Shape, skew & outliers
Skewness names which tail is longer, and the reliable memory aid is that the skew follows the tail: right-skewed (positive) data have a long right tail, which pulls the mean ABOVE the median. Outliers are conventionally flagged by the 1.5 × IQR rule, and one extreme value can shift a mean dramatically while leaving the median untouched.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Compare the mean and medianMean > median suggests right skew; mean < median suggests left skew; roughly equal suggests symmetry.
- Describe the shape in wordsSymmetric, unimodal, bimodal, uniform, or skewed — and say which direction.
- Apply the 1.5 × IQR ruleOutliers lie below Q₁ − 1.5·IQR or above Q₃ + 1.5·IQR.
- Decide what to do about themInvestigate before deleting. A genuine extreme observation is data, not an error.
Worked example
For data with Q₁ = 20, Q₃ = 32, decide whether the values 5 and 48 are outliers.
- IQR = 32 − 20 = 12, so 1.5 × IQR = 18.
- Lower fence: 20 − 18 = 2. Upper fence: 32 + 18 = 50.
- 5 > 2, so it lies inside the lower fence — not an outlier.
- 48 < 50, so it is inside the upper fence too.
Answer. Neither is an outlier by the 1.5 × IQR rule — the fences are 2 and 50.
Where marks get dropped
These are the specific errors that cost credit on shape, skew & outliers questions — QED's rubric penalises each of them separately.
- Getting the skew direction backwards. The skew is named for the TAIL, not for where the bulk of the data sits.
- Deleting outliers automatically. They may be the most informative observations, and removing them without justification is data manipulation.
- Using 1.5 × IQR as though it were a law. It is a convention producing roughly 0.7% flagged points for normal data.
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Shape, skew & outliers — frequently asked questions
Why does right skew pull the mean up?
Because the mean is sensitive to magnitudes and the long right tail contains a few very large values. The median only counts positions, so it stays put.
Is an outlier always an error?
No. It may be a recording mistake, a genuine rare event, or evidence the model is wrong. Investigate the cause before deciding.
What is kurtosis?
A measure of tail heaviness relative to a normal distribution. High kurtosis means more extreme values than normal, which matters for risk modelling.
The rest of Statistics
Describing data, distributions, estimation and hypothesis tests. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Mean, median & mode
- 2Variance, standard deviation & spread
- 3Shape, skew & outliers
- 4Boxplots, histograms & quartiles
- 5The normal distribution & z-scores
- 6Sampling, bias & the sampling distribution
- 7The central limit theorem
- 8Confidence intervals for a mean
- 9Hypothesis testing & p-values
- 10t-tests & comparing two means
- 11Chi-square tests for independence
- 12Correlation & least-squares regression
- 13Type I / Type II errors & power
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