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I have created cognitive summary scores from a battery of 17 neuropsychological measures. I standardized the raw scores of my patient group to the mean and standard deviation of a matched healthy control group and then averaged these standardized scores to create a composite score. The resulting composite scores were normally distributed. A journal reviewer is implying that the raw scores on each of the neuropsych measures had to be normally distributed in the control group in order to use this approach. Is this so? Could you suggest a reference that speaks to this issue?

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  • $\begingroup$ Are you sure you mean: standardize and not normalize? $\endgroup$ – Adam Ryczkowski Aug 1 '13 at 22:19
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Several issues seem bundled up together here.

  1. Why you are standardizing (Value - mean) / SD naturally does nothing to the shape of a distribution, and the reason(s) for standardizing must be something else, e.g. washing out different measurement units or producing comparable numbers.

  2. Why is anything being normally distributed important to your analysis? You don't spell out why it is important. To me, the distributions are what they are: they will depend on your patient samples, and if not normal that could be very interesting or important. For example, you may have some very unusual patients in there, possibly even outliers.

  3. The more you average, the more the resulting averages will be normally distributed You don't need references for this: it is the central limit theorem. If the individual standardized scores are normal, the fit of the overall averages to a normal will be better, but starting out with normal distributions isn't essential here. You could do some simulations averaging random samples from various distributions to show that.

  4. Normal or not is too coarse There is an overarching assumption that being normal is a yes or no matter, but it is better to think that there are just degrees of approximation to normality, poor or good as the case may be. I'd recommend that you use normal probability plots throughout in keeping track on how close distributions are to the normal.

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