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Testing 2.0 – A Crash Course in Statistics

Despite knowing that your child is bright, you are starting to see their grades slip, even with hard work and dedication in their classes.  They are growing increasingly anxious, frustrated, withdrawn, and school resistant.  The behavior is beginning to take a turn for the worse, teachers are getting worried, administrators have been calling home, and your child has been recommended for educational testing to provide metrics to help your child – not only to access the curriculum, but to feel better about themselves and re-engage in the school community.  


The testing results come back, and at first glance everything looks great.  They have a Full Scale IQ of 123 (Superior Range/Very High, 16th Percentile).  This is great news!  Your student has more intellectual horsepower than 94% of the population.  But then why are they struggling in school?!?  The answer may lay buried further down in the sub-test results, and a crash-course in statistics might help you understand their struggle, and what you can do to help.


Let’s start with the good, old-fashioned, Full Scale IQ Bell Curve.

testing your childs iq


This Bell Curve above demonstrates how data points in IQ testing spread out over the continuum, when the entire population is considered.  (It also shows the test’s normed Standard Deviation of 15 points, but more on that later…).  This is why a Full Scale IQ score of 100 means that you are “smarter” than 50% of the population, but also that you are “generally as smart as” 68% of the population, “still within intellectual striking distance” of almost 95% of the population, and are only “statistically significantly smarter” than roughly 2.1% of the population.  Access to the Brain Trust of Mensa requires beating out 98% of the population (above 131).  Hopefully this helps demonstrate why Einstein (160+) was so rare – scores like that only occur in less than .001% of the population.  That’s not a lot.


Now let’s look at how this Bell Curve is applied in the Weschler Intelligence Scale for Children, Edition 4, (WISC IV), which is likely a component of the testing your child has received.

standard deviation of iq scores


As you can see, the Average score is 100, just as was demonstrated above.  In fact, the “Average” category extends from 90 – 109.  Your child’s score of 123 falls comfortably into the “Superior/High” range .  But don’t get caught up in the score ranges, because their trouble in school might be hidden somewhere in the sub-tests, and in the statistical complexity of “Standard Deviations.”


Without taking you all the way back to your STATS 101 Class (and the all-nighters it took to get through it), let’s leave your student for a moment and use a simpler example, comparing the heights of two families of four.   


In Family #1 all four members are exactly 4’ tall.  The average is 4’, and there is a Standard Deviation from that height of zero.  The data points are well grouped, and nobody is going to stand out as either short or tall when the Family Holiday Card photo is taken.


Now let’s take a more complex family makeup.  In Family #2 Dad is 6’ tall, Mom is 5’ tall, the Daughter is 3’ tall, and the Baby Boy is only 2’ tall.  (6+5+3+2=16 and 16/4 = 4.) The average (”mean”) height for Family #2 is the same as Family #1 (4′), but with very simple observation of the photo on the front of the card, you will know which family is which, without needing to read the back. 


In Family #2, when you calculate each member’s height above or below the family average, the average difference in Family #2 from the mean is about 1’6”, which is their Standard Deviation (for you math geeks, I do “explain” this process at the bottom).  In Family #2, both Dad and Baby Boy are over one Standard Deviation from the Mean.  They are clearly Tall and Short, whereas Mom and Daughter look pretty average, due to their proximity to the mean.  Also notable, Mom is basically two full Standard Deviations taller than her Son, and the Dad is even taller than that!  That’s why Dad looks soooo tall in the Family Holiday Photo, and Baby Boy looks sooo tiny in his mother’s arms.  This is the power of Standard Deviations.


This is also why only using the average height of the family is a terrible way to decide how tall door frames should be.  While a 5’ door frame may actually be OK for Family #1, there are bound to be a lot of bonked heads if the same decision was made for Family #2, who has the exact same average height.  While the Average is an important number, understanding the Standard Deviations in the spread of the data is also incredibly valuable.


Now that we understand Standard Deviations, let’s get back to your child’s WISC IV testing, which (always) has a standard deviation range of 15 points.  Your child scored a Full Scale IQ of 123.  That is (still) exciting, but further investigation unearths a sub-test score of 93 in their Processing Speed.  Now, 93 isn’t bad.  In fact, it’s still well within the range of Average when normed against the entire population, and in the 45th Percentile overall.  If their Full Scale IQ was 98, or even 105, there would be less of a spread, the data would be more tightly grouped, and his scores would look common, more average, less variable – analagously like the Family #1 Holiday Card.


But that’s not your child.  In hopes of clarifying things, let me transpose the Intelligence scores to our height metaphor: Your child has a Full Scale IQ of 123 – his IQ is 6’ tall, and it’s standing next to a 3’ tall Processing Speed Score.  The rub for your child isn’t that the Processing Speed doesn’t work well (in fact, it’s still working better than a significant percentage of the population and well within range of most of the population), the rub is that his Processing Speed is two full Standard Deviations off from his Full Scale IQ (remember: when evaluating the WISC IV, 15 points is always considered one Standard Deviation).  The relational difference between the two scores is statistically significant, and it is this relational difference which is contributing to the challenges – both academically and socially that your child is experiencing.  This is visually represented below:

understanding your teens iq score

When your student is forced to perform time-pressured  tasks, juggle multiple incoming data streams, and make quick decisions, he is limited by the top end capacity of the most stressed portion of his neurology – in this case, the Processing Speed.


What does all of this mean for your child, as an academic student?


Allow them to take tests untimed, ask teachers to provide class notes before the lecture, and give your child with this type of “learning difference” a little bit more time to think about answers before they’re called on to speak.  In other words, remove the governor their Processing Speed has on their Full Scale IQ, and their performance will likely jump to the 6’ level.  In fact, when two or more data points are two Standard Deviations apart in educational testing measures, your child becomes eligible for Special Education accommodations, and in this case of a Full Scale IQ of 123 your child is actually what is termed Twice-Exceptional (they are both Gifted by nature of their 123 “Superior/High” intelligence, and they have an identified Learning Difference by nature of the 30 point, statistically significant, two Standard Deviations spread between the Full Scale IQ and the Processing Speed of 93).  Getting the accommodation this child needs to access innate abilities could be one of the greatest gifts you ever give them.  This is the power of reading the sub-test scores and understanding Standard Deviations.


And for you Math Geeks:  

Finding the Mean: 6+5+3+2=16 and 16/4 = 4

Individual Data Point Variances From the Mean: 6-4=2, 5-4=1, 4-3=1, 4-2=2

All Variances Squared and then Added: 2*2+1*1+1*1+2*2=10

Total Variance Found by Dividing by the # Of Data Points and Then Taking the Square Root of the Results: 10/4 = 2.5, 2.5=1.6 (‘ish)


About the Author

headshot of Jake WeldJake Weld holds a masters degree in education and has over twenty years of experience in traditional, LD, and therapeutic schools, adolescent and young adult programs, and conventional, wilderness, and residential settings. He has served as the Executive Director of a therapeutic boarding school, the Assistant Headmaster of a specialized LD boarding school, and as the Academic and Program Director of various schools and programs.  He is currently the Director of Admissions and Business Development for Mansfield Hall, a specialized college support program in Burlington, VT, and Madison, WI.