Outliers

Unusual relative to what?

A journey of 48 minutes may look unusual in a set concentrated between 8 and 22 minutes. It may be ordinary in a population travelling much farther. A value is not an outlier because it is large in absolute terms.

Context also matters in two-variable data. A result can be ordinary on each separate axis but unusual relative to a relationship. Conversely, an extreme predictor value may lie close to a fitted line. Scatter Plots and Linear Regression develop these bivariate distinctions.

First establish what each record represents, whether units are consistent and which population or model provides the comparison.

Apply an explicit IQR screen

Consider the synthetic journey times

Using the median-of-halves convention in Quartiles and Interquartile Range, $Q_1=12$ and $Q_3=20$. The IQR is 8. The inner fences are

The 48-minute value lies above 32 and is flagged. The 22-minute value is not. The rule says that 48 is unusual according to this particular ordered-data screen. It does not calculate the probability of a recording mistake.

An observation exactly on a fence is retained as unflagged in this set. If the IQR is zero because many values coincide, this rule can flag every different value. That situation requires context, not automatic deletion of all variation.

Trace the record before altering it

Check the original entry and the variable definition. Was 48 entered instead of 18? Were seconds accidentally recorded as minutes? Was the person describing a round trip while others described one-way journeys? Or did the learner genuinely experience a longer trip?

Correct a confirmed data-entry error using a documented source. If the value is valid, retain it or use a method suited to the distribution. If its status remains uncertain, report that uncertainty and consider a transparent sensitivity analysis.

Never replace an unusual observation with the mean simply to make a graph look neater. Such a change alters the recorded values and their variability without justification.

Compare conclusions with and without the observation

For the full illustrative dataset, the total is 180 minutes, the mean is 18 and the median is 15. Excluding 48 leaves nine values with total 132:

The remaining median is 14 minutes. The change is not just a tidier version of the same answer. It is a different dataset, with a different number of observations and a changed centre.

A defensible report might state both summaries and explain why the observation was investigated. If it was retained as a valid long journey, the full-data mean remains the relevant mean of all ten records. If it was corrected from a verified original entry, report the correction rather than pretending the first value never existed.

Sensitivity analysis asks whether an interpretation depends heavily on a choice. It is not permission to select whichever version gives the preferred conclusion.

Robust summaries are not substitutes for explanation

The median and IQR are generally less sensitive to extreme magnitudes than the mean and standard deviation. They can describe a central experience in a skewed dataset while preserving unusual values in the underlying records.

A trimmed mean deliberately excludes a stated proportion from both tails before averaging. Winsorisation replaces extreme values according to a declared rule. These are defined statistical procedures, not informal cleaning. Their use needs to fit the question, and the rule should be reported.

In a small learning dataset, show the values whenever practical. Readers can then see whether a mean, median or model adequately represents what occurred.

Outliers, leverage and influence

In regression, an observation with a large vertical residual is unusual relative to the fitted prediction. A point far from the other predictor values has high leverage because its horizontal position gives it the potential to alter the line. Influence concerns how much the fitted result actually changes when that point is included.

These are not equivalent labels. A high-leverage point can follow the relationship closely, while a large residual near the centre of the predictor range can have a different effect. Inspect the scatter plot and compare the fit using a documented procedure rather than diagnosing everything from one numerical threshold.

This page introduces the distinction; it does not prescribe an automatic influence cutoff for every regression problem.

Avoid rules that assume a model without checking it

Rules based on standard deviations or formal outlier tests often assume a particular distribution. A strongly skewed population can produce values far above its mean without those values being mistakes. Several unusual observations can also mask one another by changing the mean and spread used in the screen.

Do not apply a normal-model tail probability merely because a z-score was calculated. Standardisation changes the scale, not the shape of the distribution. Z-Scores explains that limitation.

Related topics

Box Plots displays IQR-based flags. Scatter Plots reveals unusual pairs, and Linear Regression shows how residuals help assess a fitted relationship.