The Normal Curve

It is very common to picture the scattering of data around an average data point using a graph called "frequency distribution."  This is accomplished by converting a BAR GRAPH to a smooth curve connecting the midpoints along the tips of the bars.  (Since  the standard deviation measures scattering, a COMPACT frequency distribution with a tall peak has a relatively small standard deviation, while a wide distribution with a lower peak would have a relatively large standard deviation.)  As luck would have it, many physical human characteristics have graphs with symmetrical mounded appearances around the mean (average) value.  Such data can be acquired by charting weights, heights, shoe sizes, IQ's, and athletic abilities.

This distribution is called a NORMAL frequency distribution with the corresponding curve called the NORMAL (or bell-shaped) CURVE.  In this scenario, the mean ,median, and mode all have the same value and a specific percent of the total data lies within one, two, or three standard deviations of the mean.

Specifically...68% of the data points lie within 1 standard deviation of the mean (+/-)
95% of the data points lie within 2 standard deviations of the mean (+/-)
99.7% of the data points lie within 3 standard deviations of the mean (+/-)

Normal distributions are a family of distributions that have the same general shape. They are symmetric with scores more concentrated in the middle than in the tails. Normal distributions are sometimes described as bell shaped.  The height of a normal distribution can be specified mathematically in terms of the mean (m) and the standard deviation (s)  For normal distributions the mean is the most efficient measure of central tendency and less subject to fluctuations in the sample.

z-scores are related to the normal distribution through the use of a table of values for area under the curve.

Recall the formula for z-scores is (data value - mean value)/standard deviation.

A very "kewl" normal curve interactive applet.
http://www.ruf.rice.edu/~lane/stat_sim/binom_demo.html