Statistics for Environmental Engineers

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Definition of Quantile and Percentile

The population distribution is the true underlying pattern. Figure 8.1 shows a lognormal population distribution of у and the normal distribution that is obtained by the transformation x = ln(y). The population 50th percentile (the median), and 90th, 95th, and 99th percentiles are shown. The population pth percentile, yp , is a parameter that, in practice, is unknown and must be estimated from data. The estimate of the percentile is denoted by yp. In this chapter, the parametric estimation method and one nonparametric estimation method are shown.

The pth quantile is a population parameter and is denoted by yp. (Chapter 2 stated that parameters would be indicated with Greek letters but this convention is violated in this chapter.) By definition, a proportion p of the population is smaller or equal to yp and a proportion 1 — p is larger than yp. Quantiles are expressed as decimal fractions.

Quantiles expressed as percentages are called percentiles. For example, the 0.5 quantile is equivalent to the 50th percentile; the 0.99 quantile is the 99th percentile. The 95th percentile will be denoted as y95.

A quartile of the distribution contains one-fourth of the area under the frequency distribution (and one-fourth of the data points). Thus, the distribution is divided into four equal areas by y0250 (the lower quantile), the median, y0.5 (the 0.5 quantile, or median), and y075 (known as the upper quartile).

Parametric Estimates of Quantiles

If we know or are willing to assume the population distribution, we can use a parametric method. Parametric quantile (percentile) estimation will be discussed initially in terms of the normal distribution. The same methods can be used on nonnormally distributed data after transformation to make them approximately normal. This is convenient because the properties of the normal distribution are known and accessible in tables.

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