QQ Plots Stat 342, Spring 2014 Prof. Guttorp - TA Aaron Zimmerman

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1 QQ Plots Stat 342, Spring 2014 Prof. Guttorp - TA Aaron Zimmerman To get you started, remember that that a q-q-plot plots (Fn 1 (p), F0 1 (p)) for p (0, 1), where Fn 1 (p) = inf{y : F n (y) p}, where F n (y) is the empirical CDF F 1 0 (p) is the inverse of the CDF of fixed CDF F 0. You may find the following table useful. The X values in the table correspond to the expected order statistics drawn from a standard normal distribution with n = 7. That is, X (i) is the expected value of the i th smallest observation (out of n observations) drawn from a standard normal distribution. X (1) X (2) X (3) X (4) X (5) X (6) X (7) x P(Z < x) In words, what is a q-q-plot illustrating? If the observed data came from a distribution similar to F 0, we should be able to match quantiles. That is, we d expect our smallest observation from our n-sample to be similar to the expected smallest observation (from a sample of size n) from F 0. In this case, the q-q-plot will be linear. It shows us the relationship between our sample quantiles and the quantiles from F 0. If the empirical CDF was the same as F 0, we would see all the the points on the q-q-plot fall on a straight line. You re given the following sample of 7 datapoints: {0.48, 1.39, 0.60, 1.47, 0.82, 0.46, 0.02}. Plot a normal q-q-plot with this data. What does the picture tell you? Figure 1: The plot seems linear and it appears as if the sample could be from a standard normal distribution. 1

2 Here are seven qqplots from a sample of size 144. What can you say about the distribution of each of the samples. Be as detailed as possible. It may be useful to draw a potential distribution that could have created the sample. The theoretical quantiles come from a standard normal. The lines on the plots are all the function y = x

3 Here are the histograms of the samples used to plot the q-q-plots. Notice how heavy-tails, light-tails, and different types of skew affect the q-q-plots. N( 4, sd=1) N( 4, sd=5), slope= x.shift x.scale t(df=2) Unif( 1.5, 1.5) x.heavytail x.lighttail Log Normal Flipped Log Normal x.posskew x.negskew Bimodal mixture of 2 normals (x.bimodal mean(x.bimodal)) 3

4 At least some of the samples (n = 6 for each of them) came from a standard normal distribution. Circle which plots you think came from normal samples. They all come from a standard normal! I didn t look for crazy ones either. This is just 16 samples of size 6 from a standard normal

5 We could also use a q-q-plot to check if the two samples came from the same distribution. Describe how you might do that. Instead of comparing each sample against a reference distribution, we can compare the quantiles from each sample against each other. This let s us explore whether or not the two samples could have come from the same distribtion. To do this, we don t even have to hypothesize a specific distribution that they each came from, we just need to plot (Fn 1 1 (p), Fn 1 2 (p)). Draw a possible q-q-plot that that could arise from comparing a Unif(-1,1) sample against a Unif(-2, 2) sample. Unif( 1, 1) Unif( 2, 2) Draw a possible q-q-plot that that could arise from comparing a Unif(0, 1) sample against a χ 2 (1) sample. Chi Sq(1) Unif(0, 1) 5