The "normal curve" is very important and often referred to in the process of measuring human abilities and attitudes.

 

A colorful, easy-to-understand website with many examples is here (visited in July 2016).

 

A more typical technical reference is found at Wikipedia (visited in July 2016).

 

Before taking a close look at the normal curve, I need to remind you of what "standard scores" are.

 

Say I have a set of test scores with a mean of 75 and standard deviation of 5.  (Remember that "the mean" is the average test score.)

 

The most common standard score is the "z-score".

 

I can convert any test score to a "z-score" by subtracting the mean from the test score, and dividing the result by the standard deviation.

 

For example, a student named Marisol got a test score of 80.

 

Her corresponding z-score is (80-75)/5, or 1.00.

 

Andres got a test score of 65.  His corresponding z-score is (65-75)/5, or -2.00.

 

If I calculate a z-score for all of the students, the mean of the z-scores will be zero, and the standard deviation of the z-scores will be one.  This will always be true as long as the standard deviation of the test scores is greater than zero.

 

z-scores are not the only type of standard score.  There are also "T scores", "CEEB scores", "IQ scores", and others.

 

Because the the normal distribution is so important, we'll look at it in a bit of detail using a bmp file saved on my local hard disk.

 

When we look at the scores from a test, or from an attitude scale, many times it is convenient to convert them to z-scores.  If the scores have a normal distribution, over 99% of them will fall in the range z = -3.00 to z = +3.00, while about 68% of them will fall in the range z = -1.00 to z = +1.00;  95% of the scores will fall in the range z = -1.96 to z = +1.96.

 

Many times a z-score less than -3 or greater than +3 is referred to as an "outlier".  This term is used to express the fact that, under a normal distribution, these z-scores are very, very rare, occurring less than 1% of the time.

 

At times outliers indicate that there may have been a problem with the original test scores.  For example, for medical reasons a student may have been excused after answering just a few test questions, resulting in a very low test score.

 

Three score quartiles are frequently used in measurement.  They're Q1 (the 25th percentile), Q2, commonly called the median (50th percentile), and Q3 (the 75th percentile).  The "inter-quartile range" is Q3 - Q1.  If scores are normally distributed, half of the scores will be found in this range.