Honors Math 3
Name:
Date:
The Normal Distribution
We have observed that probability histograms can be approximated by a “bell curve”
given enough trials. Today we are going to explore the properties of this curve, which is
often called the normal distribution.
The equation on the left gives the unit normal distribution, which has a mean of 0 and a
standard deviation of 1. This distribution has the property that the total area between the
curve and the x-axis is 1, very similarly to probability histograms where the total of all
the bars is exactly 1. It may help to picture many small-width bars on the above graph.
Since the total area under the curve is 1, the unit normal distribution measures
probability. Other normal distributions are transformed in ways that preserve this total
area, so that they are useful in answering questions about probability.
The equation on the right gives a normal distribution with mean μ (Greek letter mu) and
standard deviation σ. You do not need to memorize these formulas or where they come
from. If you are interested in understanding the transformations that give you this general
normal distribution from the unit normal distribution, however, you may wish to read
pages 287-288 more closely.
Examples:
1. Adult women’s heights are normally distributed, with a mean of 63.5 inches and a
standard deviation of 2.5 inches.
a. Approximately what percent of women have heights between 61 inches and
68.5 inches?
b. Give a range in which approximately 99.7% of all women’s heights should lie.
We can use our calculators (under 2ND VARS) to help us with problems that deal with
values that aren’t exactly whole numbers of standard deviations away from the mean.
Probability density function – normalpdf ( x, m, s) – approximates values in a matching
probability histogram
Cumulative density function – normalcdf ( x1, x 2, m, s ) – finds the area under the normal
distribution between two values (gives the percentage of values that fall in that range)
c. Approximately what percent of women are 5 feet, 5 inches tall?
d. Approximately what percent of women are between 5 feet and 5 feet, 5
inches?
2. An experiment consists of tossing a coin 10,000 times.
a. Write an expression that gives the exact probability of getting exactly
5,000 heads.
b. The above expression includes a combination that is too large for our
calculators to handle. Therefore, let’s use the normal distribution to
approximate the probability of getting exactly 5,000 heads.
c. What is the approximate probability of getting between 4,900 and 5,100
heads?
d. What is the approximate probability of getting more than 5,100 heads?