S1 2.4 - Skewness and outliers

Reading skewness

Can a few unusually long waits change which service looks quicker? To answer fairly, first look at the direction and shape of the spread, rather than relying on one average.

On a histogram, look along the horizontal scale for the direction in which the tail stretches. On a box plot, compare whisker lengths and the median's position inside the box. For example, $Q_3-\text{median}>\text{median}-Q_1$ is evidence of more spread above the median within the middle half, suggesting positive skewness. Use the whole available display: quartiles alone do not establish the exact shape of the entire distribution.

[DIAGRAM: asset_name: Skewness from box-plot evidence; asset_slug: skewness-box-evidence; caption: Reflecting the distribution reverses the skew but preserves its spread.; description: Two horizontal box plots. The positively skewed example has summary zero, eight, twelve, twenty-four, forty-two. The negatively skewed reflected example has summary zero, eighteen, thirty, thirty-four, forty-two. Each of the five summary positions is labelled.]

A long right tail often pulls the mean above the median; a long left tail often pulls it below. If both mean and median are supplied, that comparison can support a judgement. Do not invent a mean from a box plot: it is not one of the plotted summaries. Neither a single summary comparison nor symmetry alone determines every detail of a distribution.

Change only the direction of the scale by reflecting every value through $21$, using $x\mapsto42-x$. Distances stay the same but left and right swap.

Use evidence you can actually calculate or see.

Interpret a fresh quartile pattern.

Applying a stated outlier rule

An outlier rule supplies a consistent way to flag observations that are unusually far from the main body of the data.

One possible rule flags values below $Q_1-1.5\,\mathrm{IQR}$ or above $Q_3+1.5\,\mathrm{IQR}$. Those two thresholds are called fences. This rule is an example, not a rule to impose on every question. Another question may use a different multiplier or a criterion based on the mean and standard deviation. For example, if the stated rule is “more than two standard deviations above the mean”, with mean $40$ and standard deviation $3$, its upper threshold is $40+2(3)=46$. A value of $47$ is flagged, whereas $46$ is not. The rule alone supplies the criterion; no assumption about a particular distribution shape is needed for this classification.

If the wording says “below” or “above”, equality with a fence does not count as an outlier. For a box plot with separate outliers, draw each outlier as an individual point or cross on the same scale. Extend whiskers to the smallest and largest observed values that are not outliers, rather than automatically to the fences.

[DIAGRAM: asset_name: Outlier fences and observed whisker endpoints; asset_slug: outlier-fences-and-whiskers; caption: The outlier is plotted separately; the whiskers stop at actual non-outlying observations.; description: A box plot with lower quartile ten, median fourteen and upper quartile eighteen, non-outlying endpoints four and twenty-five, and an outlier at thirty-five. The stated one-point-five-IQR fences are minus two and thirty, drawn as dashed reference lines rather than whisker endpoints.]

Change only the rule's multiplier from $1.5$ to $3$, leaving the data unchanged.