Mathematic Analyses about Error Bars using SD, SE and CI in Psychological Research Results

Error bars using the Standard Deviation (SD), the Standard Error (SE) and the Confidence Interval (CI) as measurement respectively can often be found in psychological research papers. However, some meaningful questions should be asked about these three kinds of error bar. Are there any differences between these three kinds of error bars? Or, are they just some equivalents that can substitute with each other? If there are some differences, then what are the differences? Moreover, is it sensible that all error bars of SD, SE and CI are taken as exchangeable in all situations? Are there any independent situations that they should be applied? In what situation should which error bar be used? Also, when error bars are presented, by what criteria can statistical significance be judged based on those error bars? Can error bars be used in research results of repeated measures? In fact, multiple issues about error bars still remain unclear. Thereby, clarification of these issues via mathematic theory analyses will be the main purpose of the current paper. Controversies over error bar applications have long been an unsettled issue in psychological field and other scientific domains. The current paper is devoted to clarify multiple misconceptions and confusions in error bar applications, through construction of triple trinity theories, especially the triple implicit metastases model in it. Based on construction of these models, not only the measure methods and application situations of error bar are clarified, but also the three implicit metastases processes are made clear which are easily neglected by some researchers. Ignoring the three implicit metastases processes plus some equivocal conceptualizations on dimensions of error bar measure and application situations may together contribute to the multiple misconceptions in error bar applications. By using some mathematical analyses based on these triple trinity models in the current paper, multiple misunderstandings and confusions in error bar applications can be uniformly integrated and interpreted.