SL 4.3—Summary statistics

The mean is the arithmetic average, the median is the middle ordered value, and the mode is the most frequent value. Range, variance and standard deviation describe how far scores vary around a centre.

The mean uses every value but is pulled by outliers; the median is more resistant. Standard deviation is small when scores cluster near the mean, but neither statistic alone tells you whether the measurement is valid.

For 2, 3, 3, 4 and 18, mean=6 while median=3. Reporting both exposes the outlier and prevents a reader from mistaking the arithmetic average for a typical observation.

Do not choose a statistic by habit. Match it to the measurement scale and distribution, and never delete an extreme value without explaining how it arose.

For grouped data, estimate the mean with class midpoints: $\bar x\approx\sum fm/\sum f$; the modal class has greatest frequency when class widths are equal. If every value becomes $aX+b$, the mean and median become $a$ times the old value plus $b$, while standard deviation becomes $|a|$ times the old standard deviation and variance becomes $a^2$ times the old variance. Quartile conventions can differ, so report the technology method when relevant.