STATISTICAL INFERENCE

STATISTICAL CONTROL CHARTS

Case 1: µ, s known

The usual American and Japanese practice in making charts is to place the control limits at
µ ± , where

µ is the mean of the distribution of individual measurements on items produced by the process in its present state, it describes the centre or aim of the process;

s is the standard deviation of the process;

n is the size of the regular samples that are taken over time to produce the data.

The probability that a particular falls outside the µ ± limits when the process is in control is about 0.003, based on the assumption that we expect to have a distribution that is nearly normal. (Recall that the Central Limit Theorem suggests that the sample mean will be closer to normal than the individual measurements - approximate normality is good enough as the control chart provides only a warning that further investigation may be required.)

Example - Air conditioner thermostats

= 74.25° and 75.75°

For known µ and s, an x-bar control chart for continuous data, constructed with subgroups size n, has Centre line CL = µ Upper Control Limit UCL = µ + Lower Control Limit LCL = µ -

To produce an x-bar control chart using Minitab, the command is

MTB> xbarchart c E;

SUBC> mu k1;

SUBC> sigma k2.

where c is the column containing the data, E is the sample size and k1 and k2 are the values of µ and s.

Case 2: µ, s not known

Example (continued) - Variation In Percent Solids

Percent Solids (60 Subtracted) of Seventeen Samples in Triplicate

Subgroup
1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17
8.8	4.3	5.8	7.5	8.2	5.1	7.5	6.6	6.3	4.9	6.0	7.6	7.0	5.8	7.5	5.0	7.5
9.2	3.3	6.1	6.2	7.3	7.0	7.7	7.9	7.3	4.7	6.2	7.3	6.5	7.1	5.9	6.1	6.3
9.1	3.8	7.6	6.0	7.2	6.9	7.6	5.7	7.3	5.4	6.8	6.2	5.7	5.2	6.0	5.7	7.4
Average
9.03	3.80	6.50	6.57	7.57	6.33	7.60	6.73	6.97	5.00	6.33	7.03	6.40	6.03	6.47	5.60	7.07
Standard Deviation (s)
0.21	0.50	0.96	0.81	0.55	1.07	0.10	1.11	0.58	0.36	0.42	0.74	0.66	0.97	0.90	0.56	0.67
Range (R)
0.40	1.00	1.80	1.50	1.00	1.90	0.20	2.20	1.00	0.70	0.80	1.40	1.30	1.90	1.60	1.10	1.20

These data are used to estimate the process mean and standard deviation.

Just as we denote a mean by placing a bar over the top (), the average of the averages, or the grand sample mean, is denoted with a bar over the bar: .

The average of all of the observations is = 6.53; this is an estimate of the process mean µ (if the process is in control).

We can estimate the process standard deviation from the average of the standard deviations of the subgroups:

Again, we must assume the process is in control, at least with respect to its standard deviation, for this estimate to be meaningful. The estimate based on is not an unbiased estimate of the process standard deviation. Constants (usually referred to as c₄) have been computed which yield an unbiased estimate of the process standard deviation, s, (at least for normally distributed data) given by /c₄. The value of c₄ for various subgroup sizes is shown below.

 n   2      3      4      5      6      7      8      9     10 


c4 .780   .886   .921   .940   .952   .959   .965   .969   .973

Since we obtained = 0.66, the unbiased estimate of process standard deviation is /c₄, so = 0.66/0.886 = 0.74. The standard error of each subgroup mean is thus . The mean of the distribution of subgroup averages is the same as the mean of the distribution of single observations. Appealing to the Central Limit Theorem, for a stable process we should find about 99.7% of the subgroup averages within three standard errors of .