Mean of grouped data; properties; weighted mean

For a discrete series (each value with its frequency f), multiply each value by its frequency:

X̄ = ΣfX ÷ Σf

For example, in a survey of families: 10 families have 2 children, 20 have 3, and 10 have 4. ΣfX = 20 + 60 + 40 = 120, and Σf = 40, so the mean is 120 ÷ 40 = 3 children.

For a continuous series (data in classes), we do not know the exact values in each class, so each class is represented by its mid-point, m:

X̄ = Σfm ÷ Σf

The assumed mean and step deviation methods work here too, using the mid-points, and give the same result. Remember: if the classes are inclusive (10-19, 20-29…), the mid-points can still be found directly, but check that classes are set out correctly before you start.

Two properties of the arithmetic mean:

• The sum of the deviations of the values from their mean is always zero: Σ(X – X̄) = 0. The values above the mean exactly balance those below it.
• The mean is affected by extreme values. If one or two students from a college get very high packages while most get modest ones, the mean package looks much better than what most students actually receive. For the values 3, 5, 7 and 85, the mean is 25, although three of the four values are below 10.

Sometimes items are not equally important. A weighted mean gives each item a weight (w) to show its importance:

Weighted mean = Σwx ÷ Σw

For example, to find the average price change in a family's shopping, a rise in the price of rice (a big part of the budget) should count more than a rise in the price of salt.