Quantile Plot

Mean: 1

Variance: 1

Skew:

In order to check our statistical model it is often useful to use a quantile plot to assess how well the model fits the data. A quantile plot or Q-Q plot compares model quantiles with sample quantiles.

Define order statistics

y_{(1)} \leq . . . \leq y_{(n)}

to be the re-ordered data points

y_{1}, . . ., y_{n}

sorted in ascending order.

The

i^{t h}

th order statistic is defined to be the sample p-quantile with

p = (i - 0.5) / n

.

Now we have the sample quantiles we need the theoretical quantiles of the model. We denote the theoretical quantiles with Qi where

P (Y_{i} \leq Q_{i}) = \frac{i - 0.5}{n}

We can then produce a scatter plot of the theoretical quantiles against the sample quantiles.

        R-Code
        y <- rnorm(200,mean=0,sd=1)
        ordered_y <- sort(y)
        p <- ((1:200)-0.5)/200
        Q <- qnorm(p,mean=0,sd=1)
        dat <- data.frame(Q=Q,y=ordered_y)
        ggplot(dat, aes(x=Q, y=y)) +
        geom_point() +
        geom_abline()