Thus, if you're teacher asks for the standard deviation and you calculate the MAD, you will probably get the wrong answer.

The important thing to remember, however, is that, in general, the MAD and the standard deviation will NOT be equal.

the variance is easier to work with than the MAD-though the exact reasons why this is true is beyond the scope of the video). However, statisticians usually prefer the variance/standard deviation versus the MAD because the MAD is not as 'mathematically tractable' as the variance (i.e. Just as there are different 'measures of central tendency' of a set of observations (such as mean, median, mode, etc.) there are also different 'measures of dispersion' (besides the MAD and standard deviation, another common 'measure of dispersion' is the 'Interquartile Range'). In fact, what you're describing sounds like what statisticians usually call the 'Mean Absolute Deviation', or 'MAD', for short (other names sometimes used are the 'average deviation' or 'mean deviation'). You're right-we could instead take the absolute value.