# IQ scores and normal distributions

1. Jul 10, 2007

### polomint

Hello

I would like to hear your opinions on the normality of scores on an IQ test. The test had 30 questions and apart from the general IQ score separate subtest scores such as mathematics, verbal and spatial IQ were also calculated.

Here is a list of results that were obtained from the data.

http://www.gtest.net/normalcurve/histograms.htm
http://www.gtest.net/normalcurve/plots.htm
http://www.gtest.net/normalcurve/descriptives.htm
http://www.gtest.net/normalcurve/normalitytests.htm
http://www.gtest.net/normalcurve/lilliefors.htm

I would like to report standardized scores and percentiles for the general score and all subtest scores. However I was told that all scores and subtest scores had to follow a normal distribution before this could happen.

Mark

Note: The skewness and kurtosis results seem to be in the required range. I then read that they should be divided by the standard error, but when this happens the results became way too large. Similarly the Shapiro-Wilk statistic is close to 1 but the significance is 0. How does this affect the normality result?

2. Jul 10, 2007

### EnumaElish

There is nothing holding you from reporting standardized scores. If the "raw" scores are normal, then your standardized scores will be standard normal. If the "raw" scores are non-normal, then the standardized scores will not be standard normal.

3. Jul 11, 2007

### polomint

So far I have been transforming the general IQ score (labeled 'score' in the diagrams) to a conventional IQ score (mean 100, SD15) and also calculating and reporting the percentile from the z score.

However I have been reporting the subtests as percentages. For example the general knowledge score is out of 16. If a user gets 8 points then the general knowledge score will be reported as 50. The problem is that for some 'easy' subtests there are high averages and standard deviations. This leads to some people being awarded perfect scores but percentiles of only 90 or 91. Is this by itself an indication of scores not following a normal distribution?

I would like to report subtests scores the same way that I am reporting the general IQ score in order to maintain consistency. However since some subtest raw scores didn't follow a normal distribution as closely as others I wasn't sure if standardizing all subtests scores was the right thing to do.

Regards

Mark

4. Jul 11, 2007

### EnumaElish

Can you illustrate this with one or two examples?

5. Jul 16, 2007

### polomint

Hello

Yes here is an example. The general knowledge subtest has an average of 70 and SD 23. If someone gets a perfect score of 100 then the z score would be 1.3. Therefore the highest percentile that they could score is the 90th.

similarly the verbal subtest has an average of 68 and SD 22. The z score for a perfect score of 100 would be 1.45 and this would be the 93rd percentile.

Other subtest scores produce higher percentiles for higher scores. For example the mathematics subtest has average 53 and SD 21. A perfect score produces a z score of 2.24 (99 percentile)

The general IQ score also reports better percentiles. For example the average score is 19 (out of a total score of 30) and the SD is 5.3. A perfect score produces a z score of 2.08 (98 percentile)

6. Jul 16, 2007

### EnumaElish

Last edited: Jul 17, 2007
7. Jul 18, 2007

### polomint

I read the following paragraph from wikipedia about normal curves and intelligence
Distribution in testing and intelligence

Sometimes, the difficulty and number of questions on an IQ test is selected in order to yield normal distributed results. Or else, the raw test scores are converted to IQ values by fitting them to the normal distribution. In either case, it is the deliberate result of test construction or score interpretation that leads to IQ scores being normally distributed for the majority of the population. However, the question whether intelligence itself is normally distributed is more involved, because intelligence is a latent variable, therefore its distribution cannot be observed directly.​

Do you agree with the part where it says that the raw tests scores can be converted to IQ values by fitting them to the normal distribution? With this way can non normal raw scores be converted to normal IQ scores?

Last edited: Jul 18, 2007
8. Jul 18, 2007

### mathwonk

the phenomenon of perfect score and relatively low percentile occurs all the time on the math SAT, now that it has been dumbed down enough for more people to get 800.

9. Jul 18, 2007

### EnumaElish

I am not aware of any meaning attributable to this sentence other than a description of the formula "normalized" scores = (raw score - test mean)/(test standard deviation). (http://www.eskimo.com/~miyaguch/MCReport/mcreport3.html)

This does not normalize non-normal data, it just expresses them in a standardized metric.

A related description is: "IQs are calculated by multiplying the z-score by 15 (16 on some tests) and then adding 100. SAT scores are calculated by multiplying the z-score by 100 and then adding 500." (http://www.gifted.uconn.edu/siegle/research/Normal/Interpret Raw Scores.html)

These are a series of linear operations applied to raw scores:
raw score ---> z ---> IQ. Each "--->" stands for a linear operation ("affine," if one is technically correct and picky). I am skeptical that non-normal data can be converted into normality in this way.

This is not to say that non-normal data can never be converted to normal. Lognormal data can easily be converted into normal by applying the logarithmic function, which is a nonlinear operation. There are other instances. But they do not seem to be what the Wiki article or the above-quoted excerpts are referring to.

Last edited: Jul 19, 2007