Non-normal distribution but also a symmetrical one?

Laplace distribution can be symmetrical.

The skewness/error results

1) The best answer is not: A failure to reject the null hypothesis that a distribution is normal is not equivalent to proving that the distribution is normal. I am not sure how you "prove" that a distribution is normal.

2) With a large sample size it should be possible to reject the assumption of normality (at least in biological data). Increasing the sample size makes normality tests more sensitive to minor deviations from normality.

3) There are several distributions that are symmetrical. Uniform distribution is symmetrical. One could design a bimodal distribution that is symmetrical with the mean and median midway between the two modes. Any multimodal distribution could be symmetrical for that matter: the shape of the distribution to the left of the mean is a mirror image of that to the right of the mean.

Testing the normality of your data may not be the issue. Testing the normality of the residuals in parametric models is more commonly the issue.

I agree with Timothy, you may also wish to read a new aspect;

https://www.researchgate.net/publication/314032599_TO_DETERMINE_SKEWNESS_MEAN_AND_DEVIATION_WITH_A_NEW_APPROACH_ON_CONTINUOUS_DATA

Hello Johnathan,

I'd like to follow up on the last paragraph in the reply from Timothy Ebert. " normality of the residuals in parametric models is more commonly the issue." If the reason you are interested in normality is evaluating the assumptions for commonly used statistical analysis, read on.

Some texts (Quinn and Keough), publications (Tacha et al 1982), and statistical packages treat the evaluation of assumptions of homogeneity and normality as preliminary to statistical analysis of data. This points to evaluation of the response variable. While this can be a useful guide to initial choice of error distribution (normal, something else) it is short of sufficient. The assumptions for the general linear model (including regression, t-tests, ANOVA, and ANCOVA) are that the errors (residuals) are normal and homogeneous (Eisenhart 1947, Seber 1966, Neter et al 1983 pp 31& 49, Quinn and Keogh 2002 pp 110 & 280). Evaluation of assumptions in advanced texts, where it occurs, often entails a residual versus fit plot (for homogeneity) and a normal score plot or Quantile-Quantile plot (for normality of the residuals).

The statistical literature warns against statistical tests to evaluate assumptions and advocates graphical tools (Montgomery & Peck 1992; Draper & Smith 1998, Quinn & Keough 2002). Läärä (2009) gives several reasons for not applying preliminary tests for normality, including: most statistical techniques based on normal errors are robust against violation; for larger data sets the central limit theory implies approximate normality; for small samples the power of the tests is low; and for larger data sets the tests are sensitive to small deviations (contradicting the central limit theory).

That preliminary evaluation of assumptions is insufficient can come as a surprise, as can advice against statistical tests to evaluate assumptions. But I think best practice, as in the literature, needs to be said.

Good luck with your research,

~David S

Eisenhart, C. 1947. Biometrics 3:1-21

Läärä, E. 2009. Statistics: reasoning on uncertainty, and the insignificance of testing null. — Ann. Zool. Fennici 46: 138–157.

Neter, JW, MH Wasserman, MH Kutner.1983. Applied linear regression models. Homewood Illinois, Richard D. Irwin, Inc.

Quinn, G and MJ Keough. 2002. Experimental Design and Data Analysis for Biologists. p 110, 280

Seber, GAF. 1966. The Linear Hypothesis: A General Theory. London, Griffin.

Tacha, TC, WD Warde, KP Burnham. 1982. Use and interpretation of statistics in wildlife journals. Wildlife Society Bulletin 10: 355-362.

I think symmetric distribution happens when there is the existence of both extremely low and extremely high values.

All thanks for those valuable sharings.

Hello Jonathan,

Returning to your question about non-normal and symmetrical,

and leaving aside the issue of whether we should use hypothesis testing on the response variable to check assumptions, one answer is that obviously non-normal distributions that are symmetrical can arise in several ways. First of all, kurtotic.

The distribution is too "flat" (platykurototic). Or it is too thin in the shoulders (leptokurtotic). And sometimes, we see similarly long tails to the left and right, which do not show up in a skewness measure.

b.t.w, long tails in the response variable usually show up in the residuals. So long tails are a case where looking at the distribution of the response variable usually gets it right. But of course better to let the residuals speak, rather than interviewing the response variable.

Good luck with your research,

~David S