Assessing Normality

The two important terms discussed in this article is normality and transformation.

Normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.

PS. If the sample size is more than 30. Then the distribution of the data tends to be normal. This term is called Central Limit Theorem.

Transformation is done if the data is not normally distributed.

But how do we assess normality? It is essential to consider both graphical and numerical techniques in evaluating normal distribution.

Normal Quantile Plot

Normal quantile plots are a way of looking at a data set to see if it seems plausible that it may be a sample from a normally distributed population or procedure. If the data is normally distributed, then the points will fall exactly along the straight line.

Skewness and Kurtosis

Skewness and Kurtosis are both terms used to describe two different characteristics of normal distributions.

• Skewness describes where the data is relative to the mean.

• Kurtosis looks at how the data falls in the tails of the distribution.

What does positively/negatively skewed mean?

Positive value for skewness indicates that most of the data is above the mean, and the opposite is described as negatively skewed. The first graph shows a positively skewed data, and the second graph shows a negatively skewed data.

Skewness

Graph 1: Positively skewed

• Histogram: Look at the tail, if the tail is at the right-side of the graph it is positively skewed.

• Q-Q plot: Look at the side where the data is sparse, if the data is sparse at the right-side then it is positively skewed.

Graph 2: Negatively skewed

• The opposite of positively skewed.

Types of Kurtosis?

• Positive kurtosis (leptokurtic) = thicc tails and tall peaks

• Negative kurtosis (platykurtic) = thin tails and short peaks

• Normal kurtosis (mesokurtic) = just right

Graph 3: Types of Kurtosis

SPSS: a skewness and kurtosis of 0 indicates a symmetrical distribution.

Numerical Examinations to Test Normality

There are two numerical examinations that can test normality:

• Kolmogorov Smirnov test (K-S test)

• Shapiro Wilk test

Null hypothesis (p-value > 0.05) = normally distributed

Alternative hypothesis (p-value < 0.05) = not normally distributed

In conclusion, if the p-value is equal or less than 0.05, we will reject the null hypothesis (basically means that the data is not normal).

• Limitations for both test is that significant (p-value < 0.05) does not inform if the data is not normal in data sets with large sample size. (in other words, false positive if the data is large)

Summary in Assessing Normality

1. Use both numerical and graphical examination in testing for normality

2. Numerical examination: Kolmogorov-Smirnov and Shapiro Wilk test

3. Graphical examination: histogram and Q-Q plot

4. If the data is more than 100 (or large sample size) put more emphasis on graphical examination for normality