I am working my way through Chapter 3 in the Applied Predictive Modeling by Kuhn and Johnson. In section 3.2 the discussion values close to zero indicate symmetry. My question is - how close to zero? In the end of chapter exercises the preprocess function transforms some from 1.7 to .9 - is that close enough?


1 Answer 1


Different references will tell you different cut-off points for skewness.

I use a cut-off of 2 for the absolute value of the skewness, so I assume that is reasonably close to zero if: $$ -2 \leq skew \leq 2 $$

Some citations with references (some of them are incomplete, sorry). I'm sorry but I have these in some notes, but I don't have recorded which skewness formula and which kurtosis formula the authors are referring to. Following Nick Cox's comments I've added some remarks to some of the quotes.

"The values for asymmetry and kurtosis between $-2$ and $+2$ are considered acceptable in order to prove normal univariate distribution."
George, D., & Mallery, M. (2010). SPSS for Windows Step by Step: A Simple Guide and Reference, 17.0 update (10a ed.) Boston: Pearson.
In my opinion, prove normal univariate distribution should be replaced by assume a univariate distribution close to the normal in the above sentence.

"The acceptable range for skewness or kurtosis below $+1.5$ and above $-1.5$."
Tabachnick & Fidell (2013).

"Most of the researchers in the field of social science are following a less stringent [criterion] based on the suggestion by Kline (1998, 2005). Data with a skew above an absolute value of $3.0$ and kurtosis above an absolute value of $8.0$ are considered problematic."
It is possible that Kline is referring to kurtosis and not to excess kurtosis, while Tabachnick & Fidell may be referring to excess kurtosis. Even so, Kline provided the largest cut-offs that I've seen so far.

"It can be consider[ed] normal when $-1<$skewness$<1$."
Bulmer M. G. (1979), Principles of Statistics.
I would say close to the normal distribution.

"Indices for acceptable limits of $\pm 2$" (of skewness and excess kurtosis).
Trochim & Donnelly, 2006; Field, 2000 & 2009; Gravetter & Wallnau, 2014.

Trochim, W. M., & Donnelly, J. P. (2006). The research methods knowledge base (3rd ed.). Cincinnati, OH:Atomic Dog.

Gravetter, F., & Wallnau, L. (2014). Essentials of statistics for the behavioral sciences (8th ed.). Belmont, CA: Wadsworth.

Field, A. (2000). Discovering statistics using SPSS for Windows. London-Thousand Oaks- New Delhi: Sage publications.

Field, A. (2009). Discovering statistics using SPSS. London: SAGE.

  • $\begingroup$ This is awesome! Thank you so much!! $\endgroup$ Oct 24, 2019 at 13:50
  • 2
    $\begingroup$ But watch out: skewness is measured in different ways, and although in several fields moment-based skewness is the standard, and perhaps even the only measure discussed, other measures exist. For example, (mean $-$ median) / SD is a handy (and underused) measure, but it is limited to $[-1, 1]$, a fact that often surprises and is intermittently rediscovered. $\endgroup$
    – Nick Cox
    Oct 24, 2019 at 13:58
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    $\begingroup$ Some of the advice above is seriously confused. Far from being a proof of normality, in principle any skewness not zero disproves it! (In practice, samples from a normal won't show zero skewness, but even so, proof is quite the wrong word to wave at naive readers.) Also, the main reasons to transform are not to get closer to normal distributions, but to get closer to linear and additive relationships and homoscedasticity. I will often transform even if skewness is within any or all of these bounds. $\endgroup$
    – Nick Cox
    Oct 24, 2019 at 14:03
  • $\begingroup$ If you ignore kurtosis < 8 (is that before or after subtracting 3?) you will often pay the price in a poor analysis. $\endgroup$
    – Nick Cox
    Oct 24, 2019 at 14:05
  • $\begingroup$ Thanks Nick for the comments. Your comment about the purpose of transformation is getting closer to linear and additive relationships & homoscedasticity especially resonates. $\endgroup$ Oct 24, 2019 at 17:03

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