Univariate Plot
Study guides: Exam PA, Exam MAS-I
A Univariate Plot displays the distribution of one variable at a time — its centre, spread, shape, and outliers — without reference to any other variable. Univariate plots are the first step of Exploratory Data Analysis: they are what tells you which response distribution and which transformations a model can plausibly use.
\text{one variable} \ \longrightarrow \ \text{shape, centre, spread, outliers}
| Plot | Shows | Best at |
|---|---|---|
| Histogram | Binned counts | Overall shape, multimodality |
| Box Plot | Five-number summary | Outliers, comparing groups |
| Dot plot / strip plot | Every observation | Small samples, ties |
| Empirical CDF | \hat{F}(x) = \frac{1}{n}\#\{x_i \le x\} |
Quantiles, no binning choice |
| QQ Plot | Sample vs theoretical quantiles | Tail behaviour, distributional fit |
| Bar chart | Category counts | Rare levels of a Categorical Predictor |
- For a continuous variable, start with a Histogram for shape and a Box Plot for outliers; for a categorical variable, a bar chart of level frequencies is the univariate plot, and it is how thin levels get found before they blow up a GLM's standard errors
- The empirical CDF is the one univariate plot with no tuning choice — no bin width, no whisker rule — so it is the honest check when a histogram's shape looks bin-dependent
- Univariate plots of the response narrow the choice of distribution and Link Function; univariate plots of each predictor find skew, spikes at zero, and impossible values before they contaminate the fit
- They cannot show relationships: a variable can look perfectly well behaved on its own and still be badly related to the response, which is what a Scatter Plot and the diagnostic plots are for
- Plot exposures too, not just losses — a univariate plot of exposure by level shows where the data is thin enough that a fitted coefficient will be unstable
Choosing Plots for a New Data Set An actuary receives a data set with claim count (integer, mostly zero), claim severity (positive, right-skewed), and territory (12 levels). Which univariate plot fits each?
Answer
- Claim count: a bar chart of the counts
0, 1, 2, \ldots— a histogram of a variable that is85\%zeros just shows one bar. Compare the sample mean and variance while you are there; variance\ggmean means overdispersion, so negative binomial rather than plain Poisson Regression.- Severity: a Histogram of
\ln(\text{severity})plus a Box Plot on the raw scale, so the shape and the outliers are both visible.- Territory: a bar chart of exposure by level, to find levels with too little data to fit their own coefficient.
When the Histogram and the Box Plot Disagree A box plot of a variable looks symmetric with no outliers, but the histogram shows two clear humps. Which is telling the truth?
Answer Both — they measure different things. The five numbers behind a box plot (
\min, Q_1, Q_2, Q_3, \max) can be identical for a symmetric unimodal sample and a symmetric bimodal one, and the box draws nothing between them, so a box plot cannot show two humps. The histogram is right about the shape, and the two modes usually mean a missing categorical variable (two classes of risk mixed in one column) that should be split out.