2.5 The Normal Distribution
The normal distribution is arguably the most famous distribution in all of statistics. Why is it so
important?
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Normal distributions are good descriptions for some distributions of real data. Distributions that
are often close to Normal include:
o Repeated measurement of the same ____________ (like the diameters of the tennis
balls we measured in class)
o Characteristics of _________________ (such as lengths of animal pregnancies and
scores of SAT exams)
Normal distributions are good approximations to the results of many kinds of chance outcomes,
such as tossing a coin many times.
Most importantly, many statistical inference procedures based on Normal distributions work
well for other roughly ______________ distributions.
The normal distribution is really a family of distributions with the same shape but different ________
and _________________________. Each of them has these properties:
1.
2.
3.
4.
The total area under the curve is _____
The curve is ______________ so the mean, median and mode all fall together
The curve is bell-shaped
The greatest proportion of scores lies close to the mean
The mean describes where the corresponding curve is ___________. Half the scores lie to the left of the
mean and half to the right.
The standard deviation describes how much the curve spreads out around that center. The bigger σ is
the more spread out or __________ the curve.
1 | Section 2 . 5
The Standard Normal Distribution
Because all normal distributions have the same basic shape, you can use rescaling and ______________
to change any normal distribution to the one with a mean = 0 and standard deviation = 1. This process is
known as ____________________, and the resulting distribution is called the standard normal
distribution. Because all normal distributions are the same when you standardize, you can easily find
probabilities associated with any of them using a single table or compare different distributions.
The standard normal distribution has a mean 0 and standard deviation 1. The following formula changes
a variable x to the standard normal variable z (z-score):
z
x  mean
standard deviation
This z-score simply represents the number of standard deviations a value x is from the mean.
This formula can also be rearranged to find an x value when you know z:
x  mean  z stan d ard
d ev iatio n
Finding Probabilities
In a normal distribution the area under the curve is used to answer questions about proportions or
________________ of observations. The empirical rule describing the approximate proportion of scores
in various areas is:
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68% of the scores fall within 1 standard deviation of the mean
95% of the scores fall within 2 standard deviations of the mean
99.7% of the scores fall within 3 standard deviations of the mean
http://www.stat.tamu.edu/~west/applets/empiricalrule.html
2 | Section 2 . 5
A more accurate area is found using the standard normal table or by using the following function built
into your calculator:
normalcdf(lower bound, upper bound, mean, standard deviation)
Here is an outline of the method for finding the proportion of the normal distribution in any region.
Step 1: Draw a picture of the distribution in terms of the observed variable x, and shade the area of
interest under the curve.
Step 2: Standardize x to restate the problem in terms of a standard normal variable z.
Step 3: Use the table (or your calculator) to find the required area under the standard normal curve.
Note: Although you may use your calculator to find the area you must not use any “calculator speak” in
the AP exam, i.e. don’t write what calculator steps you used in your solution.
Step 4: Write your conclusion in the context of the problem.
Examples:
1. Find the percentage of values below the z-score –2.23 in a standard normal distribution.
2. Find the z-score that has 32% of values below it.
3. What percentage of values in a standard normal distribution fall between –1.46 and 1.46?
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4. The heights of 18 to 24 year-old males in the United States are approximately normal, with
mean 70.1 in. and standard deviation 2.7 in. The heights of 18 to 24 year-old females are also
approximately normally distributed and have mean 64.8 in. and standard deviation 2.5 in.
a. Estimate the percentage of US males between 18 and 24 who are 6 ft tall or taller.
b. How tall does a US woman between 18 and 24 have to be in order to be at the 35th
percentile of heights?
c. Which of the following heights are outside the middle 95% of the distribution? Which are
outside the middle 99%?
i. A male who is 79 in. tall
ii. A female who is 68 in. tall
iii. A male who is 65 in. tall
iv. A female who is 65 in. tall
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