting lower boundaries from upper boundaries. This topic is important because GCSE statistics questions often ask students to complete a histogram, find a missing frequency, or estimate the number of values in a class. A student may need to use frequency = class width times frequency density. This is the reverse of the formula used in the study case. I teach both directions together so the student can move flexibly between frequency, width and density. A useful extension is to give a histogram bar with width 8 and height 2.5, then ask for the frequency. The student should calculate 8 times 2.5 = 20. Another extension is to ask students to compare two bars and decide which interval contains more data. These questions help students understand that the larger area, not necessarily the taller bar, represents the larger frequency. In one-to-one tutoring, I would finish by asking the student to create a small frequency density table from grouped data. They would calculate class widths, frequency densities and then describe how each bar should be drawn. This builds both calculation skill and graph interpretation. With clear explanation, histograms become logical rather than confusing. For support with this topic, students can book one-to-one tutor services: https://www.mastermathstutoring.co.uk/services. For the wider course page, visit GCSE and IGCSE Maths tutoring: https://www.mastermathstutoring.co.uk/gcse-igcse. To ask about lessons or availability, use contact MasterMaths Tutoring: https://www.mastermathstutoring.co.uk/contact-me.