Distributions in practice
Real data rarely announce their distribution. This lesson shows how to check assumptions, handle skew, and use distributions responsibly.
Checking a model against data
Compare a histogram of your data with the shape the model predicts. For a normal model, a normal probability plot should be roughly straight. For count data, compare the observed mean and variance: if the variance is much larger than the mean, a simple Poisson model is probably wrong.
Transforming skewed data
Right-skewed data, like incomes or reaction times, can often be made more symmetric by taking logarithms. Analysing \(\ln(\text{income})\) instead of income can make normal-based methods appropriate. Results can be transformed back for interpretation, for example as percentage changes.
Outliers and heavy tails
Some phenomena, like stock returns or city sizes, have heavier tails than the normal: extreme values occur far more often than the normal predicts. Assuming normality there underestimates risk, a lesson learned painfully in financial crises.
Percentiles and quantiles
The \(p\)th percentile is the value below which \(p\)% of the distribution falls. Growth charts for children, standardised test reports, and medical reference ranges all use percentiles. They describe position without assuming any particular shape.
Mixtures
Sometimes data come from two groups mixed together, such as heights of adult men and women combined, producing a bimodal shape. Recognising a mixture is important: summarising it with one mean can describe nobody well.
Worked example: choosing a model
A café records the number of customers arriving per 10 minutes. The mean is 4.1 and the variance 4.3, with counts from 0 to 11. Mean close to variance and a count story suggest a Poisson model. The gaps between arrivals should then look roughly exponential with mean about 2.4 minutes.
Models are approximations
George Box famously said that all models are wrong, but some are useful. A distribution is a simplified description that helps you calculate and predict. Judge a model by whether it is good enough for the decision at hand, and check it against new data.
Common mistakes
Assuming normality without checking. Ignoring heavy tails in risk calculations. Summarising a bimodal mixture with a single mean. Forgetting to transform back after a log transformation.
Key takeaways
Check distributional assumptions with plots and simple summaries. Log transformations can tame right skew. Heavy tails and mixtures demand care. Percentiles describe position without shape assumptions. Models are useful approximations, not exact truths.
Flashcard
Straight pattern suggests normal data
Check your understanding
Customer arrival counts have mean 3.9 and variance 4.1. Which model fits best?