Histograms

With equal class widths, a bar chart of frequencies is honest: a class with twice the frequency looks twice as big. With unequal widths it lies — a class covering 0–50 with frequency 20 would dwarf a class covering 50–55 with frequency 15, even though the second is far more crowded. Dividing by the width fixes this: area = width × density = width × (frequency ÷ width) = frequency. Equal areas, equal frequencies, honest picture.

The examiner-proof way to work: always make a table with four columns — class, width, frequency, density — and fill it in before drawing or reading anything.

1. Write the class width of every interval (upper bound − lower bound).
2. Divide each frequency by its width to get the density: $\text{fd} = f \div w$.
3. Label the vertical axis frequency density — not “frequency” — and choose a scale that fits the largest density.
4. Draw each bar exactly spanning its class interval, to the computed height, with no gaps between adjacent bars.

Going from histogram to frequencies is the same triangle of values rearranged:

A question that says “show that there are 10 items in class D” wants the one-line calculation: width × height. A question giving you a frequency and asking for the bar height wants frequency ÷ width. Everything on this topic is one of these three rearrangements.

Histograms with unequal widths are Extended content, typically one question worth 4–8 marks on Paper 2 or Paper 4. The standard parts: a “show that” using area = frequency (1 mark), completing frequencies from bar heights or heights from a table (2–3 marks), then a follow-on that reuses the frequencies — an estimated mean, or a probability of picking from particular classes. The follow-on is where table discipline pays: an estimated mean needs class midpoints (including both boundaries), and the wrong-answer chains almost always start from a mis-read frequency.

1. 1.Plotting frequency as bar height

   The single most common error on this topic. If class widths differ and your tallest bar belongs to the largest frequency, stop — you’ve drawn a bar chart, not a histogram.

2. 2.Reading class width off the wrong boundaries

   Width is upper bound minus lower bound of that class. For 1.5 < m ≤ 2.5 the width is 1, not 2.5. Write widths into your table before dividing anything.

3. 3.Using class width (or boundary) instead of midpoint in the estimated mean

   The estimated mean is $\sum f \times \text{midpoint} \div \sum f$. The midpoint is halfway between the boundaries — for 1.5–2.5 that’s 2, not 1 and not 2.5.

4. 4.Leaving the vertical axis labelled 'frequency'

   Label it frequency density. On drawing questions the scale and labelling can carry a mark, and the wrong label signals the wrong method even when the bars are right.