In summary, the normal distribution has the following 6 characteristics:
1. It is symmetrical around the mean.
2. The tails of the curve are asymptotic, that is, they approach but never touch the base line.
3. The mean, median, and mode all lie at the center of the curve.
4. The area under the curve is 1.00 or 100%.
5. The mean determines the location of the center of the distribution and the standard
deviation determines the shape of the distribution.
6. The Empirical rule relates areas under the curve to horizontal distances on the baseline by
using the following approximations:
a. u + 1σ encompasses about 68%.
b. u + 2σ encompasses about 95%.
c. u + 3σ encompasses about 99.7%.
Schematically, the Empirical rule would appear as follows:
The Empirical rule only reflects approximations whereas the normal curve table is more
precise. The Empirical rule provides a quick analytical tool when the normal curve table is not
available, such as in a meeting or a review and analysis. The z equation can be used to convert
distances from the mean in any unit of measure to our standard unit of measure, standard
deviations from the mean. The equation is z = X - u divided by σ, where z is the number of
standard deviations from the mean. Expressed symbolically, z = X - u
We use the table of areas under the standard normal curve, or the Empirical rule, to find the
area under the curve between the mean and z. The Normal Curve Table is the more preferred
Proceed to the Practice Exercise for Lesson 3.