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  farther to the left of the peak. A normal distribution is symmetric and has a skewness of zero. Later in this chapter we will see several examples of skewed distributions. A rule of thumb is that the distribution is reasonably symmetric if the skewness is between -0.5 and 0.5, and the distribution is highly skewed if the skewness is <-1 or >1.

If a data distribution is definitely non-normal, it might still be possible to transform the dataset into one that is normally distributed. Such a transformation is worthwhile, because it permits use of the parametric statistics above, and we shall soon see that parametric statistics are more efficient than non-parametric statistics. In some fields, transformations are so standard that the ordinary untransformed mean is called the arithmetic mean to distinguish it from means based on transformations.

The most pervasively suitable transformation is logarithmic: either take the natural logarithm of all measurements and then analyze them using techniques above, or simply calculate the geometric mean ($\overline{g}$): $\overline{g}$ = Σ(xi)1/N. The geometric mean is appropriate for ratio data and data whose errors are a percentage of the average value. If data are positively skewed, it is worth taking their logarithms and redoing the histogram to see if they look more normal. More rarely, normality can be achieved by taking the inverse of each data point or by calculating a harmonic mean ($\overline{h}$): $\overline{h}$ = N/Σ(1/xi).

Rejecting Anomalous Data
Occasionally a dataset has one or more anomalous data points, and the researcher is faced with the difficult decision of rejecting anomalous data. In Chapter 6, we consider the potential pitfalls of rejecting anomalous data. In many scientists’ minds, data rejection is an ethical question: some routinely discard anomalous points without even mentioning this deletion in their publication, while others refuse to reject any point ever. Most scientists lie between these two extremes.

My own approach is the following:
 * publish all data,
 * flag points that I think are misleading or anomalous and explain why I think they are anomalous,
 * show results either without the anomalous points or both with and without them, depending on how confident I am that they should be rejected.

In this way I allow the reader to decide whether rejection is justified, and the reader who may wish to analyze the data differently has all of the data available. Remembering that sometimes anomalies are the launching point for new insights, no scientist should hide omitted data from readers.

Here we will consider the question of data rejection statistically: are there statistical grounds for rejecting a data point? For example, if we have 20 measurements, we can expect about one measurement to differ from the mean by more than 2σ, but we expect (Table 2) that only 0.13% of the data points will lie more than three standard deviations below the mean. If one point out of 20 differs from the mean by more than 3σ, we can say that such an extreme value is highly unlikely to occur by chance as part of the same distribution function as the other data. Effectively, we are deciding that this anomalous point was affected by an unknown different variable. Can we conclude therefore that it should be rejected?