Statistics for Business
Because Q1 is the 25th percentile and Q3 is the 75th percentile, the interval from Q1 to Q3 contains the middle 50% of observations.
Because the box represents values between the mean minus one standard deviation and the mean plus one standard deviation, which approximately equals the middle 50% in large normal samples.
Because the box marks the smallest 50% of values when ordered from smallest to largest, which is equivalent to the middle 50% only if the sample is symmetric around the mean.
Because Q1 and Q3 are the 10th and 90th percentiles respectively, and so they bracket the central half of the distribution when combined with the median at the center of the box.