Standard Error vs Standard DeviationOne of the most important, yet sometimes misunderstood, statistical distinctions is that of the difference between standard error and standard deviation, especially when using them as error bars in graphs. The main, practical distinction between the two is that standard deviation gives you an idea of how the data is spread in an experimental data set (these are used in descriptive error bars) while standard error is an estimate of how variable the mean will be after the experiment has been repeated multiple times (these are used in inferential error bars) . To state the difference more practically, standard deviation should be used to visualize the distribution of the data in a single experimental (and can be used for comparing single data points to the experimental data spread), while standard error should be used when you want to compare means of experimental groups, such as treatment vs placebo. Including standard error bars in figures is helpful because it gives the viewer an estimate of statistical significance for the differences between means (a lack of overlap in standard error bars between groups suggests statistical significance). Finally, no matter what bars are included in graphs, they should always be labeled in the figure legend so as to properly inform the viewer. See reference  for a more comprehensive review.
|A table of commonly used error bar calculations. This is found in reference .|
Normally Distributed Data: t-test vs Wilcoxon-Mann-WhitneyAnother important aspect to considering when performing statistical tests is whether or not the data set is normally distributed. This is important because many of the common elementary statistical tests, such as the t-test, assume a normal distribution and will yield inappropriate results if used on a data set that is not normally distributed. Because they make assumptions on the distribution shape of the data, these tests are considered parametric statistical tests. In order to justify the use of such statistical tests, I test the normality of my dataset. There are tests that can be run to determine whether a set of data is normally distributed, with one common test being the Shapiro-Wilk Normality Test. Histograms can also be used to visualize the data set distribution and gain an idea of whether or not the data is normally distributed (it is often considered "normal" is the data is distributed in a bell shaped curve).
While there are multiple ways to deal with non-normal data sets, I am going to briefly mention one. It is common, when dealing with non-normal data sets, to use non-parametric statistical methods, which are methods that do not assume normally distributed data (briefly, non-parametric statistical methods involve ranking of the values so that the order of the values is used instead of the values themselves). In most cases, elementary parametric statistics have non-parametric equivalents. An example of this is the Wilcoxon-Mann-Whitney test which can be thought of as a non-parametric t-test. Both of these tests make certain assumptions and have advantages over the other, such as how the t-test is more sensitive than the Wilcoxon test to revealing subtle statistical significance but requires a normally distributed data set . Overall, it is important to pay attention to the assumptions, limitations, and benefits of the statistical tests you are using. I also want to reiterate how these topics are more involved than what I am discussing here, so please refer to my Works Cited, and especially reference  for this section, for further reading.
Normally Distributed Data: Data Visualization
|An annotated example of a notched box plot.|
ConclusionsStatistics can be a confusing field with even some of the most common tests being complicated and involved. Especially in fields like the sciences, which rely on even basic statistics, it is important to understand the assumptions different tests make and the limitations they have. Hopefully this post provided you with some increased insight into the common statistics that you can now apply. If you want to call me out on any mistakes or typos, or if you have any questions/comments, please feel free to leave a comment below.
1. Cumming G, Fidler F, Vaux DL. (2007). Error bars in experimental biology J Cell Biol. DOI: 10.1083/jcb.200611141
2. Fay MP, Proschan MA (2010). Wilcoxon-Mann-Whitney or t-test? On assumptions for hypothesis tests and multiple interpretations of decision rules Stat Surv DOI: 10.1214/09-SS051
3. Olsen CH (2003). Review of the use of statistics in infection and immunity. Infection and immunity, 71 (12), 6689-92 PMID: 14638751
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