These functions are used to compute statistics required by the xbar chart for one-at-time data.

1 2 3 | ```
stats.xbar.one(data, sizes)
sd.xbar.one(data, sizes, std.dev = c("MR", "SD"), k=2)
limits.xbar.one(center, std.dev, sizes, conf)
``` |

`data` |
the observed data values |

`center` |
sample/group center statistic. |

`sizes` |
samples sizes. Not needed, size=1 is used. |

`k` |
number of successive pairs of observations for computing the standard deviation based on moving ranges of k points. |

`std.dev` |
within group standard deviation. Optional for |

`conf` |
a numeric value used to compute control limits, specifying the number of standard deviations (if |

Methods available for estimating the process standard deviation:

Method | Description |

`"MR"` | moving range: this is estimate is based on the scaled mean of moving ranges |

`"SD"` | sample standard deviation: this estimate is defined as as(x)/cd(n), where n = number of observations x. |

The function `stats.xbar.one`

returns a list with components `statistics`

and `center`

.

The function `sd.xbar.one`

returns `std.dev`

the standard deviation of the statistic charted.

The function `limits.xbar.one`

returns a matrix with lower and upper control limits.

Luca Scrucca luca@stat.unipg.it

Montgomery, D.C. (2005) *Introduction to Statistical Quality Control*, 5th ed. New York: John Wiley & Sons.

Ryan T.P. (2000) *Statistical Methods for Quality Improvement*, New York: John Wiley & Sons.

Wetherill, G.B. and Brown, D.W. (1991) *Statistical Process Control*. New York: Chapman & Hall.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 | ```
# Water content of antifreeze data (Wetherill and Brown, 1991, p. 120)
x <- c(2.23, 2.53, 2.62, 2.63, 2.58, 2.44, 2.49, 2.34, 2.95, 2.54, 2.60, 2.45,
2.17, 2.58, 2.57, 2.44, 2.38, 2.23, 2.23, 2.54, 2.66, 2.84, 2.81, 2.39,
2.56, 2.70, 3.00, 2.81, 2.77, 2.89, 2.54, 2.98, 2.35, 2.53)
# the Shewhart control chart for one-at-time data
# 1) using MR (default)
qcc(x, type="xbar.one", data.name="Water content (in ppm) of batches of antifreeze")
# 2) using SD
qcc(x, type="xbar.one", std.dev = "SD", data.name="Water content (in ppm) of batches of antifreeze")
# "as the size inceases further, we would expect sigma-hat to settle down
# at a value close to the overall sigma-hat" (Wetherill and Brown, 1991,
# p. 121)
sigma <- NA
k <- 2:24
for (j in k)
sigma[j] <- sd.xbar.one(x, k=j)
plot(k, sigma[k], type="b") # plot estimates of sigma for
abline(h=sd(x), col=2, lty=2) # different values of k
``` |

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