Significance of Differences between Means

The unpaired t-test is an example of a parametric method, which means that it is based on the assumption that the two samples are taken from normal, or approximately normal distributions. Generally, parametric tests should be used where possible because they are more powerful (effectively, more sensitive) than the alternative non-parametric methods [32]. However, significance levels obtained from parametric tests may be inaccurate, and the true power of the test may decrease, if the assumption of normality is poor. The non-parametric alternative to the unpaired t-test is the Mann-Whitney test [32]. In this test, a rank is assigned to each observation (1 = smallest, 2 = next smallest, etc.), and the test statistic is computed from these ranks. Obviously, the test is less sensitive to departures from normality, such as the presence of outliers, since, for example, the rank assigned to the smallest observation will always be 1, no matter how small that observation is. 

In the studies described immediately above, the samples are unpaired. This means that there is not a one-to-one correspondence between the observations in one sample and those in the other. Occasionally, such a pairing exists. For example, we might search the CSD for crystal structures that have been determined both by X-ray and neutron diffraction [43]. This will give rise to paired samples, since each observation from an X-ray study will have a counterpart from the corresponding neutron study. In comparing paired samples, the tests mentioned above (unpaired t-test, Mann-Whitney) can be replaced by their paired equivalents (paired t-test, see [31] Wilcox-on test, see [32]). When applicable, paired tests are more effective than unpaired tests, provided that the pairs of observations (x j, ) are positively correlated (i.e. x, tends to be large when j is large, and vice versa). This makes the variance of (.Xj-yi) less than the sum of their individual variances, and paired tests exploit this property. 

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The experiments described here are principally diagnostic in nature where cellular biomass was significantly enhanced in bottles after resource (iron or light) amendment, relative to control (or other) treatments, we infer that algal growth rates in the control (or other) treatments were limited by a deficiency in that resource. The statistical significance of differences between mean values of parameters measured in different treatments were assessed using a two-tailed r-test for comparisons between two treatments, or a one-way analysis of variance (ANOVA) for comparisons between three or more treatments, at a confidence level of 95% (P = 0.05). 

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