Microsoft Office Tutorials and References

In Depth Information

**Deviating from the Middle**

✓
The highest point of the curve is the mean.

✓
The width and height are determined by the standard deviation. The

larger the standard deviation, the wider and flatter the curve. You can

have two normal distributions with the same mean and different

standard deviations.

✓
68.2 percent of the area under the curve is within one standard

deviation of the mean (both to the left and the right), 95.44 percent of the area

under the curve is within two standard deviations, and 99.72 percent of

the area under the curve is within three standard deviations.

✓
The extreme left and right ends of the curve are called the
tails.
Extreme

values are found in the tails. For example, in a distribution of height,

very short heights are in the left tail, and very large heights are in the

right tail.

Figure 9-7:

Displaying a

normal

distribution

in a graph.

Different sets of data almost always produce different means and standard

deviations and, therefore, a different-shaped bell curve. Figure 9-8 shows two

superimposed normal distributions. Each is a perfectly valid normal

distribution; however, each has its own mean and standard deviation, with the

narrower curve having a smaller standard deviation.

Analysis is often done with normal distributions to determine probabilities.

For example, what is the probability that a 10-year-old child’s height is 54

inches? Somewhere along the curve is a discrete point that represents this

height. Further computation (outside the scope of this discussion) returns

the probability. What about finding the probability that a 10-year-old is 54

inches high
or greater
? Then the
area
under the curve is considered. These

are the type of questions and answers determined with normal distributions.

A good amount of analysis of normal distributions involves the values in the

tails
— the areas to the extreme left and right of the normal distribution curve.