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The asymmetry index is the ratio of the skewness to the standard error. It is an indication of the asymmetry of a distribution.

Table 1. Interpretation of asymmetry index
Asymmetry Index (x)	Interpretation
0 < x ≤ 2.0	The asymmetry is weak. The distribution is relatively symmetrical.
2.0 < x ≤ 4.0	The asymmetry is moderate. The distribution is relatively asymmetrical.
x > 4.0	The asymmetry is strong. The distribution is asymmetrical.

 

Normal distribution

The normal distribution is a theoretical distribution of values. It is often called the bell curve because the visual representation of this distribution resembles the shape of a bell. It is theoretical because its frequency distribution is derived from a formula rather than the observation of actual data.

Even though the normal distribution is theoretical, the distributions of many fields in the real world resemble the normal distribution. An example is the traditional bell curve for ranking students. Most students have average grades (the mean), while there are a few with the poorest grades and a few with the highest grades.

The assumption of normality is important for many statistical tests. If the shape of a field’s distribution is not bell-shaped, some statistical tests might not be valid.

The normal distribution is symmetric about the mean. That is, the distributions of values to the right and left of the mean are mirror images. 68% of the values in the distribution fall within one standard deviation of the mean (to the left and right). 95% of values fall within two standard deviations, and 99.7% within three.

Skewness

Skewness is a measure of the asymmetry of a distribution. The normal distribution is symmetric and has a skewness value of 0.

A distribution with a significant positive skewness has a long, thin tail to the right. A distribution with a significant negative skewness has a long, thin tail to the left.

The asymmetry index is used to evaluate the skewness value.

Symmetry

Symmetry describes how values are distributed on either side of the central value. If both sides are distributed equally, the distribution is symmetrical.

The normal distribution is a good example of a symmetrical distribution. The left side of the bell curve looks like the right side of the bell curve. The two sides are mirror images.

Skewness is a measure of symmetry. Calculating the ratio of the skewness to the standard error generates a value to ascertain the asymmetry. A larger value indicates greater asymmetry.

Kurtosis

Kurtosis is a measure of the extent to which observations cluster around the mean. For a normal distribution, the value of the kurtosis statistic is 0.

Positive kurtosis indicates that the observations cluster about the mean more, relative to the normal distribution. The distribution has thinner tails until the extreme values of the distribution, at which point the tails are thicker, relative to a normal distribution.

Negative kurtosis indicates that the observations cluster less relative to a normal distribution. The distribution has thicker tails until the extreme values of the distribution, at which point the tails are thinner relative to a normal distribution.