how is a normal probability plot used to detect outliersSolu

how is a normal probability plot used to detect outliers

Solution

Normal probability plot can be used to detect outliers only if the underlying distn is known to be normal. So we have a data set X1, X2, X3, ... , Xn which is knownto be normal. Now we want to test if there is any outlier in the distribution. Firstly arrange the observations in ascending order. Now calculate the probabilities Pi = (i-0.5)/n where i = 1(1)n. now pi gives the probability that a value is less than the ith value xi. Using the inverse cdf of normal distribution we find the yi value where N(0,1) cdf is pi. Now if the distribution of Xi is normal then Xi should be very close to Yi. We proceed to plot all such pair(Xi,Yi). We also plot a 45 degree line. Now if the plotted points all lie close to he 45 degree line then it means that the underlying distn of Xi s is normal and there are no outliers. However if a particular point in the plot lies wayoff the plot of the others then that point is taken to be an outlier. However if there is no underlying knowledge that the data set is normal then plotted points may come hayward just because of plotting a different distn against normal distn values. However if the underlying distn of Xi is known to be normal and a particular pair (Xj,Yj) is way outside the 45degree line then that Xj is said to be an outlier.