Skewnessandkurtosissummary PDF

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Skewness, Kurtosis, and Normality Robert Trevethan

It is important to realise that there are differing views about what is satisfactory, and what is unsatisfactory, skewness and kurtosis. Furthermore, there are some misleading or confusing sites on the web, e.g., one that says that normal kurtosis is 3. That might be the case for “unadjusted”, or “proper” kurtosis, however, and some statistics packages (including SPSS) apply an adjustment to make “normal” kurtosis zero. Below is some information from the following article: Kim, H-Y. (2013). Statistical notes for clinical researchers: Assessing normal distribution (2) using skewness and kurtosis. Restorative Dentistry and Endodontics 38, 52–54. https://doi.org/10.5395/rde.2013.38.1.52

According to Kim, the Kolmogorov–Smirnov and Shapiro–Wilk tests are “unreliable” with big samples, i.e., > 300. In essence, they are too “sensitive”. His recommendations regarding skewness and kurtosis are the following: Sample size

Strategy

Criteria

< 50

Convert to z by dividing by the std error

If z > |1.96|, data are not normally distributed

50–300

Convert to z by dividing by the std error

If z > |3.29|, data are not normally distributed

300+

Don’t convert to z values. Instead, look at the histograms and the absolute values of skewness and kurtosis without considering z values.

Either an absolute skewness value larger than 2 or an absolute kurtosis (proper) larger than 7 [RT: which I think, for many stats packages, means kurtosis of 3] may be used as reference values for determining substantial non-normality. Note that some packages (including SPSS) convert kurtosis such that values should not exceed |3|.

Also note that Mayers (2013, p. 53) suggested that a cutoff of ±1.96 should be used for samples smaller than 50, a cutoff of ±2.58 for samples from 51 to 100, and a cutoff of ±3.29 for samples larger than 100 when used in conjunction with the examination of histograms. Mayers, A. (2013). Introduction to statistics and SPSS in psychology. Harlow: Pearson Education Limited.

Obviously, there are discrepant views. On the next page, I have provided my (RT) recommendations, based on the above, but with an additional row for sample sizess between 50 and around 175. This is partly supported by the following article: Ghasemi, A., & Zahediasl, S. (2012). Normality tests for statistical analysis: A guide for non-statisticians. International Journal of Endocrinology and Metabolism, 10, 486-489. https://www.ncbi.nlm.nih.gov/pmc/articles /PMC3693611/

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My (RT’s) recommendations, based on publications on the previous page Sample size

Strategy for skewness & kurtosis values

Criteria

< 50

Convert to z by dividing by std error

If z > |1.96|, data are not normally distributed

50 to ~175

Convert to z by dividing by std error

If z > |2.58|, data are not normally distributed

~175 to 300

Convert to z by dividing by std error

If z > |3.29|, data are not normally distributed

300+

Don’t convert to z values. Look at the histograms and the absolute values of skewness and kurtosis without converting to z-values.

Either an absolute skewness value larger than 2 or an absolute kurtosis (proper) larger than 7 [RT: which I think means kurtosis of 3] may be used as reference values for determining substantial nonnormality. Note that some packages (incl. SPSS) convert kurtosis such that values should not exceed |3|.

Here is another view from Peter Samuels on ResearchGate https://www.researchgate.net/post/How_do_we_know_which_test_to_apply_for_testing_normality:

What matters is the absolute size of the skewness and kurtosis relative to their standard errors: A better rule of thumb is that they should be less than twice their standard errors (twice is incidentally a better estimate than 1.96 because the exact value actually increases above 2 for small sample sizes). A personal note from experience: When sample sizes are < 300, statistics generated for skewness and kurtosis can appear to be either acceptable despite histograms departing noticeably from normality, or, alternatively, the statistics for skewness and kurtosis can appear to be unacceptable when histograms indicate no obvious departure from normality. Therefore, at times it can be best to simply inspect histograms. Wheeler (2004) strongly recommended this procedure for identifying skewness and kurtosis when sample sizes are small—a procedure that subsequently received firm endorsement (“Are the skewness and kurtosis useful statistics?”, 2016). Are the skewness and kurtosis useful statistics? (2016). https://www.spcforexcel.com /knowledge/basic-statistics/are-skewness-and-kurtosis-useful-statistics Wheeler, D. J. (2004). Advanced topics in statistical process control: The power of Shewhart’s charts (2nd ed.). Knoxville, TN: SPC Press.

Tests for normality There are very divergent views about appropriate tests for normality. By way of summary, it might be worth noting Kim’s advice (in red on the previous page) and also that the Shapiro– Wilk test is often preferred for small samples and the Kolmogorov–Smirnov test for large samples. However, some methodologists draw the line between small and large at 50, and others at 2,000, so uncertainty reigns. It might be advisable to inspect P–P and Q–Q plots. 2...