In a linear regression where variables have been standardized, a change of one SD in X is associated with a change of β*1 SD in Y, and I have seen people interpreting this in units since we know what the SDs are (e.g. I saw this in a statistics textbook: "Standardized β = 0.192 indicates that a band rated one standard deviation (1.40 units) higher on the image scale can expect additional album sales of 0.192 standard deviations units. This is a change of 15,490 sales (0.192 × 80,699). A band with an image rating 1.40 higher than another band can expect 15,490 additional sales."). But aren't all SDs equal to 1 after standardizing? I'm very sure I'm missing something because this doesn't seem plausible...
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$\begingroup$ What you're missing is that the standard deviation for the album sales is apparently $80,\!699$. Also, this type of interpretation quickly falls apart in all but the simplest models. $\endgroup$– Frans RodenburgCommented Aug 14 at 16:08
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$\begingroup$ I saw that but this is the issue - I thought if we standardize all variables then they all have a mean of 0 and an SD of 1, but in this example that would lead to "a band rated one SD (1 unit) higher ... can expect additional sales of 0.192 SD units (= 0.192*1)" which seems wrong. This is just a theoretical question because I realized I must be thinking wrong about this, of course it's more complex models this is a moot point anyway, practically. Thanks for the link, that's an interesting question! $\endgroup$– user20501139Commented Aug 14 at 16:16
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1$\begingroup$ The $0.192$ change is on the scale of "standard deviations away from the mean." Multiplying that by one is still on the scale of SDs. Multiplying by whatever was divided by during scaling converts back to the original scale. $\endgroup$– Frans RodenburgCommented Aug 14 at 16:18
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