# How to Construct an Empirical Quantile-Quantile Plot

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Tags: , How to Construct an Empirical Quantile-Quantile Plot

Purpose: To provide a graphical comparison of two sets of data.

Procedure:

1. Rank order the data in both sets of data.
2. If the sample sizes are the same, plot ordered values from each set of observations against each other.
3. If the sample sizes are not equal, convert to quantiles and plot the corresponding quantiles against each other. Start with the group with the smaller sample size and try to match the scores with the same or similar quantiles. We will have to use interpolation to match the quantiles. The maximum number of points will be the smaller sample size. [See Chambers, Cleveland, Kleiner, and Tukey (1983) for more on this procedure.
4. Draw a line with a standardized slope of 1. This line represents completely overlapping distributions.
5. Compare our data to the line from step 4.
1. If our data overlap this line then the two sets are similar.
2. If our data are parallel to this line but displaced either above or below it, then the treatment effect is additive. When our data are above the line, the vertical displacement from the line is the size of the treatment effect. When our data are below the line, the horizontal displacement from the line is the size of the treatment effect.
3. If our data are on a straight line which is not parallel to this line, then the treatment effect is multiplicative. This also means that one set has a smaller variance than the other.
4. We can easily detect outliers as values that deviate from the straight line for our data.

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