Complete study about the use of control charts

1. Background

Within manufacturing use of control charts is a daily business to track and control special characteristics of geometry. Due to cost and measuring time, the most extended chart is X-Rm and X-R. It is usual to do adjustments to geometry as part of continuous improvement process.

                When an adjustment is done in a piece of equipment or tooling, to check the result some parts or sub-assemblies are manufactured which will be measured afterwards to validate the change. Manufacturing processes and more specifically geometry characteristics usually have a behavior that can be statistically assimilated to a normal distribution, and therefore controlled by using control charts of the type previously commented. This point means that such distribution is explained by normal parameters - , which are estimated by .

                Geometry adjustments usually affect parameter , since those are changes in  the relative location of different parts and subassemblies. Eventually, due to maintenance interventions or improvements, variability reductions are also produced and therefore affect parameter s. This study is focused on the most usual type of changes which are those that affect the mean location.

2. Study objective

                The objective of the present study is to develop and deliver a tool for the different departments which are involved in manufacturing and therefore have to validate geometry changes.

                By means of the methodology suggested in this study, it could be stated with a confidence level of 99.9% that the change in the geometry has been made.

3. Statistics’ basics

                The aim is to find out the sample size to have a confidence level of 99.9% within a mean shift at the manufacturing process.

                To produce a mean shift is to move Gauss Bell which can be parameterized by  or graphically represented by lines within a control chart, as follows:

                                                                                                                            
                UCL and LCL stand for Upper Control Limit and Lower Control Limit which are placed from average +3.3S (3.3 times the standard deviation) and -3.3S. Although the use of 3S is more common, it does not make a practical difference in terms of sample size. The use of 3 or 3,3 is a decision based on process stability.

                Within X-Rm control charts, average value is represented by  since it is the mean of the individuals, therefore n=1.

                When a mean shift is produced, to have enough certainty that the change has been made we need that measurements after the adjustment surpass old control limits. This condition could be represented graphically as follows:

                This could also be analytically explained by means of the expression LCL₁≥UCL₀. If we select a sample size that ensures us the measurement points fall outside the old control limits when the change is made, then, we can state with a certainty of 99.9% (1-α) that the change has been made.

                At the same time if the point fall inside the old control limits, so the probability of having so is also 99.9% (1-β) as we have placed H₁ for our hypothesis test as shown on the picture above. Note that the process of the alternate hypothesis is represented on the right by LCL1- 1-UCL1 and is completely displaced from the current (old) process that is LCLo- o-UCLo for the Null hypothesis.

3.1. - Method development

                According to AIAG SPC manual, control charts  & X – Rm, are obtained from the following equations:

                Within the control charts used we do not normally have , but parameter R_total is available (total range). R is the difference between maximum value and minimum within a sample. If such sample has enough size (between 30 and 50) and the normality of data is previously checked, we can state that s ≈ R total / 4. Such relationship is an approximation (rough estimation) and therefore we cannot use symbol ‘=’. Moreover, it would depend on distribution and sample size. With sample sizes from 30 to 50, the error made is quite acceptable and does not improve as the sample size increases, since for n>150 → R/S >> 4.

                SPC manual estimates s using the expression

 ->

And therefore

Equation (1) can be expressed as

 If we need that at least LCL1≥UCLo is true, then (from now on we will use symbol = instead of ≈ to simplify notation)

This drives to

Also expressed as

Parameter  corresponds to the adjustment or change that we want to check, it is therefore a known parameter. Total R and s represent the variability of the process they are also known, and A2 and d2 are corrective coefficients which depend on the sample size n which is what we want to find out. These values are found in tables. For this study we have used constants tables as a source for these values, although they can be easily found in the statistical technical literature, occasionally they can be found with another nomenclature.

Therefore, the problem boils down to find out what value of n makes it possible to comply with the following equations:

 

NOTE: When X-Rm charts are used, we only have to use E2 instead of A2, although it would only work to confirm if the charts are valid to check the change made, since these charts are only run with n=2 (individual measurements compared to previous one to obtain Rm).

            3.2. Tables

By combining the different tables of constants the following one is obtained:

n	A2	d2	E2*	A2*d2	E2*d2
2	1,880	1,128	2,659	2,121	3
3	1,023	1,693	1,772	1,732	3 (n=2)
4	0,729	2,059	1,457	1,501	3 (n=2)
5	0,577	2,326	1,290	1,342	3 (n=2)
6	0,483	2,534	1,184	1,224	3 (n=2)
7	0,419	2,704	1,109	1,133	3 (n=2)
8	0,373	2,847	1,054	1,062	3 (n=2)
9	0,337	2,970	1,010	1,001	3 (n=2)
10	0,308	3,078	0,975	0,948	3 (n=2)

Table 1.1

 

From previous point expressions we calculate the following tables. From the second half of the expression we work out these values:

		R total
	0,1	0,5	1	1,5	2	2,5	3
0,1	1,818	0,364	0,182	0,121	0,091	0,073	0,061
0,25	4,545	0,909	0,455	0,303	0,227	0,182	0,152
0,5	9,091	1,818	0,909	0,606	0,455	0,364	0,303
1	18,182	3,636	1,818	1,212	0,909	0,727	0,606
1,5	27,273	5,455	2,727	1,818	1,364	1,091	0,909
2	36,364	7,273	3,636	2,424	1,818	1,455	1,212
2,5	45,455	9,091	4,545	3,030	2,273	1,818	1,515
3	54,545	10,909	5,455	3,636	2,727	2,182	1,818
3,5	63,636	12,727	6,364	4,242	3,182	2,545	2,121
4	72,727	14,545	7,273	4,848	3,636	2,909	2,424
4,5	81,818	16,364	8,182	5,455	4,091	3,273	2,727
5	90,909	18,182	9,091	6,061	4,545	3,636	3,030

Table 1.2

		S
	0,05	0,1	0,2	0,3	0,4	0,5	0,6	0,7
0,1	0,909	0,455	0,227	0,152	0,114	0,091	0,076	0,065
0,25	2,273	1,136	0,568	0,379	0,284	0,227	0,189	0,162
0,5	4,545	2,273	1,136	0,758	0,568	0,455	0,379	0,325
1	9,091	4,545	2,273	1,515	1,136	0,909	0,758	0,649
1,5	13,636	6,818	3,409	2,273	1,705	1,364	1,136	0,974
2	18,182	9,091	4,545	3,030	2,273	1,818	1,515	1,299
2,5	22,727	11,364	5,682	3,788	2,841	2,273	1,894	1,623
3	27,273	13,636	6,818	4,545	3,409	2,727	2,273	1,948
3,5	31,818	15,909	7,955	5,303	3,977	3,182	2,652	2,273
4	36,364	18,182	9,091	6,061	4,545	3,636	3,030	2,597
4,5	40,909	20,455	10,227	6,818	5,114	4,091	3,409	2,922
5	45,455	22,727	11,364	7,576	5,682	4,545	3,788	3,247

Table 2.2

4. Method for estimation of sample size using the tables

The result from tables 1.2 and 2.2 are the inputs for table 1.1. Therefore to check if the change is done, we would look for the value of , and in the case of not finding it, it is always recomendable to use the next lower value. It is the same with the variability parameter, either with R (table 1.2) or s (table 2.2), although in this case the most restrictive value for the inference is the next upper value. This value is  in the case of  charts or  in the case of X – Rm. Therefore, for a given variability (s or R) and a given  tables 1.2 and 2.2 gives us a number to be used on table 1.1 as explained below.

Once we have the value from tables 1.2 or 2.2 as explained above, we look for such value within table 1.1 (column A2*d2 for  charts or E2*d2 for X-Rm) that would correspond to a specific simple size (n) enough to say that the change has been made with a probability of 99,9%. In the case of not finding the exact value (that is what usually happens) we have to look for the value that gives us more certainty which is always the next lower, this obviously means a bigger sample size.

Note that expression from table 1.1 is always 3. This is because for charts X-Rm, n is always 2 (moveable range). Therefore, we have highlighted values from 3 and higher within tables 1.2 and 2.2 meaning that control charts X-Rm only in those cases of average shift and variability can be used with the sufficient reliability to state that the change has effectively been made.

As an example, in processes with a total range of 1, only in the cases where we want to produce a change of 2 or bigger, we could say that the change has been made or not with an almost total certainty by using a chart X-Rm. As we can see, the first value of the table that is equal to or bigger than 3 is 3,636 and corresponds to  = 2, as mentioned before.

In other words, we can also use the tables in a reverse way. Therefore, to know what type of chart and sample size is needed to make an inference o prediction with an almost total certainty. For instance, in a process with R=0.5 and a geometry change of 0.5, we should use n=3. Therefore, the most recommendable is to use a chart type .

We can also state that we should measure 3 parts, sub-assemblies or units to be almost sure that the change has been made when our Process has a total Range of 0.5 and we want to adjust the mean in 0,5.