a normal distribution is a continuous, symmetrical distribution defined by
two parameters: \(\mu\) (mean) and \(\sigma\) (standard deviation), and
the probability function
\( \displaystyle{f(x)=\frac{1}{\sigma\sqrt{2\pi}}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}} \)
this is a relatively special function, but it is also built-in to most
calculators and computer applications, for example, in a TI-84 it is the
built-in function normalpdf(\(x\),\(\mu\),\(\sigma\)), where you supply
the variable name, the mean, and the standard deviation of your particular
distribution.
We write \( X \sim N(\mu,\sigma) \) to represent that "\(X\) is a normally
distributed random variable with mean \(\mu\) and standard deviation \(\sigma\)."
the standard normal distribution is the normal distribution with
\( \mu=0 \) and \( \sigma=1 \)
draw several normal distributions on your calculator and paper to get accustomed
to this famous shape
for a normally distributed random variable \(X\), the z-score of a
data element \(x\) of \(X\) is the number of standard deviations
of the data element from the mean
\(\displaystyle{z=\frac{x-\mu}{\sigma}}\)
z-scores allow us to compare the data in normal distributions with
different means and standard deviations
The Empirical Rule
If \( X \sim N(\mu,\sigma) \), then
\(68\%\) of the data lies within one standard deviation of the mean,
\(95\%\) of the data lies within two standard deviations of the mean, and
\(99.7\%\) of the data lies within one standard deviation of the mean
6.2 Using the Normal Distribution
recommended:
43-59, 68-88
for a normal distribution \(X \sim N(\mu,\sigma)\), the area under the probability
distribution function to the left of the value \(x\) represents the probability
\(P(X < x)\).