This is clearly approximately symmetric and mound shaped. We are going to model this with a curve
that idealizes what we see in this sample of 100. That is, we will model this with a continuous curve that
“describes” the shape of the distribution for very large samples. That curve is the graph of the normal
distribution . A normal curve , when superimposed on the above histogram, looks like this:
The function that yields the normal curve is defined completely in terms of its mean and standard
deviation. Although you are not required to know it, you might be interested to know that the function that
defines the normal curve is:
One consequence of this definition is that the total area under the curve, and above the x-axis, is 1 (foryou calculus students, this is because .
This fact will be of great use to us later when we consider areas under the normal curve as
probabilities.
68-95-99.7 Rule
The 68-95-99.7 rule, or the empirical rule, states that approximately 68% of the terms in a normal
distribution are within one standard deviation of the mean, 95% are within two standard deviations of the
mean, and 99.7% are within three standard deviations of the mean. The following three graphs illustrate
the 68-95-99.7 rule.