http://www.ck12.org Chapter 5. Normal Distribution
count, there are 9 years for which thez−scores are between−1 and 1. As a percentage of the total data, 9/13 is
about 69%, or very close to the ideal value. This data set is so small that it is difficult to verify the other percentages,
but they are still not unreasonable. About 92% of the data (all but one of the points) ends up within 2 standard
deviations of the mean, and all of the data (Which is in line with the theoretical 99.7%) is located betweenz−scores
of−3 and 3.
Lesson Summary
Anormal distributionis a perfectly symmetric, mound-shaped distribution that appears in many practical and real
data sets and is an especially important foundation for making conclusions about data called inference. Astandard
normal distributionis a normal distribution in which the mean is 0 and the standard deviation is 1.
Az−scoreis a measure of the number of standard deviations a particular data value is away from the mean. The
formula for calculating az−score is:
z=x−x ̄
sdZ−scores are useful for comparing two distributions with different centers and/or spreads. When you convert an
entire distribution toz−scores, you are actually changing it to a standardized distribution. A distribution has
z−scores regardless of whether or not it is normal in shape. If the distribution is normal, however, thez−scores
are useful in explaining how much of the data is contained within a certain distance of the mean. Theempirical rule
is the name given to the observation that approximately 68% of the data is within 1 standard deviation of the mean,
about 95% is within 2 standard deviations of the mean, and 99.7% of the data is within 3 standard deviations of the
mean. Some refer to this as the 68− 95 − 99 .7.
There is no straight-forward test for normality. You should learn to recognize the normality of a distribution by
examining the shape and symmetry of its visual display. However, anormal probabilityornormal quantile plot
is a useful tool to help check the normality of a distribution. This graph is a plot of thez−scores of a data set against
the actual values. If the distribution is normal, this plot will be linear.
Points To Consider
- How can we use normal distributions to make meaningful conclusions about samples and experiments?
- How do we calculate probabilities and areas under the normal curve that are not covered by the empirical rule?
- What are the other types of distributions that can occur in different probability situations?
Review Questions
- Which of the following data sets is most likely to be normally distributed? For the other choices, explain why
you believe they would not follow a normal distribution.
a. The hand span (measured from the tip of the thumb to the tip of the extended 5thfinger) of a random
sample of high school seniors.
b. The annual salaries of all employees of a large shipping company.
c. The annual salaries of a random sample of 50 CEOs of major companies, 25 women and 25 men.