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I'm solving this multiple choice question on the properties of a standardized variable. Two of the possible options (which are wrong but look right to me) are 1. It is always normally distributed and 2. it has a bell shaped distribution.

I've been googling and apparently, standardizing a variable does not change the shape of the curve. So if we have a non normal distribution- standardizing it wont change its shape. However, I thought one of the defining characteristic of a Z distribution was that it was symmetric.

If a normal distribution is always symmetric, and a Z distribution is always symmetric, how can we have a non normal standard distribution? Can we have a distribution that is not bell shaped and still standardized? I'm trying to think of examples but can't.

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    $\begingroup$ Is the variable normally distributed before the standardization? $\endgroup$ – Cagdas Ozgenc Mar 13 '15 at 12:54
  • $\begingroup$ Its a general multiple choice question. Is it possible to standardize a non normally distributed variable? Because our teacher told us that we convert a variable to Z value only if its normally distributed. $\endgroup$ – dexter Mar 13 '15 at 13:04
  • $\begingroup$ Standardization is a general term. But as analog to normal distribution, any distribution in the class of distributions called location-scale distributions can be location centered and set to unit scale. Therefore if the variable is not normal before standardization both #1 and #2 are false. $\endgroup$ – Cagdas Ozgenc Mar 13 '15 at 14:10
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Standardizing a set of scores—that is, converting them to z-scores—that is, subtracting the mean and dividing by the standard deviation—indeed will not make a distribution any more or less normal. It won't make an asymmetric distribution symmetric, either. A set of z-scores is not necessarily a "z-distribution", whether the term "z-distribution" means a normal distribution or something more exotic like Fisher's z-distribution.

An easy example of a standardized set of scores that is obviously neither normal nor otherwise bell-shaped is (-1.46, -0.88, -0.29, 0.29, 0.88, 1.46). This dataset has mean 0 and standard deviation 1 but is uniformly distributed. I produced it by standardizing (0, 1, 2, 3, 4, 5).

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