6.1 Bell Curves
By Susan, Bennett, and John
What Is A Bell Curve?
A Bell curve is a type of graph that includes a highest point, downward cup, and upward cups. The curve consists of points called inflection points, which transition when going up or down.
Properties of A Bell Curve
It is Bell-shaped, with highest point over the mean.
It is symmetrical with a vertical line going through the mean.
The curve approaches the horizontal axis but never touches or crosses it.
Transition points occur between upward and downward cups, about negative and positive means.
It is symmetrical with a vertical line going through the mean.
The curve approaches the horizontal axis but never touches or crosses it.
Transition points occur between upward and downward cups, about negative and positive means.
Examples
Example 1: Curve too skewed to the right.
Example 2: Curve too skewed to the left and touches axis.
Example 3: Correct curve
How to find the Mean and Standard Deviation on A Curve
On a bell curve the mean is the highest point. The lower point which does not touch the horizontal axis, usually equals 1.
The Empirical Rule
68% will lie within 1 standard deviation.
95% will lie within 2 standard deviation.
99.7% will lie within 3 standard deviation.
95% will lie within 2 standard deviation.
99.7% will lie within 3 standard deviation.
Rule Continued
Going further into the curve,
34% will lie between negative and positive 1 standard deviation.
13.5% will lie between negative and positive 2 standard deviation.
2.35% will lie between negative and positive 2 and 3 standard deviation.
0.15% will lie between negative and positive 3 standard deviation.
34% will lie between negative and positive 1 standard deviation.
13.5% will lie between negative and positive 2 standard deviation.
2.35% will lie between negative and positive 2 and 3 standard deviation.
0.15% will lie between negative and positive 3 standard deviation.
What Does Normal Distribution Mean?
Area under the entire distribution is 1.
Area over specific value randomly selected falls between A and B for example:
The distribution is symmetrical and mound-shaped and is centered over the mean.
Area over specific value randomly selected falls between A and B for example:
The distribution is symmetrical and mound-shaped and is centered over the mean.