Lack of Symmetry
Skewness is the lack of symmetry (lack of normality) in a probability distribution. The skewness is usually quantified by the index as given below
$$s = \frac{\mu_3}{\mu_2^{3/2}}$$
where $\mu_2$ and $\mu_3$ are the second and third moments about the mean.
This index formula described above takes the value zero for a symmetrical distribution. A distribution is positively skewed when it has a longer and thin tail to the right. A distribution is negatively skewed when it has a longer thin tail to the left.
Any distribution is said to be skewed when the data points cluster more toward one side of the scale than the other. Creating such a curve that is not symmetrical.
Skewed Data
The two general rules for Skewed Data are
- If the mean is less than the median, the data are skewed to the left, and
- If the mean is greater than the median, the data are skewed to the right.
Therefore, if the mean is much greater than the median the data are probably skewed to the right.
Misinterpretation of Mean and Median: The mean can be sensitive to outliers in skewed distributions and might not accurately represent the “typical” value. The median, which is the middle value when the data is ordered, can be a more robust measure of the central tendency for skewed data.
Statistical Tests: Some statistical tests assume normality (zero skewness). If the data is skewed, alternative tests or transformations might be necessary for reliable results.
Identifying Skewed Data
There are a couple of ways to identify skewed data:
- Visual Inspection: Histograms and box plots are useful tools for visualizing the distribution of the data. Skewed distributions will show an asymmetry in the plots.
- Skewness Coefficient: This statistic measures the direction and magnitude of the skew in the distribution. A positive value indicates a positive skew, a negative value indicates a negative skew, and zero indicates a symmetrical distribution.
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