Q–Q plot

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normal exponential qq

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Q–Q (quantile-quantile) plot is a probability plot, which is a graphical method for comparing two probability distributions by plotting their quantiles against each other. First, the set of intervals for the quantiles are chosen. A point (x, y) on the plot corresponds to one of the quantiles of the first distribution (x-axis) plotted against the same quantile of the second distribution (y-axis). If the two distributions being compared are similar, the points in the Q–Q plot will approximately lie on the line y = x. If the distributions are linearly related, the points will lie on a line, but not necessarily on the line y = x.

Usage

Q–Q plots are used to visually check the assumption of a distributional family (e.g., normality) against a sample data set. They can also be used to compare the distribution of a sample to a theoretical distribution or to compare two empirical distributions. This makes Q–Q plots a valuable tool in statistical analysis for assessing assumptions, models, and hypotheses.

Construction

To construct a Q–Q plot, the quantiles of the distributions to be compared are calculated. Often, one of the distributions is a theoretical distribution, such as the normal distribution, and the other is the empirical distribution of the sample data. The quantiles of the theoretical distribution can be calculated or obtained from a table, while the quantiles of the empirical distribution are calculated from the data. The points plotted on the Q–Q plot are then the ordered pairs of these quantiles.

Interpretation

The interpretation of a Q–Q plot depends on the conformity of the data points to the line y = x. If the points closely follow the line, this indicates that the distributions are similar. Deviations from this line indicate differences between the distributions. For example, if the points lie above the line at the lower tail, it suggests that the empirical distribution has more mass in the lower tail than the theoretical distribution. Conversely, if the points lie below the line at the upper tail, it suggests that the empirical distribution has less mass in the upper tail than the theoretical distribution.

Applications

Q–Q plots are widely used in various fields such as finance, meteorology, and medicine to validate models or assumptions about the underlying distribution of data. In finance, they can be used to assess the assumption of normality in returns. In meteorology, they can help in understanding rainfall patterns. In medicine, they can be used to analyze the distribution of biomarkers or other clinical measurements.

Advantages and Limitations

The main advantage of Q–Q plots is their simplicity and ease of interpretation. They provide a visual means to assess the similarity of distributions that is more intuitive than statistical tests. However, their effectiveness is limited by the subjective nature of visual interpretation, and they may not always clearly distinguish between distributions with subtle differences.