Summarising data: averages and spread
Averages, spread and comparing distributions
There are three averages. The mean = sum ÷ count and uses every value; the median is the middle value once data are ordered, at position (n + 1)/2; the mode (or modal class for grouped data) is the most common value. The mean is affected by extreme values, the median is resistant to them.
Spread measures how spread out the data are. Range = maximum − minimum, easy but driven by outliers. Interquartile range = upper quartile − lower quartile, covering the middle 50% of the data. Standard deviation measures spread about the mean and uses all values. Quartiles can be read from a cumulative frequency curve at n/4, n/2 and 3n/4, or found by interpolation within a class.
Trap: do not confuse the median position with the median value - for n = 20 the median lies halfway between the 10th and 11th values. For grouped data you can only find an estimated mean using midpoints; say so. When asked to compare two distributions, comment on both an average and a measure of spread, in context - one average alone is never a full comparison.
Worked example. For 3, 5, 6, 8, 9, 11, 14: median position = (7 + 1)/2 = 4th value = 8. Lower quartile = 5, upper quartile = 11, so IQR = 11 − 5 = 6. Range = 14 − 3 = 11.