The Xcross-R control chart

Introduction

1. Calculate the range R

For the range control chart, one starts by calculating the average of the R-_cross ranges. This is the average of the range of each sample.

Example:

We have the results of 5 samples of 3 parts. We get the following table:

We obtain the range calculating the average of the range of each group, in our example 1.46.

2. Calculation of limits and construction of the R-chart

To obtain the high and low limits at 3 σ, they are estimated from an array of coefficients. The formulas are as follows:

• UCL: Upper limit = R_cross * D₄
• LCL: Lower limit = R_cross * D₃

By taking the data from the previous table, we get

• UCL = 3.7595 (D4 being at 2.575)
• LCL = 0 (D3 is equal to 0 for subgroups of 3 units).

3. Calculate the average X_cross and build the chart of averages

3.1 Calculation of X_cross

This is the average of our measures.

3.2 Calculation of limits

To obtain the high and low limits at 3 σ, we use the following formulas:

UCL: Upper limit = X_cross + A₂ * R_cross

LCL: Lower limit = X_cross -a₂ * R_cross

Example:

The previous example follows:

The average of each sample is calculated

We get:

• X_cross = 10.5
• UCL = 11.99 (A2 being equal to 1.023 for subgroups of 3 units)
• LCL = 9.01

Interpretation

Average control chart

The X-_cross chart reflects the fact that the process is centered. If it is normal, the center of the process does not drift. If one of the criteria is validated, the process will drift or change erratically and quickly. It then becomes necessary to refocus it by looking first:

• The setting of the machines
• The characteristics of the materials used
• The techniques practised by the operators.

Range control chart

It reflects uniformity and consistency. If the graph is narrow, it is that the product is uniform. If on the contrary a criterion is validated, then it is said out of control, the process contains something that does not work regularly.

Attention is drawn to the fact that in terms of reading the criteria, they differ according to whether we are at the upper or lower limit.

If we are at the low limit, our dispersion is weak, we are improving and there is no need to act. On the other hand one can investigate to understand why one improves.

Conversely, if the criteria are validated ” from the top “, our dispersion increases and we have to act. Usually this can be improved by better maintenance and good training of the operators.

The factors to be applied

The formulas for calculating the 3-σ limits depend on coefficients. This is because in reality, calculating the standard deviation is statistically complex if you have little value.

To overcome this problem, the creators of the method have tabbed coefficients to simplify the calculations and estimate the standard deviation from the scope. The reduced extent is calculated from the formula:

W = R/σ

With:

• R: range Values
• Σ being the standard deviation

We’re going to deduct 2 coefficients, calculated from the normal law, that are:

• D₂ which represents the average W
• D₃ which represents the standard deviation of W

The limits therefore become for the scopes:

R_Bar ± 3 * (d₃/d₂) * R_Bar

That is written as follow

R cross + D3 * r cross & R cross + D4 * r_cross

With:

• D3 = 1 – 3 * (d₃/d₂)
• D4 = 1 + 3 * (d₃/d₂)

And in the same way, the limits for the averages are:

Xcross ± 3 * (d₃/d₂) * R_cross

That is written as follow

Xcross ± A2 * R_cross

With: A2 = 3/(√ (n) * d₂)

The different values above depend on the size of the different samples. It is simply recalled that if we collect more than one sample each time:

• The overall average X_cross is calculated against the average of each sample
• The overall average R_cross is also calculated relative to the extent of each subgroup.

From the table

Source

S. M. Zimmerman, M. L. Icenogel (1999) – Statistical quality control

J. Ledolter, C. Burril (1999) – Statistical quality control

D. H. Stamatis (2003) – Statistical Process Control