Mean, median and mode, averages from frequency tables, estimated mean from grouped data, and weighted means.

What this dot point is asking

AQA wants you to find the mean, median and mode, calculate averages from frequency tables, estimate the mean from grouped data using midpoints, and find a weighted mean. On the AQA GCSE Statistics (8382) papers this material appears on both Paper 1 and Paper 2 as short calculation questions and as part of longer "compare the data sets" tasks, so accuracy and clear method are essential.

Mean, median and mode

Each average answers a slightly different question, and AQA expects you to choose the right one for the context:

• The mean uses every value, so it carries the most information, but it is pulled toward outliers. It is the right choice for symmetrical, outlier-free data such as repeated measurements.
• The median is the central value, so half the data lies below it and half above. It resists outliers and is the standard choice for skewed data such as incomes, house prices or reaction times.
• The mode is the only average that works for qualitative (categorical) data such as favourite colour, and it is useful for discrete data where the most common value matters, for example the most popular shoe size a shop should stock.

For an even number of ordered values the median is the mean of the middle two. For $n$ ordered values the median sits at position $\frac{n+1}{2}$: with $n = 7$ that is position $4$, and with $n = 8$ it is position $4.5$, meaning halfway between the $4$th and $5$th values.

Averages from a frequency table

The reason you multiply by frequency is that each value occurs $f$ times, so $fx$ is the running total contributed by that row. Dividing by $\sum f$ (the total number of data items) rather than by the number of rows is the single most common slip on this question type.

To find the median from a frequency table, build a cumulative frequency column and locate the $\frac{n+1}{2}$th item: the value in whose cumulative band that position falls is the median. The mode is simply the value (the $x$, not the frequency) with the highest frequency. AQA often asks all three from one table to test that you read the columns correctly.

Estimated mean from grouped data

When data is grouped into classes, the original values are lost, so you cannot find an exact mean. Instead you assume every value in a class sits at the class midpoint and proceed as for a frequency table.

The estimate is reliable only if values are spread reasonably evenly within each class. If the data clusters at one end of a class, the midpoint assumption introduces bias, which is why grouped means are quoted as estimates.

Weighted mean

A frequency-table mean is itself a weighted mean in which the "weights" are the frequencies. The same idea lets you combine the means of two groups: if class A has $30$ students with mean $60$ and class B has $20$ students with mean $70$, the overall mean is $\frac{30 \times 60 + 20 \times 70}{30 + 20} = \frac{3200}{50} = 64$, not the plain $\frac{60 + 70}{2} = 65$, because the larger group counts more.