Chapter 6
The Normal Distribution
In this handout:
• Probability model for a continuous random variable
• Normal distribution
• The idea of a continuous probability
distribution draws from the relative frequency
histogram for a large number of
measurements (recall chapter 2).
• Group data in class intervals, compute the
relative frequencies of the intervals, and build
a histogram (figure (a)).
• The total area under the histogram is 1.
• For two boundary points a and b of a class,
the relative frequency of measurements in
interval [a,b] is the area above this interval in
the histogram.
• With increasing number of observations,
refine the histogram by having more class
intervals with smaller widths (fig. (b)).
• By proceeding in this manner, the jumps
between consecutive rectangles tend to
dampen out, and the top of the histogram
approximates the shape of a smooth curve, as
illustrated in figure (c).
• This curve is called probability density curve.
Boxes on Page 223
Probability density function; continuous random sample
• For important distributions, areas
have been extensively tabulated.
• In most tables, the entire area to the
left of each point is tabulated.
• To obtain the probabilities of other
intervals, we must apply the
following rules:
P[a < X < b] =
(Area to left of b) – (Area to left of a)
Figure 6.2 (p. 225)
Different shapes of probability density curves. (a) Symmetry and deviations from
symmetry; (b) different peakedness.
Figure 6.3 (p. 225)
Mean as the balance point and median as the point of equal division of the probability
mass.
Figure 6.4 (p. 226)
Quartiles of two continuous distributions.
Statisticians often find it convenient to convert random variables to a
dimensionless scale.
E.g., if a random variable X has mean 400 and standard deviation 60,
then the standardized variable is Z = (X - 400)/60
Normal distribution
A bell-shaped distribution has been
found to provide a reasonable
approximation in many
situations.
The normal distribution with a
mean of μ and a standard
deviation of σ is denoted by
N(μ, σ) .
The curve never reaches 0 for any
value of x, but because the tail
areas outside (μ-3σ, μ+3σ) are
very small, we usually terminate
the graph at these points.
Figure 6.6 (p. 230)
Two normal distributions with different means but the same standard deviation.
Figure 6.7 (p. 230)
Decreasing  increases the maximum height and the concentration of probability about
.