Basic Analysis and Graphing - SAS

Chapter 2 Performing Univariate Analysis 79
Statistical Details for the Distribution Platform
Normal Mixtures
The Normal Mixtures option fits a mixture of normal distributions. This flexible distribution is capable of
fitting multi-modal data.
Fit a mixture of two or three normal distributions by selecting the Normal 2 Mixture or Normal 3 Mixture
options. Alternatively, you can fit a mixture of k normal distributions by selecting the Other option. A
separate mean, standard deviation, and proportion of the whole is estimated for each group.
pdf:
k π i
σ ---- φ x – μ i
------------
i σ
i=
1 i
k
E(x) = π i
μ i
i=
1
k
k 2
Var(x) = π i
μ2 i
σ2
( + i
) –
π i
μ i
i = 1
i=
1
where μ i , σ i , and π i are the respective mean, standard deviation, and proportion for the i th group, and
φ ( . )
is the standard normal pdf.
Smooth Curve
The Smooth Curve option fits a smooth curve using nonparametric density estimation (kernel density
estimation). The smooth curve is overlaid on the histogram and a slider appears beneath the plot. Control
the amount of smoothing by changing the kernel standard deviation with the slider. The initial Kernel Std
estimate is formed by summing the normal densities of the kernel standard deviation located at each data
point.
Johnson Su, Johnson Sb, Johnson Sl
The Johnson system of distributions contains three distributions that are all based on a transformed normal
distribution. These three distributions are the Johnson Su, which is unbounded for Y; the Johnson Sb,
which is bounded on both tails (0 < Y < 1); and the Johnson Sl, leading to the lognormal family of
distributions.
Note: The S refers to system, the subscript of the range. Although we implement a different method,
information about selection criteria for a particular Johnson system can be found in Slifker and Shapiro
(1980).
Johnson distributions are popular because of their flexibility. In particular, the Johnson distribution system
is noted for its data-fitting capabilities because it supports every possible combination of skewness and
kurtosis.
If Z is a standard normal variate, then the system is defined as follows:
Z = γ+
δfy ( )