Raising the bar

Chapter 3. Describing Data: Numerical

Unit 3.3. Shape of the distribution

The definitions of symmetry and skewness must be given in terms of the histogram.

Symmetric distribution. The distribution is called symmetric if the histogram is symmetric about the mean (that is, observations equidistant from the mean to the left and right must have equal frequencies).

Empirical distributions rarely are exactly symmetric, and judgments about approximate symmetry are subjective, see Figure 3.2. The histogram in that figure represents a sample from a normal distribution, which is symmetric; however, the histogram is not symmetric. This illustrates how far a sample distribution can be from the population distribution (both are discussed in detail later).

Figure 3.2. Histogram example.

We don't need the mathematical definition of skewness. The approximate definition of positive skewness given in NCT (that the mean should be greater than the median) is too often at variance with the precise definition. As a rule of thumb, it is better to talk about tails.

Skewed distribution. We say that a distribution is positively skewed if the right tail is heavier than the left, see Figure 3.3 (think about the bars of the histogram as made of a heavy substance).

Figure 3.3. Distribution skewed to the right.

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