15.6. The Normal Curve http://www.ck12.org
normalcdf(-3, 3, 0, 1) = 0.997300 or 99.73%
The quality control technician would decide if this is a high enough success rate for producing a usable widget.
Concept Problem Revisited
Height, weight and other measures of people, animals or plants are normally distributed.
Vocabulary
Astandard normal distributionis a normal distribution with mean of 0 and a standard deviation of 1.
Theempirical rulestates that for data that is normally distributed, approximately 68% of the data will fall within
one standard deviation of the mean, approximately 95% of the data will fall within two standard deviations of the
mean, and approximately 99.7% of the data will fall within three standard deviations of the mean. It is a good way
to quickly approximate probabilities.
Normalcdfis the normal cumulative distribution function and calculates the area between any two values for data
that is normally distributed as long as you know the mean and standard deviation for the data. Your calculator has
this function built in, and it produces an exact answer as opposed to the empirical rule.
Guided Practice
- What is the probability that a person in Texas is exactly 6 feet tall?
- Two percent of high school football players are invited to play at a competitive college level. How many standard
deviations above the average player would someone need to be to have this opportunity? - On average, a pumpkin at your local farm weighs 10 pounds with a standard deviation of 6 pounds. You go and
find a pumpkin weighing 26 pounds. Of all the pumpkins at the farm, what percent weigh less than this enormous
pumpkin?
Answers: - Since height is a continuous variable, meaning any number within a reasonable domain interval is possible, the
probability of choosing any single number is zero. Many people may be close to 6 feet tall, but in reality they are
5.9 or 6.0001 feet tall. There must be someone in Texas who is the closest to being exactly 6 feet tall, but even that
person when measured accurately enough will still be slightly off from 6 feet. This is why instead of calculating the
probability for a single outcome, you calculate the probability between a certain interval, like between 5.9 feet and
6.1 feet. For continuous variables, the probability of any specific outcome, like 6 feet, will always be 0. - This situation is the inverse of the previous questions. Instead of being given the standard deviation and asked to
find the probability, you are given the probability and asked to find the standard deviation.