Mean of grouped data

The mean is the ordinary average: add everything and divide by how many there are. When values repeat, multiply each value by how often it occurs (its frequency f) first:

x̄ = Σfᵢxᵢ / Σfᵢ (Σ, 'sigma', just means 'add them all up')

Lots of data? Put it in classes, like marks 10–25, 25–40, 40–55. Then you don't know each pupil's exact mark any more, only how many fall in each class.

So we use the middle of each class, its class mark: (lower limit + upper limit)/2. For 10–25 it is 17.5.

Where does a value on the boundary go? A mark of exactly 40 goes into 40–55, the class it starts, not 25–40.

The direct method, column by column:

Then mean = Σfx ÷ Σf.

Why a grouped mean is only an estimate. In the book, 30 pupils' exact marks have a mean of 59.3. Grouped into classes, the mean comes out as 62. The difference is there because we pretend every pupil in a class scored exactly the class mark. That's the price of grouping.