Frequency
distributions

Analyzing
single
variables

Presenting univariate
data

Measures of
central
tendency

Measures of
variability

Standard
deviation and
the normal distribution

Computer
analysis
reminder

Aids

Standard deviation and the normal distribution

Properties

Everything you will learn about probability and testing hypotheses rests on use of properties of the normal distribution or what is generally known as the normal curve. A normal curve is shown in Figure 17.6. A normal curve has four unique properties. These are:

• A symmetrical, bell-shaped form;
• The mean, median and mode are identical and coincide at the middle of the distribution;
• A fixed proportion of observations lie between the mean and specific units of the standard deviation of the distribution; and
• A precise mathematical relationship exists between the scores and their frequencies, which specifies the shape of the curve.

The last point, although critical, can only be understood in mathematical terms. Understanding the math behind the normal curve, however, is not necessary for learning how the normal distribution is used in statistical analysis. Understanding the other properties is very important.

In a symmetrical distribution, the frequencies for scores are spread out at fixed and identical distances to the left and right of the mean. This is why the shape of the curve is the same above and below the mean. If you folded the one side over the over, they would overlap perfectly. This match identifies a symmetrical distribution. In these distributions, scores tend to concentrate in the middle of the distribution and to fall off progressively toward either end of the distribution..