1.5 Shape of a Distribution

A histogram shows the shape of the distribution of a quantitative variable. The shape of a distribution includes the following three aspects:

  • Overall shape: what the distribution looks like, e.g., bell-shaped, J-shaped, triangular, and uniform. (See examples in the figures below.)
  • Modality: number of peaks (highest points). A distribution is called unimodal if it has only one peak, bimodal if it has two peaks, and multimodal if it has more than two peaks.
  • Symmetry and skewness. If you fold a distribution in the middle and the two parts can match, the distribution is called symmetric. If it has a longer left tail, it is called skewed to the left (or left skewed), and skewed to the right (or right skewed) if it has a longer right tail.

The following figure shows some special shapes of distributions.

Nine special shapes of distributions presented in three rows and three columns. Image description available.
Figure 1.8: Some Special Shapes of Distributions [Image Description (See Appendix D Figure 1.8)]

Distributions in Figure 1.8 can be described respectively as follows:

(a) bell-shaped, unimodal, and symmetric

(b) triangular, unimodal, and symmetric

(c) rectangular, no peak, and symmetric

(d) unimodal and right-skewed

(e) unimodal and left-skewed

(f) J-shaped, unimodal, and left skewed

(g) reversed J-shaped, unimodal, and right-skewed

(h) bimodal and symmetric

(i) multimodal and asymmetric

Example: Shape of a Distribution

Comment on the overall shape and modality of the following histogram.

Histogram of grade, the y-axis is frequency and x-axis is grades. Image description availableFigure 1.8.1: The histogram of the grades data above shows its distribution is bell-shaped, unimodal, and symmetric. [Image Description (See Appendix D Figure 1.8.1)]


Exercise: Shape of a Distribution

Figure 1.9 is the histogram of survival time in years after cancer diagnosis. Comment on the overall shape and modality of the histogram.

Histogram of survival time after diagnosis of cancer, the y-axis is frequency and x-axis is survival time in years. Image description available
Figure 1.9: Histogram of Survival Time (in years) [Image Description (See Appendix D Figure 1.9)]
Show/Hide Answer

We can see the distribution of the survival time is unimodal and right-skewed.


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