Symmetric or Skewed?

A curve is said to be symmetric about Y axis if right side of the curve is the mirror image of the left side. The frequency distribution of a symmetric data is represented by the diagram on left hand side below. Here the frequency of attributes equidistant from the mean is same, so f(x₁) = f(x₂). The mean (average value), the mode (most frequent value), and median (lies in the middle of ordered data) coincide. Mean = Mode = Median. Income distribution of a European Country with most of people of that country with income around the average value can be described as symmetric. Since the values overlap the variance is low. A small sample can serve the purpose.

The divergence from symmetry is called skewness.  The frequency curve below on right hand side is positively skewed.  Here lower values have high frequency and higher values have low frequency. Mode < Median < Mean.f(x₁) is not equal to  f(x₂). Income distribution of countries like Nepal and India is positively skewed. Here most of the people exist on very low income but some people are one of the richest in the world. Since the values are scattered from very low to very high, the variance is high. A relatively large sample has to be collected to reflect the inherent variability.