Mode of grouped data

The mode is the value that occurs most often. In 2, 6, 4, 5, 0, 2, 1, 3, 2, 3, the value 2 appears three times, so the mode is 2.

For grouped data we can only find the modal class, the class with the highest frequency. The mode lies somewhere inside it. To pin it down, use:

Mode = l + (f₁ – f₀) / (2f₁ – f₀ – f₂) × h

Write the three frequencies in a row: f₀, f₁, f₂, the one before, the modal one, the one after.

Why the formula makes sense: if the class before is busier than the class after, the mode is pulled towards the start of the modal class, and the other way round.

Example: family sizes 1–3 (7), 3–5 (8), 5–7 (2) … The modal class is 3–5. Mode = 3 + (8 – 7)/(16 – 7 – 2) × 2 = 3 + 2/7 ≈ 3.29.

Mode and mean answer different questions. The mean is the average; the mode is the most common value. 'Most pupils scored about 52, but the average was 62' is a perfectly sensible pair. The mode can be smaller than, equal to, or bigger than the mean.