A general problem with selection of children for gifted programs or the academic "top track" is that parents tend to overestimate the abilities of their children and will be displeased by a finding that their offspring are not in the top 5% or top 20% of the distribution of academic ability. Here is a recent article describing how people overestimate their own intelligence.

65% of Americans believe they are a...of two nationally representative surveys
Patrick R. Heck, Daniel J. Simons, and Christopher F. Chabris
Abstract
Psychologists often note that most people think they are above average in intelligence. We sought robust, contemporary evidence for this “smarter than average” effect by asking Americans in two independent samples (total N = 2,821) whether they agreed with the statement, “I am more intelligent than the average person.” After weighting each sample to match the demographics of U.S. census data, we found that 65% of Americans believe they are smarter than average, with men more likely to agree than women. However, overconfident beliefs about one’s intelligence are not always unrealistic: more educated people were more likely to think their intelligence is above average. We suggest that a tendency to overrate one’s cognitive abilities may be a stable feature of human psychology.

I think the study would have been much better with the option of "I think I'm about average." As is you are forced to say you are above or below, which I tend to think would lead to more people choosing above than otherwise would have.

Yes. Since they were effectively asking people if they think they are stupid (I think it would have come across that way) I am surprised even 35% took that option. And impressed by those people.

Interesting point. Of the remaining 35%, I wonder what fraction think they are about average and what fraction thinks they are below average.

If the statement that you are asked to agree or disagree with is “I am more intelligent than the average person.”, and you think you are average in intelligence, you can say you disagree, which does not mean you think you are below average in intelligence.