Proving an upward trend in line graphs with inconsistent increase Do you think this trend is increasing? I want to know if the proficiency test affects the performance of participants throughout the years.

How about this one?

If you think the trend is increasing, how do i prove it?
 A: I would suggest, you fit some (non-)linear model to your data in order to capture the general nature of the trend. I.e. you must beforehand specify what you call trend. For example you could say it is a linear trend. In this case you could fit a relationsship like
$$Y= \beta_0 + \beta_1*t+u $$
by emplyoing a linear model.
After that you can check how your resiudals change with time and infer how your data points deviate from the global trend. Furthermore you can fit the model without one point and look how the inclusion of this data point changes the slope of your trend. Be aware that you do not have much data points and hence no real hard statments are possible.
In your first picture a linear trend seems appropriate. The analysis could look like the figure below, showing the fitted trend on the left and the residuals on the right.

Here, a linear trend (red line) seems to apply pretty well for the first four observations and no strong indication for a deviation from the trend can be found, except for the second observation, in the residual plot on the right. However, if you compare the fit for the first four observations (red line) with the fit of all observations (the black line) it seems like the curve became steeper because of the additional data point in 2015. This gives some evidence, that the global trend is increasing.
This means, you can answer your question by checking how new data points alter the global trend. However, this is not a real proof, but only one way of thinking about the trend and its changes. BTW, the tag "graphical-model" seems misleading to me as already mentioned.
A: Your data can be represented as years (explicit) and ranks (1 $=$ lowest) allowing calculation of rank correlation. 
  +-----------------------+
  | year   first   second |
  |-----------------------|
  | 2009       .        1 |
  | 2010       .        4 |
  | 2011       2        2 |
  | 2012       1        3 |
  | 2013       3        6 |
  | 2014       4        5 |
  | 2015       5        7 |
  +-----------------------+

A Spearman (or separately a Kendall) rank correlation can be regarded as measuring how far the data trace out a monotonic increase (always rising) or decrease (always falling). Always rising implies rank correlation of exactly 1. I get Spearman correlations of 0.90 and 0.86 for these series, which match strong tendencies to increase, as are visually apparent. Some might want to decorate with observed significance levels or P-values. Given the preconception of increase, a one-tail test is arguably applicable if any is. 
It's a judgment call on whether such small datasets showing simple patterns need or much benefit from even this extra analysis. Conversely, more insight might be gained from any substantive knowledge of events or factors influencing the outcome. 
