4 · IB Maths
Box plots and histograms
The same data drawn two ways. Skew the distribution and watch the mean slide away from the median, then add an outlier and see which statistics react.
Predict first
Start from the default settings. After you press Add an outlier once, which of these three statistics changes by the greatest proportion?
The standard deviation uses squared distances from the mean, so one far-away value inflates it a lot. The median and the quartiles depend only on rank order, so a single extreme value barely moves them.
Check it in the model: Note the Standard deviation, Median and IQR readouts, press Add an outlier, then compare the new values.
histogram and box (Q1 to Q3) median mean · red dots are outliers
outlier if x < Q1 − 1.5 IQR or x > Q3 + 1.5 IQR
mean = Σx ÷ n ; σ = √[Σ(x − mean)² ÷ n]
Try this
- Start with the Symmetric preset and note the mean and median. Raise Skew to 1.0 and compare how far each moves.
- Press Add an outlier three times. Which of mean, median, IQR and standard deviation change most, and which hardly change?
- Choose Tight cluster, then Add an outlier. Compare the standard deviation and the IQR before and after, and explain which one the outlier affects more.
Exam tip: whiskers stop at the furthest values within 1.5 × IQR of the box and anything beyond is an outlier; for skewed data quote the median and IQR, and when comparing two sets comment on centre, spread and shape.
More on 4: Venn diagrams and conditional probability · Discrete random variables · Continuous random variables · Bayes' theorem: medical test · Standard deviation and outliers · Binomial distribution · Markov chains · The normal distribution · Poisson distribution · Scatter and regression · Tree diagrams and Bayes’ theorem