Basic Analysis and Graphing - SAS

Chapter 2 Performing Univariate Analysis 73
Statistical Details for the Distribution Platform
s = standard deviation and g ( 1 – α ⁄ 2;
p,n)
is a constant that can be found in Table 4 of Odeh and Owen
1980).
To determine g, consider the fraction of the population captured by the tolerance interval. Tamhane and
Dunlop (2000) give this fraction as follows:
Φ---------------------
x+ gs – μ –
σ
Φ
x ---------------------
– gs – μ
σ
where Φ denotes the standard normal c.d.f. (cumulative distribution function). Therefore, g solves the
following equation:
P Φ X ----------------------
+ gs–
μ Φ
X ----------------------
– gs – μ
– ≥ 1 – γ
σ σ =
1 – α
where 1-γ is the fraction of all future observations contained in the tolerance interval.
More information is given in Tables A.1a, A.1b, A.11a, and A.11b of Hahn and Meeker (1991).
Statistical Details for Capability Analysis
All capability analyses use the same formulas. Options differ in how sigma (σ) is computed:
• Long-term uses the overall sigma. This option is used for P pk statistics, and computes sigma as follows:
σ
=
n
( x i
– x) 2
-------------------
n – 1
i = 1
Note: There is a preference for Distribution called Ppk Capability Labeling that labels the long-term
capability output with P pk labels. This option is found using File > Preferences, then select Platforms >
Distribution.
• Specified Sigma enables you to type a specific, known sigma used for computing capability analyses.
Sigma is user-specified, and is therefore not computed.
• Moving Range enables you to enter a range span, which computes sigma as follows:
σ
=
R
-------------
d 2
( n)
where
R is the average of the moving ranges
d 2 (n) is the expected value of the range of n independent normally distributed variables with unit
standard deviation.
• Short Term Sigma, Group by Fixed Subgroup Size if r is the number of subgroups of size n j and each
ith subgroup is defined by the order of the data, sigma is computed as follows: