Averages and measures of spread - CCEA GCSE Statistics guide to mean, median, mode, range, quartiles, standard deviation and standardised scores

Averages and spread are the core of the analysis stage of the enquiry cycle. This CCEA GCSE Statistics guide covers every measure of central tendency and spread, from the mean and median to standard deviation and standardised scores.

Averages

The mean is the total divided by the number of values, the median is the middle of the ordered data, and the mode is the most common value. A weighted mean multiplies each value by its weight before dividing by the total weight, used when components carry different importance such as coursework and an exam. From a frequency table you multiply each value by its frequency; from a grouped table you estimate the mean using class midpoints and take the highest-frequency class as the modal class. The mean uses every value but is pulled by outliers, the median resists them, and the mode is the only average for categorical data.

Choosing an average

The exam often asks which average is most appropriate. Use the mean when you want all the data and there are no extreme outliers, the median for skewed data or data with outliers such as house prices, and the mode for the most common item or for categories. A good answer names the average and gives a reason tied to the data.

Range, quartiles and IQR

The range is largest minus smallest, simple but distorted by outliers. The quartiles split ordered data into four parts, and the interquartile range, the upper quartile minus the lower quartile, is the spread of the middle half and ignores extremes. Percentiles divide the data into hundredths. An outlier, often defined as more than 1.5 times the IQR beyond a quartile, strongly affects the mean and range but barely moves the median and IQR.

Standard deviation

The standard deviation is the most informative measure of spread because it uses every value. Find the mean, square each deviation from the mean, average those squares to get the variance, then square root to get the standard deviation. A small standard deviation means tightly clustered data and a large one means widely spread data. It pairs with the mean, just as the IQR pairs with the median.

Box plots, standardised scores and skewness

A box plot shows the minimum, quartiles, median and maximum, and two box plots are compared by median and IQR, always interpreted in context. A standardised score, the value minus the mean divided by the standard deviation, measures how many standard deviations a value is from its mean and lets you compare across different distributions. Skewness describes shape: positive (right) skew has the mean above the median, negative (left) skew has the mean below it, and a symmetrical distribution has them roughly equal.

How CCEA examines averages and spread

CCEA rewards the grouped-mean and weighted-mean calculations, finding quartiles and the IQR, calculating standard deviation, comparing two distributions with an average and a spread in context, and using standardised scores and skewness at Higher tier. Use the dot points for specification-level detail and worked CCEA-style questions, then test yourself with the quiz.