Data: Defects per roll
\[
c = (12,\,10,\,18,\,9,\,20,\,11,\,15,\,8,\,22,\,14), \quad n_i = 100\ \text{m for all } i
\]
Average nonconformities per meter \[
\bar{u}=\frac{\sum c_i}{\sum n_i}=\frac{139}{1000}=0.139
\]
Control limits (since (n_i=100) for all samples, limits are constant): \[
UCL = \bar{u} + 3\sqrt{\frac{\bar{u}}{n}}
= 0.139 + 3\sqrt{\frac{0.139}{100}}
\approx \mathbf{0.2508}
\] \[
LCL = \bar{u} - 3\sqrt{\frac{\bar{u}}{n}}
= 0.139 - 3\sqrt{\frac{0.139}{100}}
\approx \mathbf{0.0272}
\]
Interpretation:
\(u_i\) = \(c_i/100\) are
(0.12, 0.10, 0.18, 0.09, 0.20, 0.11, 0.15, 0.08, 0.22, 0.14);
all points lie within (0.0272) to (0.2508) → process in statistical control.
[1] "List of 11"
[2] " $ call : language qcc(data = c_vec, type = \"u\", sizes = sizes, main = \"u-Chart: Defects per Meter (Roll length = 100 m)\", xlab| __truncated__"
[3] " $ type : chr \"u\""
[4] " $ data.name : chr \"c_vec\""
[5] " $ data : num [1:10, 1] 12 10 18 9 20 11 15 8 22 14"
[6] " ..- attr(*, \"dimnames\")=List of 2"
[7] " $ statistics: Named num [1:10] 0.12 0.1 0.18 0.09 0.2 0.11 0.15 0.08 0.22 0.14"
[8] " ..- attr(*, \"names\")= chr [1:10] \"1\" \"2\" \"3\" \"4\" ..."
[9] " $ sizes : num [1:10] 100 100 100 100 100 100 100 100 100 100"
[10] " $ center : num 0.139"
[11] " $ std.dev : num 0.373"
[12] " $ nsigmas : num 3"
[13] " $ limits : num [1, 1:2] 0.0272 0.2508"
[14] " ..- attr(*, \"dimnames\")=List of 2"
[15] " $ violations:List of 2"
[16] " - attr(*, \"class\")= chr \"qcc\""
