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In relation to psychological research (or normal distributions in general), would an individual score within one standard deviation of the mean be considered "average"?

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I don't see this as an especially simple question. You're raising several issues at once including

  • categorisation of quantitative scales

  • interpretation of intervals based on mean and standard deviation (SD)

  • whether distributions are in practice (approximately) normal.

Such wording may make sense in context but it's always advisable to make any such convention totally explicit. That is, your convention that "about average" means within 1 SD of the mean should be mentioned somewhere prominent in your report, regardless of whether this is common wording in your field.

Much of the point of quantifying anything is lost if you want to translate back into words immediately. Definitions such as "within 1 SD of the mean" have precision that allows others to work with similar definitions and compare results.

Note that there will be no magical distinction between values just above mean + 1 SD and those just below that level, or those just above mean $-$ 1 SD and those just below.

Also, you seem to be presuming normal distributions; if your distributions are not normal, then many bets are off, as (e.g.)

  • The mean and SD might not be the best summary statistics to use

  • The interpretation of intervals that are mean $\pm$ so many SDs will differ in other distributions, especially if they are skewed and/or have kurtosis differing from the normal.

Note. I am not addressing psychological research specifically. I am not aware that the answer is affected by the data being psychological; if it is I am not qualified to address the difference.

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