Calculate measures of location and spread: the mean, median and mode, the range and interquartile range, and the variance and standard deviation, including from frequency data.

What this dot point is asking

Summarising a data set means describing where it is centred and how spread out it is, and CCEA GCSE Further Mathematics expects the full range of measures. You must calculate the mean, median and mode as measures of location, the range and interquartile range as simple measures of spread, and the variance and standard deviation as the key measure of spread, including from frequency tables. The standard deviation in particular underpins the normal-distribution work.

Measures of location

The three averages each summarise the centre of the data in a different way, and each suits different situations.

Choosing the right average matters: the median is preferred when there are outliers that would distort the mean.

Simple measures of spread

The range and interquartile range describe how spread out the data is in a quick way. The range is just the largest minus the smallest value, but it is sensitive to a single extreme value. The interquartile range is the upper quartile minus the lower quartile, the spread of the middle half of the data, and it ignores the extremes, which makes it more robust.

Variance and standard deviation

The standard deviation is the most important measure of spread, because it uses every value and underlies the normal distribution. It measures the typical distance of a value from the mean.

Measures from frequency tables

When data is given in a frequency table, each value occurs several times, so you weight by frequency. The mean is $\dfrac{\sum fx}{\sum f}$, where $f$ is the frequency of each value $x$. For grouped data, use the midpoint of each class as the value. The variance similarly uses $\dfrac{\sum f(x - \bar{x})^2}{\sum f}$. The principle is unchanged: every observation contributes, but identical values are handled together for efficiency, which is essential for large data sets.

Why this matters

Measures of location and spread are the basic vocabulary of statistics and feed the rest of the unit. The standard deviation is the spread used to standardise values in the normal distribution, the mean appears as the parameter of the binomial and Poisson distributions, and the choice between mean and median trains the judgement that data handling requires. Computing the standard deviation accurately, and remembering to take the square root, is the skill these questions reward.