How to use boxplots to find the point where values are more likely to come from different conditions? I have plotted some data using box plots.
I am comparing Condition 1 (left) and Condition 2 (Right) values. My aim is to find a point at which we make a decision where the value changes from point Condition 1 to Condition 2. 
Will this conclusion make sense, if I say if I do the experiment again and get any value than the median of Condition 1, then it is likely that the value will be of Condition 2? 
Or is there any other way which I can represent this data  to to get conclusion that if I get random value I can say if it is from condition 1 or condition 2? 
Data presented as code for R input:
Cond.1 <- c(2.9, 3.0, 3.1, 3.1, 3.1, 3.3, 3.3, 3.4, 3.4, 3.4, 3.5, 3.5, 3.6, 3.7, 3.7,
            3.8, 3.8, 3.8, 3.8, 3.9, 4.0, 4.0, 4.1, 4.1, 4.2, 4.4, 4.5, 4.5, 4.5, 4.6,
            4.6, 4.6, 4.7, 4.8, 4.9, 4.9, 5.5, 5.5, 5.7)
Cond.2 <- c(2.3, 2.4, 2.6, 3.1, 3.7, 3.7, 3.8, 4.0, 4.2, 4.8, 4.9, 5.5, 5.5, 5.5, 5.7,
            5.8, 5.9, 5.9, 6.0, 6.0, 6.1, 6.1, 6.3, 6.5, 6.7, 6.8, 6.9, 7.1, 7.1, 7.1,
            7.2, 7.2, 7.4, 7.5, 7.6, 7.6, 10, 10.1, 12.5)

Each condition has 39 values. 

 A: 
Here is one of many possibilities. Back in 1979, Emanuel Parzen suggested hybridising the quantile plot and the box plot. Some references are given below. Clearly, the box of the box plot shows median and quartiles, which are just key quantiles. Showing all of the data, namely all the quantiles or order statistics, is entirely possible, at least with a small number of groups (as in this thread) and a small or moderate number of observations (as in this thread too). In fact the design extends quite well to larger sample sizes. Outliers, granularity, ties, grouping and gaps (whichever way you want to think about such features) are always evident as well as general level, spread and shape. The graph is not subject to artefacts or side-effects of arbitrary rules of thumb such as what is or is not within 1.5 IQR of the nearer quartile. Conversely, it may offer too much detail for some tastes, but faced with a less than ideal graph one just moves on. 
It is reasonable to point out that quantile plots are just cumulative distribution plots with axes reversed, although they are more often shown as point patterns than as connected lines. 
Cox (2012) reported one Stata implementation and his stripplot (Stata users can download from SSC) offers another. Implementation should be trivial in any major statistical or mathematical software. 
I think this kind of display offers much more detail than a conventional box plot, which here does not fully exploit the space available. A conventional box plot can be helpful for 10-100 groups or variables, where some severe reduction of the data may be needed, but it throws out possibly interesting fine structure for the common few-group or few-variable case. 
Another key virtue of this graph is that it echoes the elementary but fundamental fact that just as half the values are inside the box, so also half the values are outside the box (and often the most interesting or more important half). I've seen even experienced statistical people misled by the stark contrast between fat box and thin whiskers. The classic illustration of this is any U-shaped distribution or any distribution with two big clumps of approximately equal size. The box will then be long and fat and the whiskers short and thin. People often miss the fact that such whiskers  are hiding the highest densities. Tukey (1977) gave an example of this with Rayleigh's data. 
In this case and in many others logarithmic scale is used. In principle, the quantile-box plot is easily compatible with any monotonic transformation, as the transform of the quantiles is identical to the quantiles of the transformed values. (There is some small print qualifying that, arising because median and quartiles may be be produced by averaging of adjacent order statistics, which doesn't usually bite.)  
I don't offer herewith any kind of graphical substitute for a significance test. This is an exploratory device.  
Cox, N.J. 2012. Axis practice, or what goes where on a graph. 
Stata Journal 12(3): 549-561. .pdf accessible here
Parzen, E. 1979a. 
Nonparametric statistical data modeling. 
Journal, American Statistical Association 74: 105-121. 
Parzen, E. 1979b. 
A density-quantile function perspective on robust estimation. 
In Launer, R.L. and G.N. Wilkinson (Eds) Robustness in statistics. 
New York: Academic Press, 237-258. 
Parzen, E. 1982. 
Data modeling using quantile and density-quantile functions. 
In Tiago de Oliveira, J. and Epstein, B. (Eds) 
Some recent advances in statistics. London: Academic Press, 
23-52.
Tukey, J.W. 
Exploratory Data Analysis. 
Reading, MA: Addison-Wesley. 
