Shape of the Distribution, Normality

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Shape of the Distribution, Normality. An important aspect of the "description" of a variable is the shape of its distribution, which tells you the frequency of values from different ranges of the variable. Typically, a researcher is interested in how well the distribution can be approximated by the normal distribution (see the animation below for an example of this distribution) (see also Elementary Concepts). Simple descriptive statistics can provide some information relevant to this issue. For example, if the skewness (which measures the deviation of the distribution from symmetry) is clearly different from 0, then that distribution is asymmetrical, while normal distributions are perfectly symmetrical. If the kurtosis (which measures "peakedness" of the distribution) is clearly different from 0, then the distribution is either flatter or more peaked than normal; the kurtosis of the normal distribution is 0.
More precise information can be obtained by performing one of the tests of normality to determine the probability that the sample came from a normally distributed population of observations (e.g., the so-called Kolmogorov-Smirnov test, or the Shapiro-Wilks' W test. However, none of these tests can entirely substitute for a visual examination of the data using a histogram (i.e., a graph that shows the frequency distribution of a variable).