**The X**_{cross}-S Control Chart represents the average of the data with the standard deviation of the same data. It is used for measurable data, in the medium and large series production framework.

## Introduction

The X-_{cross}-S Control Chart represents the average of the data with the standard deviation of the same data. It is used for measurable data, in the medium and large series production framework. It replaces the **X _{cross}-R** card when the sampling size is greater than 15 individuals. Indeed, beyond this value, the scope taking into account only the extreme values, does not take into account the information contained in the intermediate values.

The value of 15 is arbitrary. If data are often **aberrant**, it is best to use it with 10 samples. It is noted that below 6 it is not possible to use it.

Compared to the **X _{cross}-R** Control Chart, it is preferred in the context where we need to have very homogeneous data, where we look for an important degree of precision.

## 1. Calculate the standard deviation S and deduct the limits

The first step is to calculate the average standard deviation of all subgroups.

To obtain the high and low limits at 3σ, we will estimate them from an array of coefficients (see bottom of the article). The formulas are as follows:

**UCL = B4 * S _{cross}**

**LCL = B3 * S _{cross}**

## 2. Calculate X_{cross} and build the average Control Chart

### 2.1 Calculation of X_{cross}

This is the average of our measures.

### 2.2 Calculation of limits

To obtain the high and low limits at 3σ, we follow the following calculation:

**UCL: Upper limit = X _{cross} + A_{3}* S_{cross}**

**LCL: Lower limit = X _{cross} – A_{3 }* S_{cross}**

## 3. Interpretation

### Average control chart

The X-_{cross}graph 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.

### Standard deviation graph

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 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 we estimate the standard deviation when we have little value.

To overcome this problem, the standard deviation is estimated from coefficient.

The coefficient, C_{4}, which depends on the law of χ^{2}is calculated. It is calculated using the following formula:

**C _{4} = √ (2/(n – 1)) * (N/2 – 1)! /((n – 1)/2 – 1)!**

From this coefficient, all the others are deducted via the formulas:

**A3 = 3/(c _{4} ***

**√**

**(n-1)**)

**B3 = 1 – 3/C _{4} ***

**√ (1 – c**

_{4}^{2})**B4 = 1 + 3/C _{4} ***

**√ (1 – c**

_{4}^{2})## Source

C. D. Montgomery (1996) – The ASTM Manuel of presentation of data and control chart analysis

S. T. Foster (2004) – Managing quality

J. A. Patterson (2004) – Tools of statistical process Control