I am trying to figure out how to interpret the "Per Base Sequence Quality"? What does Position in reads (bp) mean? Also in order to draw this box-plot graphics, more than one quality scores are needed so we can sort them and calculate the mean, median and quartiles. In this diagram, how these values were calculated for each position?
There is a pretty good description here. The position in the reads means literally the position in the reads, so position 2 means the quality of all bases on position 2 in the fastq file (all reads). In the description link, you can find how the boxplots and blue and red lines are calculated (and what they mean). They are all based on the total quality values of all bases on a certain position, all bases means all bases of all the reads on a particular position.