Histogram

Study guide: Exam MAS-I

A Histogram is an Exploratory Data Analysis display of a single continuous variable: the range is cut into bins of equal width and a bar is drawn over each bin whose area is proportional to the number of observations falling in it. It is the sample's picture of the underlying density.

\text{bar height} = \frac{n_j}{n \cdot w} \quad \text{(density scale)}

n_j = \#\{i : x_i \in \text{bin } j\}, \qquad w = \text{bin width}

• On the density scale the bars have total area 1, so the histogram can be overlaid directly on a fitted density; on the count scale the height is simply n_j
• Bin width is the whole game: too wide and real structure (a second mode, a spike at zero) disappears; too narrow and sampling noise looks like structure
• The shape read off a histogram drives the modelling choice — right skew with a long tail points to Gamma or lognormal severity, a spike at zero plus a right-skewed remainder points to the Tweedie Distribution
• Unlike a Box Plot, a histogram reveals multimodality; unlike a QQ Plot, it is poor at judging tails, because the extreme bins hold few observations
• Applied to a discrete count variable it becomes a bar chart of the PMF — useful for spotting overdispersion before fitting a Poisson Regression
• Taking \log of a positive skewed variable before plotting usually turns an uninformative spike-and-tail into a readable, near-symmetric shape

Reading a Severity Histogram A histogram of 500 claim amounts has a tall first bar covering [0, 2{,}000), steadily shorter bars out to 20{,}000, and a handful of observations past 50{,}000. The bin width is 2{,}000 and the first bin holds 180 claims. What is the density-scale height of the first bar, and what does the shape suggest?

Answer \text{height} = \frac{n_1}{n \cdot w} = \frac{180}{500 \times 2{,}000} = 1.8 \times 10^{-4} Equivalently, 36\% of the claims fall in the first bin. The long right tail with no left tail is strong right skew, so a Normal-response model is inappropriate; fit severity with a Gamma or lognormal Generalized Linear Model, or model \ln(\text{severity}).

Choosing a Bin Width An actuary plots n = 64 observations of a continuous variable using 4 bins, then again using 32 bins. The first looks like a single smooth hump; the second looks like a comb of spikes. Which is right?

Answer Neither — both are artifacts of the bin choice. A common starting rule is k \approx \sqrt{n} = 8 bins (or Sturges' k \approx 1 + \log_2 n = 7). With 64 observations, 4 bins over-smooths away any second mode and 32 bins leaves about two observations per bin, so the "spikes" are pure sampling noise. Plot several widths and only trust features that survive all of them.