What is the significance of these scores? My brain doesn't work right anymore.  I have a traumatic brain injury and one of the many areas I have trouble with is math - an area that was once naturally easy and intuitive for me.  I remember standard deviations, generally, and have read up on what they indicate broadly and how they are reached.  When I was tested for my brain injury, there were some tests in which my abilities were worse than 3 SDs below the mean! [I am a professional with a master's degree, and was one of those students who aced everything with ease.  This information on my current condition, however, matches pretty well, I think, my current state, post injury.  I struggle greatly with many varied kinds of things.]  
So far as I can recall, 1 SD is a significant enough deviation that I don't recall anyone ever talking about anything greater than 2 SDs above or below the mean.  So my question is how "bad," may I ask, are scores such as: 0.7 SD below the mean, 1.7 SD below the mean, and  a score of 3.2 SDs below the mean.  I am asking for general but concrete advice on how to interpret these numbers generally, of course, and hold no one there responsible.  I just want to better understand the significance of my scores, mathematically. The brain injury staff mince and avoid answering my questions about this, so maybe someone here will help me understand this information.
Thank you kindly for your assistance.
 A: This is pretty low.  With a normal distribution, 99.7 percent of the population lies within 3 standard deviations from the norm (between -3 and +3), that means the remaining .3 percent is split between the top and bottom. 95 percent is with 2 standard deviations and 68 percent is within one standard deviation.
So:


*

*-1 SD would have 16 percent of the population below it.

*-2 SD would have  2.5 percent of the population below it.

*-3 SD would have only .15 percent of the population below.


Standard deviations are a way to represent what is called a normal distribution, so basically you have an average and the population is distributed on both sides.  There is not really a good way to compute it by hand, but it is typically looked up in a table or done using a calculator function.  It is essentially a bell curve, and the percentages correspond to the area under certain parts of the curve.


*

*$-0.3$ would have about 38% below

*$-0.7$ would have about 24% below

*$-1.7$ would have about 14% below

*$-3.2$ would has less than 1% below

A: Theoretically it means about a 1 in 700 level of low performance.  
Practically there is some variability in the measurement.  Maybe it could have been 2.5 or 3.4 standard deviations if measured at a different time, location, amount of sleep beforehand, or any number of other factors.  This leads to variability in the "1 in 700".   Think of it as a fuzzier "one in hundreds" level of low performance, where if large numbers of (perfectly randomly selected) people take the same diagnostic tests, between 99 and 99.9 percent will obtain a better result than that.
There is also a limit to the accuracy of the normal distribution ("bell curve") assumptions on scores.  For mental tests, extremely high or low outcomes are much more common than predicted by the bell curve that is used to convert the 3 standard deviations into a percentile or probability or "1 in 700" type of summary.  Doctors could avoid this by reporting results as a percentile, based on actual populations of patients or test-takers, and not theoretical normal distributions.  An approximation to that is to ask the doctors to quantify, within their experience of patients, what percentage are better or worse than this outcome.  They may be reluctant to give precise answers so as not to discourage patients, and rightly so. Placebo effects and unjustified confidence can help patients recover.
Finally, mental diagnostic tests, especially the ones used for people with brain injuries, tend to report multiple numbers from subtests related to memory, vision, language, etc, and not a single IQ-like score for overall performance.  Part of the point of this is to help identify which brain regions are more injured than others.
Having trouble with math may be an easier thing to overcome than other deficits, since calculators and computers might be able to fill in some steps.
