Calculate the mean, median, mode and range; find the mean from a frequency table and an estimated mean from grouped data; and compare distributions using an average and the range (and quartiles at Higher tier).

What this dot point is asking

The Eduqas statistics content asks you to find the four summary measures, the mean, median, mode and range, and to calculate the mean from a frequency table and an estimated mean from grouped data. At Higher tier you also compare distributions using an average and a measure of spread, including quartiles. Averages summarise a data set in a single value, and spread measures how varied it is, so together they describe the whole distribution. These are core statistics, tested at both tiers, with the grouped-data estimated mean being a reliable multi-step question.

The four measures

Each summary measure captures a different feature of the data.

So for $3, 5, 5, 8, 9$: the mean is $\dfrac{30}{5} = 6$, the median is $5$, the mode is $5$, and the range is $9 - 3 = 6$. The mean uses every value but is affected by extreme values (outliers); the median is more robust to them, which is why house prices are often quoted as a median.

The mean from a frequency table

When data is given as values with frequencies, multiply each value by its frequency.

So if $0$ goals occurred $5$ times, $1$ goal $8$ times and $2$ goals $7$ times: $\sum fx = (0 \times 5) + (1 \times 8) + (2 \times 7) = 22$, and $\sum f = 20$, so the mean is $\dfrac{22}{20} = 1.1$ goals. The frequent error is dividing by the number of rows instead of the total frequency.

The estimated mean from grouped data

When data is grouped into class intervals, the exact values are unknown, so the midpoint of each class represents it.

It is an estimate because the midpoint assumes the values in each class are evenly spread, which they may not be.

Comparing distributions

To compare two data sets, quote both a typical value and a measure of spread.

A comparison needs an average (mean or median) to say which set is typically higher, and a measure of spread to say which is more consistent. So "Class A had a higher mean mark ($68$ versus $61$) but a larger range ($40$ versus $25$), so A scored higher on average but was less consistent." At Higher tier, the interquartile range (upper quartile minus lower quartile) is a better measure of spread than the range because it ignores extreme values, focusing on the middle $50\%$ of the data. Always make the comparison in context, naming what the figures mean.

Choosing the right average

Each average suits different data. The mean is the best summary for fairly symmetric data with no extreme values, because it uses every data point. The median is preferred when there are outliers or the data is skewed, since it is not dragged towards an extreme value; this is why incomes and house prices are usually quoted as medians. The mode is the only average that works for qualitative (categorical) data, such as the most popular colour, and it is useful for finding the most common value. Eduqas sometimes asks which average is most appropriate and why, so being able to justify the choice in context, not just calculate each one, is part of the skill.