**Little known, this map presents very many in relation to the traditional control chart of Shewhart.**

## Introduction

The EWMA control chart (Exponentially weighted moving average – moving medium and exponential weighting) is a measurement control chart. Developed by S. W. Roberts in 1959^{1}, it represents point averages.

When we have slow or weak drifts, the easiest way is to use a traditional Shewhart control chart but by increasing the size of each sample. The downside is the cost of this solution. The EWMA control chart allows you to address this problem.

This control chart is very interesting because it has the advantage of:

- Take into account more data than the traditional control chart
**I-MR**,**x**and_{cross}-R**x**._{cross}-S - Be more sensitive than the control chart
**Cusum**To detect brutal drifts, but nevertheless less effective than Shewhart control chart in this specific case. - More suitable than the control chart
**Cusum**for individual values. - It only involves a decision criterion, the one where the data exceed the upper or lower limit.
- It smoothes the vagaries due to the natural variability of the process and thus avoids unnecessary questioning.

The use of this control chart is more common for continuous low-drift processes (e.g. bottle filling). Being more effective in detecting weak or slow drifts, it is particularly suitable. For discontinuous processes more subjected to brutal drifts, there is no need to have too much sensitivity. We’d rather use Shewhart control chart.

**EWMA control chart to measurements and attributes**

The EWMA control chart is suitable for quantitative measurements as well as for non-compliant proportions. All of the formulas presented below are written for the measurements (we talk about average…). For a control chart to attributes, it will suffice simply to replace the average term with the proportion, the formulas remaining the same.

## 1. Calculate the weighted average of the samples

For each sample, which can count 1 or more samples and be variable, a weighted average of ” *history *” is calculated. The formula is as follows:** **

**M _{i} =**

**λ * x**

_{icross}+ (1 – λ) * M_{i-1}With:

**M**average value of the sample I_{i}:**m**Previous value of point M_{I-1}:_{i}**λ:**adjustment constant between 0 and 1. It represents the ”*weight*” given to previous results

Note that:

**plus λ is close to 0:**The more we take into account the past. This implies that the weak drifts will be more easily identified. On the other hand, the brutal drifts and the important detunings, being smoothed, will be less well detected.- The
**more λ is close to 1:**the less the past is considered. This implies that we will have better reactivity to identify the abrupt depositions but, on the other hand, the weak variations can be detected less well. **if λ = 1:**The resulting map is an X-_{cross}control chart

## 2. Deduct the Limits

The limits are mobile because they are dependent on the sample number. However, they converge very quickly to a right.

**UCL = M _{0} + 3 ***

**√ (variance)**

**LCL = M _{0} -3 * √ (variance)**

With:

**I:**the sample number**σ**: standard deviation of data**n:**sample size**λ:**the weighting factor.**M**the value on which you want to center the map_{0}:**Variance:**

## 3. Interpretation

The rules for the interpretation of EWMA control chart are not the same as those of SHEWHART’s control cards. Only crossing limits triggers a process drift signal.

In case of adjustment, the mid value appears as an estimate of the average value of the process. It is therefore necessary to adjust the process of the difference between the mid-value and the target. Note that if a refocusing of the process is done, the map must be restarted to 0 to no longer take into account the data history that was processed by this refocusing.

Finally, if the EWMA control chartis very efficient to detect slow drifts, it is less than the SHEWHART control chart to detect fast drifts. The ideal is to use all three control chart : EWMA, average X_{cross} (or individual values I) and range R.

## Source

1-S. W. Roberts (1959) – Controls charts tests based on geometric moving averages

2-D. M. Hawkins, D. H. Olwell (1998) – Cumulative sum charts and charting for quality improvement

K. M. Bower (2000) – Using exponentially weighted moving average

M. Pillai (2013) – Six Sigma, how to apply it

French Metrology College (2014) – Monitoring of measurement processes

Standard NF X06-031-3-control charts, Part 3: Moving average control cards with exponential weight.