Analytical Methods Committee AMCTB No. 82

Received
1st September 2017

First published on 11th October 2017

Many inferences from statistical methods use the assumption that experimental data form a random sample (using that word in the statistical, not the chemical or physical, sense) from a population with a normal (Gaussian) distribution of measurement errors or other variations. In most cases this assumption is not actually tested, so if it is not valid false deductions may be made from the data. This Technical Brief considers cases where the normal distribution is generally taken to be valid, discusses how likely that is to be true, and how it is possible to test whether a data sample might come from a normally distributed population.

The main properties of the normal distribution are too well known to require summarising, but we should identify the many circumstances in which its occurrence is explicitly or implicitly assumed. Most obvious are in estimates of confidence limits and uncertainties, which use the fact that ∼68% of the results in a normally distributed population will lie within one standard deviation of the mean, ∼95% within two standard deviations of the mean, and so on. These characteristics are also routinely used to set up Shewhart-type control charts. Rather less obvious cases where we assume that data are normally distributed are in the application of important significance tests, including tests for outliers (see TB 69); in the calculation and use of calibration graphs, including the conventional definitions and estimation of limits of detection and quantitation; and in the numerous uses of analysis of variance. Further areas of relevance include many multivariate methods, and also cases where measurements can be converted to a normal distribution by transformation, such as the log-normal data that sometimes arise in clinical data from different patients. Analytical scientists may thus need reassurance that their data samples could come from a population with normally distributed variations.

Several established methods for testing for normality are available in many software packages. A simple approach is the use of normal probability paper, in which the individual measurements are plotted against their cumulative frequencies, the latter being on a non-linear scale derived from the percentage points of the normal distribution. Normally distributed data should yield a straight line plot: the Ryan-Joiner (RJ) test provides a correlation coefficient to evaluate the linearity. Fig. 2 shows a normal probability plot for a set of 8 measurements: the points lie on a curve, hinting at a departure from normality, but the RJ correlation coefficient is high enough to provide a p-value greater than 0.05, so we cannot reject the hypothesis that the data come from a normal population. The Kolmogorov–Smirnov method (which gives a similar conclusion for the Fig. 2 data) compares plots of the cumulative distribution function of the experimental data and the distribution expected of normal data: if the distance between the two curves is too great the null hypothesis of normally distributed data is rejected. Recently the main test methods used have been the Shapiro–Wilk test (another correlation-based method) and the Anderson–Darling test, the latter being a modification of the Kolmogorov–Smirnov approach with extra weight being given to the tails of the distribution. Monte Carlo simulations show the Shapiro–Wilk test to be marginally the most useful for testing for the normal distribution, having the best power for a given significance level. Excel® add-ins to perform it have been produced. The test statistic, W, is given by:

Fig. 2 Normal probability plot (Minitab®) for 8 measurements of the quinine level (ppm) in different batches of tonic water. |

In this equation the x_{i} values are the individual measurements, with mean , and the x_{(i)} values are the ordered measurements, with x_{(1)} the smallest, x_{(2)} the next smallest, and so on. The constants a_{i} are derived from the properties of the standard normal distribution and can be obtained from published tables. Small values of W are a sign of departures from normality, so the null hypothesis is rejected if the test statistic is less than the tabulated critical value. The Shapiro–Wilk test has been used with success with quite small data samples (though it is less effective if the data contain ties, i.e., equal measurements): when applied to the data in Fig. 2 it shows that the probability that they could come from a normal population is much greater than 0.05. Again it is clear that when we use only a few measurements any departures from normality would have to be quite gross before this or other tests could reject the null hypothesis of a normal distribution. It is for this reason that the central limit theorem provides such inestimable comfort!

An example of a Shapiro–Wilk calculation using Excel® is provided at http://www.real-statistics.com. Downloaded February 2^{nd} 2017.

James N. Miller

This Technical Brief was prepared for the Statistical Subcommittee and approved by the Analytical Methods Committee on 11/07/17.

This journal is © The Royal Society of Chemistry 2017 |