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Symmetry and Skewness

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.

3smdsas1

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.

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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 ( linex - wigglyx ) / 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

linex= 2.48 as computed earlier

wigglyx= 2.45 as computed earlier

s = 0.8176

SK = 3 (2.48 - 2.45) / 0.8176

= 0.1101

 

 

 

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Comments

malinova's picture
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