Two suggestions: the first is to use standard boxplots to show your data. The whiskers should show data, not some multiple of the CI. People are used to reading boxplots, so you will confuse if you use non-standard definitions.

The second point is even more important. You should show the data as well as the boxplots. Boxplots are descriptive statistics. It is also important to show the underlying data. Here is an example, made with Stata, using the user-written command -stripplot-. It shows the data and the boxplots. 

And a question for you: are you hiding behind the paper crane or showing it off??

Hello Ronán and thank-you for your reply!

I am indeed showing the data alongside the box & whisker - absolutely agree that it is vital for proper interpretation of the statistics!

Can you clarify what the 'standard' box/whisker conventions are? I use Origin to plot data, and it allows a number of options for both (SD/SE/95%CI/min&max etc etc). Was hoping to find a way of consistently displaying data to increase ease of comparison across the different data sets on a purely visual basis.

And in answer to your crane question, a bit of both I guess! :)

Hi Rosie,

In the following paper, the authors used box-plots for determining outliers and similarly to your question they used 25-75% for IQR and the value above and below than 1.5 times on the either side of the IQR is considered as an outlier.

Ng, W.W., Panu, U.S., Lennox, W.C., 2007. Chaos based analytical techniques for daily extreme hydrological observations. Journal of Hydrology, 342: 17-41.

I prefer the Shapiro-Wilk Test of Normality to show whether the distribution of data are normal. This test is more accurate. 

@Abdollahimohammad: A test on normality is worthless.

@Rosie: The standard way to show whiskers is that the end of the whisker is at the most extreme data point withing 1.5-times the IQR away from the box (this was Tukey's way to draw it, and Tukey invented this kind of plot). However, there are other ways (see e.g. http://en.wikipedia.org/wiki/Box_plot). In any case you should describe what the whiskers show.

I personally find mixing data and parameters not good (this includes the boxplot with whiskers showing the standard deviation [or the CI for the mean]). Either show the data (really the data, if there are

@Jochen, Is it mandatory in the Boxplot to indicate the mean etc which were not displayed automatically by the program. For instance, SPSS will not indicate the mean value in the boxplot but the median value. Is it necessary to indicate it alongside the median value etc.

Is boxplot useful for parametric distribution? I read somewhere that boxplot is only used to describe non-parametric data. How true is that?

I don't know what is "neccesary" or not. Tukey's boxplot shows the median, the IQR, the whiskers, and the data outside the whiskers. These parts are "mandatory" I would say. I don't know who had the idea to additionally present the mean in the boxplots. This is surely not "mandatory", but some autors seem to think that this adds relevant information. Also, other measures could be added, like the geometric mean, the harmonic mean, etc. It simply depends on what the authors find important.

I have to add something to my last post:

it is ok to show both, data and parameter estimates (if the parameters are for the location), but I find it puzzeling when summaries for the data are shown (i.e. boxplots for instance) along with parameter estimates (that are often mistaken as summaries).

Showing the data together with the boxplot seems also funny to me. The boxplot is to summarize the data, when there is too much data to be shown in a reasonable way (either too much data per group or too many groups). If the data is shown anyway, then I do not understand why additionally a boxplot is shown at all.

@Jochen. Thank you sir. But you seems to forget my last question. I asked if it is only non parametric that boxplot can be applied? I read in a paper that boxplot is primarily to describe non parametric data. Is there any truth in that statement?

There is nothing like "nonparametric data". Data can be categorical, ordinal, or interval/ratio scaled. Whenever quantiles/quartiles are defined for the data, the data can be graphically summarized by boxplots (thus, categorical data can not be represented in boxplots, other data can).

I think the reasoning behind your question is this:

Data is often summarized by the mean and the standard deviation (sd). These are interpreted as the parameters of the normal distribution, and within this interpretation they only make sense when the distribution of the data is Gaussian. Therefore, some people seem to think that it is inappropriate to present mean and sd when the distribution of the data is non-Gaussian.

I think this in fact correct for the sd, but wrong for the mean, because the mean is still a sensible estimate (except for some pathological distributions where the mean is not defined). For many models that are not based on Gaussian data, for instance binomial or Poisson data, the mean has a well-understood "physical" meaning. In any case, the mean has a particular statistical meaning, and this is independent of the shape of the distribution.

The more problematic part is the dispersion parameter (e.g. the sd). The sd is hard to interpret when the distribution is non-gaussian, but it still has a meaning (at least for those distributions with finite variance). The problem is that there is no sensible way to graphically represent the sd when the data is non-Gaussian. More sensible, especially for non-Gaussian distributions, is it to provide the confidence interval (ci) as an uncertainty measure for the mean. The way this ci is calculated depends on the distribution of the data, what determines the statistical model (of which the mean is the maximum likelihood estimate). Note that the mean may not be in the center of the the ci!

Finally, in large samples, the likelihood profile (used to get the ci) approximates the normal distribution (central limit theorem). Then the ci can be approximated using the normal distribution. For historical reasons, often not the ci is shown in the diagrams but an intermediate result, the standard error (se) is shown. The ci is obtained by multiplying the se with the respective quantile of the t (or z-) distribution providing the desired level of confidence.

So I'd say: when the distribution has a defined mean and a finite variance, the mean and its ci can always be shown (if desired).

Thanks all for your input - I think I will stick with my box-plots as described above, with the additional mean line, and re-consider whether I always need to show both data and boxes - I understand how it could be seen as unnecessary, but equally, I feel it supports the text better if you can quickly see the summary statistics and how they fit in the overall spread of the data

@Jochen - One more question... one of the parameters I am looking at is continuous (duration measured in seconds), yet shows non-parametric distribution, so I have been stating median rather than mean - from what you said, it sounds as if the mean (95%CI) is still appropriate. Best to switch? Thanks again for your valuable input!

Thanks, Jochen.

Rosi,

the mean and its ci are statistics of any distribution*. The mean is simple to estimate. The ci may not be so simple to get. Most statistics software have some function to calculate the mean and the ci, but they often pretend or assume that the data comes from a normal distribution (i.e. the ci is calculated based on the t-distribution). If the data comes from a different distribution, the ci is likely calculated wrongly (check the documentation of the software you are using).

When you know the distribution model, there is usually also an analytic solution to get the ci (as there is one for the normal distribution). When you do not know the distribution model, the ci can be obtained by bootstrapping.

When the sample size is large (i.e., >30*), the central limit theorem ensures that the ci can be approximated using the t-distribution (no need to think about the model or about bootstrapping).***

I wonder why you carelessly (as it seems) exchange median and mean. The (sample) mean estimates the expected value. This has a particular meaning. Often, different groups are compared and one is interested in the expected difference (between the groups). The median has a completely different meaning. It is an order statistic and has, primarily, nothing to do with expectations. Switching between these statistics means switching between your research questions, and this look quite arbitrary...

If you simply want to provide a statistic for the "mass center" of the values you should better use the mode (there are method to estimate the mode even for continuous data), since this gives you the value at which you have the highest density of values (given the distribution is unimodal).

*(with finitie mean and variance; the Cauchy distribution would be an example where neither mean nor variance are defined)

** given the distribution is not "pathologic" like very heavily skewed; the higher the assymetry, the larger needs the sample size to be to get a resonable approximation; on the other side, for non-normal but symmetric unimodal distributions, a sample size of 5 is already sufficient for most purposes.

*** good bootstrapping also requires large sample sizes. There is not much fun in bootstrapping a ci from 5 values. Thus, whenever bootstrapping becomes resonable, the central limit theorem allows a good approximation using the simpler standard procedures (t-distribution).

Hi Jochen - I don't use them interchangeably ;) More that I have been working under the assumption that continuous data should be reported as mean (95%CI or SD/SE), whereas non-parametric and non-continuous data should be represented with median (IQR) when discussing in text. As mentioned, the dataset causing me issues is continuous but requires non-parametric treatment.

Is the 'mode' the most commonly found value or something a bit more complex as you seem to suggest? I'm only really familiar with mean, median, and geometric mean...

Basically, are you saying mean (95% CI) is an appropriate way of representing both 'normal' passed and failed continuous data?

This could well be another example of the assumptions non-statistically-trained biologists sometimes make!

Hi,

there still is no "non-parametric data". The term "parametric" is related to an analysis (not to data), where the analysis is based on a model and the model is determined by the values of (model-)parameters. With "model" often two things are meant that need to be distinguished: a "scientific" model describing the relationships between variables (the parameters here are also called "coefficients" of the model and measure the strengths of the effects of predictors on a response), and a "stochastic" model or error model that accounts for observed differences between the theoretic predictions (based on the scientific model) and the real observations. The error model also follows from theoretical considerations and is technically represented by a probability distribution (like the normal distribution or the binomial distribution or the Poisson distribution or the exponential distribution or the gamma distribution or...).

The reported mean and CI/SD/SE represent the results of an analysis of the data with a particular aim (the mean is an estimate of the expected value, so showing the mean indicates that this is the aim; please read about the meaning of "expected value" to get the idea behind it. The estimates can be very easily obtained from many parametric models. For a simple model with no predictor and gaussian errors this is identical to the only coefficient (also called the "intercept" of the model).

In contrast, providing some quantiles is done to give an impression how the data is distributed. This has (so far) nothing to do with a particular analysis or some scientific model and error model. It is only to show "where the data is", what range is covered, where most of the data is.

So the arbitrariness I see is that you separate continuous data from discrete data and then you provide model results for the continuous data but a data description for the discrete data. Why do you not provide a data description (or model results) similarily for all you data? Why do you make this distinction what is more important for the different kinds of data, why did you place this "borderline" between continuous and discrete data?

Regarding the mode: this is simply the "most frequent value". This is directly visible in discrete data. For continuous data this is slightly more difficult to define the mode. One can classify the data into discrete groups (classes) and then the group containing the most values is the mode. But the definition of the classes is arbitrary. It is better to use a smooth density estimator and define the mode as the quantile of maximum density (see e.g. http://en.wikipedia.org/wiki/Kernel_density_estimation).

There are more sophisticated methods available, e.g. see

DR Bickel, “Robust estimators of the mode and skewness of continuous data.” Computational Statistics & Data Analysis 39:153-163 (2002).*

Finally: "Basically, are you saying mean (95% CI) is an appropriate way of representing both 'normal' passed and failed continuous data?" - Yes, I do. But please note that the CI has to be calculated based on the correct error model, or when this is not known, it has to be determined by bootsrapping.

* a function ("half.range.mode") is available in R in the package "genefilter".

I have a technical question to all of you: I usually represent the data in form of boxplots. However, while performing satistical tests, I first perform a 1-way ANOVA with Bonferroni-Holm posthoc if the condition of equality of variance is satisfied. Otherwise, I perform the non-parametric Kruskal-Wallis test with Dunnett´s posthoc test. I utilise the posthoc test results from 1-way Anova or from KW-test to represent statistically significant differences between data samples in the box plot. Is it okay to use the significance results from 1-way ANOVA on the box plot? Or you suggest me to use always KW test and Dunett´s test for box plots and only 1-way paramteric test for bar plots?

I suggest to NEVER use barplots at all.

There is no law of nature that would dictate what kind of plot is to be used for a particular test. Test and plot have actually little to do with each other. It is always better to show as many features of the data as possible (if this does not interfere with clarity). In simple one-way designs it is often possible to show the actual data without loss of clarity, rather than summaries (be it median, mean, SEM, SD, IQR or whatever). Showing the data shows all aspects that are known - we can see what typical values are (location), how much they spread out (dispersion) and we may be able to recognize further properties of a likely distribution (symmetry, modality). Additionally, we can directly see how much data there is, and thus how (un-)certain we would be judging higher-order properties/moments of the distributions (i.e. anything else beyond location).

So I would advice showing the data, if clarity permits it (1). Otherwise, if the amount of data per group is large, I'd advice violin plots (2). For low amounts of data, catepillar plots (3) should suffice, where the error bars may indicate the total range or the IQR. Boxplots are fine if there exists a larger amount of data per group (>20-50 values, say) and if it is known or a reasonable assumption that the distribution of the values within the groups is unimodal.

https://i0.wp.com/lh6.googleusercontent.com/-pQ80HZSJVI0/TYqJztV1bMI/AAAAAAAAABU/QZbtVgv7r74/s1600/beeswarmfig00.png?ssl=1

http://www.nature.com/ng/journal/v48/n4/images/ng.3511-SF13.jpg

http://bit.ly/2lI1VJE

Picking up on this rather late how do you add the mean line in a box plot in Stata