Bowman’s Website

August 30, 2010

Statistics Notes — The Shape of Distributions

Filed under: Statistics, Statistics Notes — Tags: — bowman @ 11:01 pm

Histograms are valuable and useful tools. If the raw data came from a random sample of population values, the histogram constructed from your sample values should have a distribution shape that is reasonably similar to that of the population.

A graph reveals several characteristics of a frequency distribution.  One such characteristic is the shape of the distribution.

Symmetrical: A frequency distribution is symmetric when a vertical line can be drawn through the middle of a graph of the distribution and the resulting halves are approximately mirror images.

Uniform: A frequency distribution is uniform (or rectangular) when all entries, or classes, in the distribution have equal frequencies.  A uniform distribution is also symmetric.

Skewed: A frequency distribution is skewed if the “tail” of the graph elongates more to one side than to the other.  A distribution is skewed left (negatively skewed) if its tail extends to the left.  A distribution is skewed right (positively skewed) if its tail extends to the right.

Bimodal: This term refers to a histogram in which the two classes with largest frequencies are separated by at least one class. The top two frequencies of these classes may have slightly different values. This type of situation sometimes indicates we are sampling from two different populations.

This following figures show the relative positions of the mean and median for right-skewed, symmetric, and left-skewed distributions. Note that the mean is pulled in the direction of skewness, that is, in the direction of the extreme observations. For a right-skewed distribution, the mean is greater than the median; for a symmetric distribution, the mean and the median are equal; and, for a left-skewed distribution, the mean is less than the median.

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This following figures include position of the mode in relation to the mean and median for right-skewed, symmetric, and left-skewed distributions. The mode is always at the “hump.”

When a distribution is symmetric, the mean, median, and mode are equal.  Mean = Median = Mode

If a distribution is skewed left, the mean is less than the median and the median is usually less than the mode.  Mean < Median < Mode
If a distribution is skewed right, the mean is greater than the median and the median is usually greater than the mode.  Mode < Median < Mean

The mean will always fall in the direction the distribution is skewed. For instance, when a distribution is skewed left, the mean is to the left of the median.

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