![]() Hence, the NNPs are not very informative for small sample size I, as is the case for other diagnostic plots it may be extremely difficult to detect deviations from normality even though the deviation exists, unless departures from normality are large. In addition, the random sampling variation of the sample order statistics will also produce random deviations from a straight line. Note that some model defects such as heterogeneous variances, outlier with unusually large residuals, and model misspecification (lack-of-fit) can also produce similar shapes as nonnormality. In that case, the shape of the plot may suggest the nature of the nonnormality (see, e.g., Rawling et al., 9 p 356). ![]() A strong deviation from a straight line indicates a violation of the normality assumption on the ɛ i. Then, the slope of the line becomes σ.) If this is the case, one can conclude that the variable of interest is normally distributed. (Some references flip the axis and plot the probability scale on the horizontal axis. Then, this plot should display approximately a straight line with slope 1/σ. Hence, if the normality assumption holds, the studentized residuals will be a sample from a normal distribution and the ordered residuals should be similar to the ordered normal scores. If we are sampling from a normal distribution N(0, 1), these values are called the normal order statistics, and can be obtained as z (i) = Φ −1( p), where Φ −1 is the inverse function of the cumulative normal distribution with μ = 0 and σ = 1, and p = ( i − 0.5)/ I, i = 1,…, I is the probability that an observation from that distribution is smaller than z ( i). If we do repeated sampling, the average (more concretely, the expected value) of the ith value is the ith-order statistic for the probability distribution being sampled. ![]() If we take a sample of size I from a probability distribution and we order the I values, the ith-smallest value is called the ith-order statistic of that sample. In an NPP of residuals, on the horizontal axis we plot the data ordered in ascending order, in this case the ranked studentized residuals, and in the vertical axis the normal scores (also called normal order statistics) z ( i) for a sample of size I. Among the probability plots, the NPP 111 is used for assessing if data are approximately normally distributed. Departures from this line indicate departures from that distribution. The data are plotted against a theoretical distribution in such a way that, if the plot is a straight line, it is reasonable to assume that the observed statistical sample comes from the specified distribution. ![]() So the message here is, there are very few cases where non-normal data should stop a project from moving forward.A probability plot is a simple tool for determining whether or not a data set follows a hypothesized distribution. Once the underlying causes are understood, process redesign and process control are much greater assurances of zero defects over the long run than the fact that a sample taken from the population happened to be normal and capable at one point in time. It is far better for a team to put its energy into learning the underlying causes of variation than to get wrapped up in finding the correct distribution or transformation method to make defect-rate predictions. ![]() Statistical techniques are available for dealing with non-normal data, but we’d like to bring some “real-world” perspective into the discussion from a Six Sigma practitioner’s viewpoint – Six Sigma practitioners get paid to reduce variation, not to model variation. Note that the histograms are as indicative of normality (or non-normality) as the probability plots in these cases.ĭefect-Rate Predictions and Non-Normal Data Here are some examples of normal and non-normal data (made into histograms), and their corresponding probability plots (generated with MINITAB software). The blue line on the chart reflects a perfectly normal distribution: The normal probability plots below show data values along the x-axis, versus the cumulative percentage of data points collected, on the y-axis. Normal probability plots can take different forms, but all have one thing in common : the closer the data points are to the theoretical-normal line, the more likely it is that the data is normal. ![]()
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