Video Transcript
For a normally distributed data set
with mean 32.1 and standard deviation 2.8, between which two values would you expect
99.7 percent of the data set to lie?
Remember, a normal distribution
with a mean 𝜇 and a standard deviation 𝜎 is represented by this bell curve. The curve is symmetrical about the
mean. And the total area under the mean
represents 100 percent. For a normally distributed data
set, we say that approximately 68 percent of the data values occur within one
standard deviation of the mean. That’s the area shown.
We say that proximally 95 percent
of the data values occur within two standard deviations of the mean. And 99.7 percent lie within three
standard deviations of the mean. We are told that this data set has
a mean of 32.1 and a standard deviation of 2.8. And we expect that 99.7 percent of
them will lie within three standard deviations of the mean, three standard
deviations of 32.1. That’s 32.1 plus three multiplied
by 2.8 and 32.1 minus three multiplied by 2.8.
32.1 plus three multiplied by 2.8
is 40.5. And 32.1 minus three multiplied by
2.8 is 23.7. We would therefore expect for a
data set with a mean of 32.1 and a standard deviation of 2.8, 99.7 percent of the
data set would lie between the values of 23.7 and 40.5.
