SL 4.3—Summary statistics

Summary statistics compress data without telling the whole story.

Mean, median, quartiles and standard deviation answer different questions about centre and spread, especially when outliers or skew are present.

Example

For 2,3,3,4,18 the mean is pulled upward while the median remains 3; median and IQR better describe a typical value.

Choose a resistant summary for skewed data and state what information the statistic loses.

Standard deviation is not a universal error bar and does not imply a normal distribution.

For grouped data, estimate the mean with $\bar x\approx\sum fm/\sum f$, where $m$ is each class midpoint. Adding a constant $c$ to every value adds $c$ to the mean and median but leaves standard deviation and IQR unchanged; multiplying every value by $k$ multiplies centre by $k$ and spread by $|k|$, while variance is multiplied by $k^2$. Technology and hand methods may use different quartile conventions.