Lions and Tigers and Test Scores, Oh My!

NORMAL DISTRIBUTION AND OTHER TEST SCORES

Different types of scores can be reported on the same test, but that is not an issue if we have the following Normal Distribution chart with us. To show that, let’s look at some examples of different test scores that may be reported to you, beginning with percentiles.

PERCENTILES or PERCENTILE RANKS

Recall that the normal distribution and IQ scores had equal intervals of measure. Two test scores that I want to address are not with equal intervals, meaning that you cannot compare the difference between two sets of two numbers and get the same difference. The first is percentiles.

On the Normal Distribution graph, the second horizontal line is titled Percentile. Percentile scores, or rank, are a measure of frequency and are often reported scores, but when they are, can cause a great deal of confusion. The first thing to know is that a percentile score does not represent a measure of correct answers. The easiest way to explain this is by example. If the test percentile score is 60, it means that the child scored equal to or better than 60% of those individuals taking the same test. And here is where the simplicity of percentile scores gets to be a bit odd and why they might be difficult to make sense of when looking where percentiles fall when compared to the Normal Distribution.

First, percentiles are not of equal intervals. Looking at the shape of the distribution, or bell shape, between each set of numbers, we can readily see that some percentile ranks are closer to some than others. For example, the percentile rank 20 is closer on the bell curve to 30 than it is to 10. The distance between percentile intervals is smaller closer to the mean. The distance between intervals gets larger as you move away from the mean.

As discussed earlier, under the normal distribution curve we saw that “average” can range from 80-119 and that the intervals of measurement are equal. In other words, the distance between 90 and 100 is 10 points as is the distance or interval between 100 and 110 is also 10. The distance between both intervals is 10. But this is NOT the case with percentiles and why it makes so little sense when a report states that the percentile score is, for example, 20 and is classified in the average range. (See chart above.)

That would mean that the individual scored equal to or greater than 20% of the population which sounds pretty low, yet, when we look at the graph and run a line upward to the normal distribution from the 20 percentile, we see that it actually falls within the “average range.” In fact, the range of “normal” percentile scores is between 20 and 80 and well within the average range of the normal distribution. Not what one would expect at all and why percentile rankings can be confusing.

But, we must remember that in one case we are looking only at the score obtained and in the second we are asking what percent of children scored with that same score and lower. And, we must keep in mind that while test scores (normal distribution) are in equal intervals of measurement, percentiles are not. They are two different types of measures and this is key to understanding the differences and why they are not aligned vertically—they cannot be, since they are simply not equal intervals of measure.

The second test scores that also have unequal intervals is that of Grade Equivalents. This is yet another reason not to use GEs as measures of growth, because grade equivalents cannot be compared to other grade equivalents. The measure of difference between 5.1 and 5.9 is not the same as 7.1 and 7.9 and yet the mathematical difference is .8. It would be of value to read an earlier posting of Grade Equivalents are NOT Equal to Grade Levels at http://www.iephelp.com/2014/12/grade-equivalents-not-grade-levels/

Normal Curve Equivalents

Note that percentiles are “scrunched” together in the middle of the distribution, reflecting the fact that about 68% of the scores in a normal distribution are found within one standard deviation above and below the mean score of 50. Also, note that percentiles are “stretched” at the tails (far left and far right) of the normal distribution, reflecting the fact that fewer scores are found in these ranges.

In educational statistics, a normal curve equivalent (NCE) is a way of standardizing scores received on a test into a 0-100 scale similar to a percentile-rank, but unlike percentiles, they maintain the equal-interval distribution of a z-score (explained in the next section).

T SCORES AND Z SCORES

As can be seen these two test scores are with equal intervals and why they appear to align with the Normal Distribution standard deviations. But they are different scores with different meanings.

T Scores have a mean of and a standard deviation of 10, whereas Z Scores have a mean of 0 and a standard deviation of 1.0. You will unlikely run across either very often, but knowing where they fall against the normal distribution is very helpful.
The last item to consider is whether test scores on one test can be compared to test scores on another test. Now you know they can.

BOTTOM LINE

Regardless of our understanding of how test scores are reported, having the Normal Distribution graph, we can determine how far from the mean the score falls and THAT is what is important in understanding the meaning of the test scores.

Percentiles rank scores are not based on equal intervals of measurement as are the others scores in the graph. Still, by knowing the percentile rank score, we can see where that score falls on the normal distribution and again compare from the mean.
Descriptors mask the meaning of the test score and it is totally appropriate to ask for the score achieved to make sense of the words used to “explain” the score.

Different test scores represent different methods of calculations against the Normal Distribution. We can always carry that score vertically to see where it falls in relation to the mean and this is the value in understanding all normed test scores and the importance of having this chart with you.

For the PDF version click:  NORMAL DISTRIBUTION AND OTHER TEST SCORES Oh My

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