Data Distributions

What is Normality?

• Refers to the shape of data distribution for individual variable.

• Normality can have serious effects in small samples (<50), but the impact effectively diminishes when sample sizes > 200.

• In parametric tests, if variation from normal distribution is large, all resulting statistical tests are invalid, because F & t-statistics assume normality (Hair et al., 2010).

How to Test for Normality?

• Histogram: Compare the observed data values with a distribution approximating normal distribution.

• Normal Probability Plot: Compare the cumulative distribution of actual data values with the cumulative distribution of a normal distribution.

• Skewness and Kurtosis Statistics.

• Shapiro-Wilks (sample < 2000).

• Kolmogorov-Smirnov (sample > 2000).

Skewness

The degree of symmetry in the variable distribution.

Threshold: -2 ≤ skewness ≤ 2 (Curran et al., 1996; West et al., 1995; Gliselli et al., 1981).

Kurtosis

The degree peakedness/flatness in the variable distribution.

Threshold: -7 ≤ Kurtosis ≤ 7 (Curran et al., 1996; West et al., 1995).

The expected Mardia’s skewness is 0 for a multivariate normal distribution and higher values indicate a more severe departure from normality.According to Bentler (2005) and Byrne (2010), the critical ratio value of multivariate kurtosis should be less than 5.0 to indicate a multivariate normal distribution. 

Data is extremely not normal, how to remedy?

• Check and remove outlier cases.

• Remove non-normal item from the model.

• Bootstrapping (i.e., re-sampling process in the existing data-set with replacement).

Exercise

Please download the data here.