TEST SCORES:
WHAT DOES MEAN, MEAN?
What was read on the internet: I like GE and AEs for all test scores and Lexile scores for
reading. Parents can understand GEs and AEs a lot better than it is for them to comprehend
Scaled Scores, Standard Scores and Percentiles.
What should have been said: Part Two
The Normal Distribution and the Mean
No, you do not need to be a statistician to understand terms used in educational and
psychological testing reports. With just a little bit of information you can make test scores make
sense. The bell curve or normal distribution, for example, is important to understand when
specialists’ reports are shared and when you are reading over reports on your own. Meetings
where test scores and data are discussed will serve you and your child best only when you can
actually participate. So, let’s start with the normal distribution.
The Bell Curve or Normal Distribution
The bell curve, or normal distribution, is named to describe the shape of the curve where the
ends of the curve nearly touch the horizontal line (called the X axis) and rises in the middle
looking somewhat like a “bell” in shape.
The idea behind the bell curve is to show how any item measured would look if all of the
measures (in the world) would be distributed from the lowest to the highest, smallest to the
largest, least often to the most often, for example, with the lowest measures on the left side and
highest measures on the right, hence, the term normal distribution. In the middle would be the
average of all of the scores collected and is where most scores/measures would be found—
somewhere in the middle. You may also hear the average as the “mean.” The terms are the
same, but only describe the scores, not the child.
To help understand how the normal distribution comes about, let’s consider the following
example. We will measure the height of all children in fifth grade in a particular school where
there just happens to be 100 children. We will most certainly find that one child is the shortest
and another is the tallest. If we were to then place the shortest child on the X axis, that child
would be to the far left and represent one child, in this example. The tallest child would be
placed to the far right and represent one child as well. We would then place the next shortest to
the right of the shortest child and the next tallest to the left of the child on the far right. Looking
again at the remaining children we discover that there is one more child with the same height as
the second child (next smallest) we just placed on the line and one child the same height as the
child to the far right (next tallest) that we just place on the X axis. So, now we have one child at
the extreme ends with two children, one place on top of the other, next to them.
If we were to continue to place the children by height along the horizontal, or X axis, and those
with the same height on the shoulders of other children with the same heights, we might just find
that we have most children of the same height smack in the middle. Stepping back to take in
what we have created, we would see the shape of a bell curve with the fewest children on top of
other children’s shoulders on either end and the most children somewhere near the middle.
Distribution of Heights of Fifth Grade Class of 100 Children
25
20
Frequency
20
18
18
15
12
12
10
7
7
5
1
2
2
1
0
50.5
50.75
51
51.25
51.5
52
52.5
52.75
53
53.25
53.5
Height in Inches
And if this was a normal distribution (how the scores are typically distributed across the
horizontal or x axis), we could take the heights of all 100 children and add those heights
together and then divide by 100 (the total number of children). We would find the resulting
number to be 52”—the mathematical average of all scores. And, lo and behold, that is the same
height where we have the most children stacked upon one another in the graph—the mean. Oh,
and “normal” is used statistically to show that if all conditions were met any data set collected
would be (normally) distributed in the same manner. Individual scores, however, are not looked
at as “normal” or “abnormal.”
In truth, all we have is 100 children in a particular 5th grade class which is a very small sample of
all of the fifth graders in our country. And if we really did select, at random (or by chance)
another fifth grade class of students, it is very unlikely that we would obtain a normal distribution
of all 5th graders because their heights would unlikely represent the heights of all children in fifth
grade across the country. In fact, in this hypothetical other 5th grade class, we might just find
that all children happened to be 52.3 inches tall and that would hardly be representative of 5th
graders across the entire country.
Test developers do not select such a small sample (100 subjects) because they know that
unless those selected represented the whole population of all fifth graders, the results they
would obtain would be inaccurate measures of all fifth graders. And so, they make sure they
have a test sample with the same percentage of boys, girls and of races representing this
country. They would also ensure that the test sample was representative of the country’s
economic status, those living in urban, suburban and rural areas and more. Doing so, ensures
that the 5th grade sample used by the test developers really does represent the country. Usually
these samples are of several thousands of children.
In the next posting, the focus will be on explaining how test scores, when compared to the
mean, make sense so that you can actively participate in the discussion of test results
If you have a topic you would like to have address please send me a note with the topic
and your reason for its need to be presented.
For more information on topics related to IEP development go to www.IEPHelp.com
and to order my book When the School Says No…How to Get the Yes! hop on over to
www.amazon.com/author/vklauer