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There is an exercise which is used to illustrate how normal distribution works. The exercise starts by saying "Suppose scores on an IQ test are normally distributed..."; What does it mean for the scores to be normally distributed.

Also there is a curve associated to the Normal Distribution, what does this curve tell, what stands on the axis?

An intuitive explanation with a supporting example (optionally), would be much welcomed.

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    $\begingroup$ Wikipedia $\endgroup$
    – nrussell
    Commented Jul 14, 2014 at 13:44
  • $\begingroup$ @nrussell Thanks for the link, but my question is more related to the wording "being normally distributed". How does an IQ score become normally distributed? $\endgroup$
    – Kristof Tak
    Commented Jul 14, 2014 at 14:03
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    $\begingroup$ It's not that it "becomes" normally distributed. You're supposing that the distribution of IQ scores in the population follows a Normal Distribution as on that Wikipedia page. $\endgroup$
    – shadowtalker
    Commented Jul 14, 2014 at 14:12
  • $\begingroup$ Right, as @ssdecontrol says it's not that they are necessarily becoming normally distributed, but rather that the population of IQ data that the sample is drawn from is assumed to follow a Normal Distribution. This means that the sample should exhibit the properties characterized by the probability density function (cumulative distribution function, etc...). There's actually a section of the Wiki specifically devoted to some of these properties. $\endgroup$
    – nrussell
    Commented Jul 14, 2014 at 14:17
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    $\begingroup$ I'd recommend picking up an intermediate probability textbook that explains continuous distributions. $\endgroup$
    – shadowtalker
    Commented Jul 14, 2014 at 14:27

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Effectively, the exercise prompt states "in the presence of the assumption that IQ scoring process follows a normal distribution, answer this question..." So you're allowed to assume that all of the properties of the normal distribution hold for the process generating the sample data: the distribution is symmetric, the distribution function characterizes IQ scores, IQ scores may be any real number, and so on. Obviously some of these are impossible (for example, since, to my knowledge, IQ scores must fall in some finite interval), but you're still permitted to assume them for the purposes of the question.

For the purposes of the question, at no point do the data become normally distributed. The data-generating process simply is a normal distribution by virtue of the question prompt.

Also there is a curve associated to the Normal Distribution, what does this curve tell, what stands on the axis?

These questions are already answered elsewhere on this website. This answer might be particularly helpful.

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    $\begingroup$ Making a clear distinction between data following a normal distribution--which they never do--and the underlying population or process following a normal distribution--which is a matter of what assumptions one makes--would greatly improve this answer. $\endgroup$
    – whuber ♦
    Commented Jul 14, 2014 at 19:36
  • $\begingroup$ Nicely done! (+1) $\endgroup$
    – whuber ♦
    Commented Jul 15, 2014 at 0:36

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