Calculate and interpret the mean, median, mode and range for lists and frequency tables, estimate the mean and identify the modal class from grouped data, and compare distributions using an average and a measure of spread.

What this dot point is asking

This is the calculation core of WJEC statistics. You are asked to find and interpret the three averages (mean, median and mode) and the range as a measure of spread, to work them out from a list, from a frequency table and, at Higher tier, from a grouped frequency table where you estimate the mean and identify the modal class. The final, examined skill is to compare two sets of data using one average and one measure of spread, written in context. Choosing the right average for the situation is also assessed.

The three averages and the range

Each average summarises the data differently.

So for $2, 4, 4, 9$: the mean is $\tfrac{19}{4} = 4.75$, the median is $\tfrac{4+4}{2} = 4$, the mode is $4$, and the range is $9 - 2 = 7$.

Averages from a frequency table

A frequency table records how many times each value occurs.

To find the mean, multiply each value $x$ by its frequency $f$, add these products to get $\sum fx$, then divide by the total frequency $\sum f$. The mode is the value with the highest frequency. The median is the value of the middle item: find its position $\tfrac{n+1}{2}$ and read across the cumulative count. Reading the table carefully (frequencies, not values, are summed for the count) is where care is needed.

Grouped data (Higher)

When data is grouped into classes, individual values are lost, so the mean can only be estimated.

The mean is an estimate because the midpoint stands in for every value in the class, and the true values are unknown. The modal class is a class, not a single number, so name the interval rather than a value.

Comparing distributions

Comparing two data sets is the highest-value skill here.

Why this matters

Averages and spread are among the most frequently examined statistics topics, appearing in short calculation questions and in extended "compare" questions that carry AO2 and AO3 marks for reasoning in context. The grouped-data mean estimate is a classic Higher technique, and choosing the appropriate average (the median resists extreme values, the mean uses all the data, the mode suits categories) is itself assessed. The discipline of always pairing an average with a measure of spread in a comparison is the habit that secures full marks.