Profilplots

Profilplots, also known as profile plots or probability plots, are graphical tools used in statistics to assess how well a data set fits a specific theoretical distribution. They are particularly useful for visualizing the agreement between observed data and a hypothesized model, such as a normal distribution, exponential distribution, or Weibull distribution.

The core idea behind a profilplot is to compare the quantiles of the observed data against the quantiles of the theoretical distribution. If the data comes from the hypothesized distribution, the plotted points will lie approximately along a straight line. Deviations from this line can indicate departures from the assumed distribution, such as skewness, heavy or light tails, or the presence of outliers.

To create a profilplot, the observed data is first sorted in ascending order. Then, for each data point, its empirical quantile is calculated. Simultaneously, the corresponding theoretical quantiles for the chosen distribution are computed. These pairs of empirical and theoretical quantiles are then plotted against each other. For example, in a normal probability plot, the sorted data values are plotted against their corresponding z-scores assuming a normal distribution.

Interpreting a profilplot involves examining the pattern of the plotted points. A straight line suggests a good fit. Curved patterns can indicate specific types of distributional departures. For instance, an S-shaped curve might suggest a distribution that is more spread out in the tails than the hypothesized distribution. Profilplots are a valuable visual aid for initial data exploration and hypothesis testing in statistical modeling. They complement formal statistical tests for distribution fitting.