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misled by using what are called ‘parametric statistics’, i.e., statistics that assume a Gaussian distribution of errors. This section is organized in the same sequence that most data analyses should follow:

1. test the data for normality;
2. if non-normal, can one transform the data to make them normal?
3. if non-normal, should anomalous points be omitted?
4. if still non-normal, use non-parametric statistics.

Normality Tests

Because our statistical conclusions are often somewhat dependent on the assumption of a normal distribution, we would like to undertake a test that permits us to say “I am 95% confident that this distribution is normal.” But such a statement is no more possible than saying that we are 95% certain that a hypothesis is correct; disproof is more feasible and customary than proof. Thus our normality tests may allow us to say that “there is <5% chance that this distribution is normal” or, in statistical jargon, “We reject the null hypothesis of a normal distribution at the 95% confidence level.”

Experienced scientists usually test data for normality subjectively, simply by looking at a histogram and deciding that the data look approximately normally distributed. Yet I, an experienced scientist, would not have correctly interpreted the center histogram of Figure 2 as from a normal distribution. If in doubt, one can apply statistical tests of normality such as Chi-square (χ²) and examine the type of departure from normality with measures such as skewness. Too often, however, even the initial subjective examination is skipped.

We can use a χ² test to determine whether or not our data distribution departs substantially from normality. A detailed discussion of the many applications of χ² tests is beyond the scope of this book, but almost all statistics books explain how a χ² test can be used to compare any data distribution to any theoretical distribution. A χ² test is most easily understood as a comparison of a data histogram with the theoretical Gaussian distribution. The theoretical distribution predicts how many of our measurements are expected to fall into each histogram bin. Of course, this expected frequency [Nf(n)] for the nth bin (or interval) will differ somewhat from the actual data frequency [F(n)], or number of values observed in that interval. Indeed, we saw in Figure 2 that two groups of 50 normally distributed measurements exhibited surprisingly large differences both from each other and from the Gaussian distribution curve. The key question then is how much of a difference between observed frequency and predicted frequency is chance likely to produce. The variable χ², which is a measure of the goodness of fit between data and theory, is the sum of squares of the fractional differences between expected and observed frequencies in all of the histogram bins:

$\chi ^{2}=\sum _{n}([Nf(n)-F(n)]^{2}/Nf(n))$ (3)

Comparison of the value of χ² to a table of predicted values allows one to determine whether statistically significant non-normality has been detected. The table tells us the range of χ² values that are typically found for normal distributions. We do not expect values very close to zero, indi--