# Inferential Statistics Is an Unfit Tool for Interpreting Data
## 3 Two Examples on Interpretating \(f_{\rm o}\) Data in a Voice Training Program
In both figures, 180 Hz is illustrated as a horizontal dashed line because it has been suggested to be fundamental frequency range relevant for being perceived as a female speaker [24; 25; 26] which is the goal of the participants. Semitones are illustrated in the right-hand panel of Panel B as number of semitones above 110 Hz. To clarify the statistical significance test, the \(t\)-test formula is: \(t=\frac{\overline{x}-\mu}{\sqrt{\mu}}\) where \(\overline{x}\) is the sample mean, \(s\) is the sample standard deviation, \(n\) is the sample size, and \(\mu\) is the population mean. In this example, \(\overline{x}\) and \(s\) refer to the mean and standard deviation in the intra-individual difference scores. \(s\) and \(n\) are used to describe the dispersion in a hypothetical distribution of samples drawn from the hypothetical population described by the null hypothesis (via \(\mu\)). The \(t\)-value describes the distance between the hypothetical population mean and the obtained sample mean, using the standard error of the sampling distribution as its unit (\(\frac{s}{\sqrt{\mu}}\)). The \(p\)-value is the tail area of the hypothetical sampling distribution delimited by the \(t\)-value. The \(p\)-value describes the hypothetical probability of obtaining the sample data or something more extreme _if_ we assume that the sample is drawn from the hypothetical population described by the null hypothesis. All those hypotheticals are all there is to the statistical test. There is nothing in the \(t\)-tests formulae (or ANOVAs, linear or logistic regressions, mixed models, etc.) that describes what the values mean or what the study context is. Nothing specifies that the values are measuring\(f_{\circ}\), what \(f_{\circ}\) values are typical for male or female sounding voices, what the patients' goals were, etc. Thus, the statistical tests are completely blind to domain knowledge and the study context. A statistical test is not helpful for _interpreting_ data because \"it\" does not \"understand\" the research question. _Interpreting_ data is simply not what statistical significance testing is about. Further, \(p\)-values are unhelpful at answering even the more basic question of \"is there an effect?\" as compared to the more sophisticated question of \"is the effect of a meaningful size?\". This is because a small \(p\)-value only signals that the data does not conform with at least _one_ of the many theoretical, auxiliary, statistical, and inferential assumptions underlying its calculation. There is no reason to assume that it is the null-hypothesis-assumption that is the culprit, especially as we often know a priori that most of the many different assumptions include in the model are false to begin with [13]. To further illustrate the mismatch between statistical significance and actual data interpretation, we can use the same example study but with different example data; Figure 2, Table 2. We can see that the inter-individual differences in treatment trajectories are large; some participants had almost no change in \(f_{\circ}\), whereas others changed quite a bit. The result of a \(t\)-test applied on the _pre-post_ mean difference is \(t_{\gamma}=1.8\), \(p=0.12\). As the \(p\)-value is above 0.05 (the set \(\alpha\) level), the training effect is not statistically significant. At this juncture, the training would sometimes be interpreted as not effective _because_ the training effect was not statistically significant. However, what is striking--in the light of domain knowledge and research context--is that some of the transwomen (\(3/8=38\%\)) did have relevant changes in their \(f_{\circ}\) in the sense that they reached \(f_{\circ}\) levels typically associated