Symmetry and Skewness

A set of observations is symmetrically distributed if its graphical representation (histogram, bar chart) is symmetric with respect to a vertical axis passing through the mean. For a symmetrically distributed population or sample, the mean, median and mode have the same value. Half of all measurements are greater than the mean, while half are less than the mean.

A set of observations that is not symmetrically distributed is said to be skewed. It is positively skewed if a greater proportion of the observations are less than or equal to (as opposed to greater than or equal to) the mean; this indicates that the mean is larger than the median. The histogram of a positively skewed distribution will generally have a long right tail; thus, this distribution is also known as being skewed to the right.

On the other hand, a negatively skewed distribution has more observations that are greater than or equal to the mean. Such a distribution has a mean that is less than the median. The histogram of a negatively skewed distribution will generally have a long left tail; thus, the phrase skewed to the left is applied here.

The Pearson coefficient of skewness provides a numerical measure of the skewness of a distribution. Denoted by SK, it is calculated as follows:

SK = 3 ( m - m ) / s for a population

= 3 ( - ) / s for a sample

For a perfectly symmetric distribution, the mean and median will have the same value, and SK will have the value of 0. A distribution that is skewed to the right will have a mean that is larger than the median, and thus SK will have a positive value; thus, the distribution is also known as being positively skewed. A distribution that is skewed to the left will have a mean that is less than the median, and so SK will have a negative value; thus, the phrase "negatively skewed". In general, the values of SK will vary between -3 and 3.

EX. Given the following sorted data:

1.2, 1.5, 1.9, 2.4, 2.4, 2.5, 2.6, 3.0, 3.5, 3.8

= 2.48 as computed earlier

= 2.45 as computed earlier

s = 0.8176

SK = 3 (2.48 - 2.45) / 0.8176

= 0.1101