**The eyeball test**may be useful for medium to large sized (e.g.,

*n*> 50) samples, however may not useful for small samples.

**The formal normality tests**including Shapiro-Wilk test and Kolmogorov-Smirnov test may be used from small to medium sized samples (e.g.,

*n*< 300), but may be unreliable for large samples. Moreover we may be confused because 'eyeball test' and 'formal normality test' may show incompatible results for the same data. To resolve the problem, another method of assessing normality using skewness and kurtosis of the distribution may be used, which may be relatively correct in both small samples and large samples.

### 1) Skewness and kurtosis

**kurtosis (proper)**and West et al. (1996) proposed a reference of substantial departure from normality as an absolute kurtosis (proper) value > 7.1 For some practical reasons, most statistical packages such as SPSS provide

**'excess' kurtosis**obtained by subtracting 3 from the kurtosis (proper). The excess kurtosis should be zero for a perfectly normal distribution. Distributions with positive excess kurtosis are called leptokurtic distribution meaning high peak, and distributions with negative excess kurtosis are called platykurtic distribution meaning flat-topped curve.

### 2) Normality test using skewness and kurtosis

For small samples (

*n*< 50), if absolute z-scores for either skewness or kurtosis are larger than 1.96, which corresponds with a alpha level 0.05, then reject the null hypothesis and conclude the distribution of the sample is non-normal.For medium-sized samples (50 <

*n*< 300), reject the null hypothesis at absolute z-value over 3.29, which corresponds with a alpha level 0.05, and conclude the distribution of the sample is non-normal.For sample sizes greater than 300, depend on the histograms and the absolute values of skewness and kurtosis without considering z-values. Either an absolute skew value larger than 2 or an absolute kurtosis (proper) larger than 7 may be used as reference values for determining substantial non-normality.

### 3) How strict is the assumption of normality?

*t*test (assuming equal variances) and analysis of variance (ANOVA) with balanced sample sizes are said to be 'robust' to moderate departure from normality, generally it is not preferable to rely on the feature and to omit data evaluation procedure. A combination of visual inspection, assessment using skewness and kurtosis, and formal normality tests can be used to assess whether assumption of normality is acceptable or not. When we consider the data show substantial departure from normality, we may either transform the data, e.g., transformation by taking logarithms, or select a nonparametric method such that normality assumption is not required.