# Control charts in SPC

## Variables

## Control

## Charts

Provide verification for the impact of any change made and help ensure that the process remains stable over time. Without the use of controls charts, there’s a greater chance of wasting time and resources chasing false alarms, while being unsure of the impact of any change that’re made to the process and the process’ sustainability.

#### What is a control chart, really?

At a very rudimentary level, a **control chart** is a chart which has **points**, a **centerline**, and **control limits**. In SPC, there is a need to monitor the **accuracy** (centering) and **precision** (variability) of a process. In order to do so we need to two control charts, one for each purpose. These charts are,

**X-bar** chart (centering)

*Here the points represent the average value of individual measurements in a sample, as it plots these sample averages over time.*

*The centerline is the running average of the aforementioned sample averages referred to as x bar-bar.*

*The control limits are horizontal lines known as upper control limit and lower control limit, these will be discussed further later in the article.*

**R bar **/** s bar** chart (spread)

*Here the points represent the range of values in a sample for an R chart or the standard deviation from mean for an s chart.*

*The centerline is the running average of all range values, for each sample, or all standard deviations, for each sample.*

*The control limits are horizontal lines known as upper control limit and lower control limit, these will be discussed further later in the article.*

**Overview of the statistics involved and key definitions**

**Sample**: When dealing with large numbers, like in production, it’s rarely feasible to audit each member. We instead try to find a representative few and extrapolate our findings to the whole. Though it inevitably biases our results, if done properly it allows us to draw powerful conclusions about a population.

**Average** (aka ‘mean’): It provides a sense of the center value of a group of data. It can be calculated by simply adding the values for each member of the group and dividing by the number of values in the group.

**Range**: It is one of the most common measure of variation in a group of data. It is calculated by simply subtracting the minimum value from the maximum value. Though range is easier to calculate and understand, however it is less accurate because it only uses two data points from a sample.

**Standard Deviation**: It provides a better measure of variation because it uses all the sample data, however it is more complicated to calculate. In SPC, traditionally table values are used to estimate the standard deviation for a given range.

* Statistical theory shows that the standard deviation of small samples taken from a population is biased, i.e., it tends to be less than the true value. Thus it is better to inflate the value a bit by dividing by ‘n-1’ rather than ‘n’.

**Normal Distribution**: It is one of the most prevalent distribution in nature and specially in the field of statistics. It is often identified by the recognizable bell shape of the distribution, therefore it is sometimes called a bell curve.

* Even if a population isn’t distributed normally (eg. age), the mean of samples from a population tend to follow a normal distribution reasonably well.

#### i. Collecting Data

Why not collect everything? Why not measure everything?

Well, if it were convenient in terms of time and money then we would! But there are always constraints on both those factors. This creates a need for a more efficient way of collecting data, in SPC it is usually called “*Rational subgrouping*“.

Our approach needs to minimize variation within a sample but maximize variation between samples, because substantial variation within sample makes it harder to detect variation between samples if there is inherent variation within the process, it the process is stable then the there would be little no inherent variability in the process thus rendering the difference in time between readings inconsequential.

In SPC the process of collecting data is generally as follows,

- Collect a small sample (3-10 units) in rapid succession
- Wait for some amount of time before collecting another sample

* **NOTE: **Make sure to collect data in the order they’re produced because control charts are meant to be a time-series chart

#### ii. Construction Control Limits

#### X-bar chart

Here, we’re monitoring the averages of samples not the individual measurements. Therefore, control limits cannot simply be based on estimate of standard deviation of the individual measurements, because they do not accurately reflect the variability of the averages is much less than the variability of individual measurements. This relationship is given by the standard deviation of samples being root of number of samples times less than the standard deviation of individual measurements.

Also, there is a well established relationship between the standard deviation of a normal distributed process and the range of a normal distributed sample, also given that we’ve already calculated the overall range of the samples given by R-bar, therefore the estimate of standard deviation of a process is given by,

therefore,

Upon condensing the constant parts into a single value we get,

#### R chart

Now we need the control limits for the R chart (which is more common than the s chart). The centerline for this chart is going to be the overall average range, R-bar, we used previously. The upper and lower limits control limits depend on the sample size, defined as,

where, D_{3} and D_{4} are constants defined in the table below.

* Figure.* X-bar and R chart in

*FSWorks*by Factory Systems

#### Mathematical Constants

* **Trend analysis** is a highly time sensitive process. If you wait to draw the control charts and perform trend analysis on them there will be always a lag between the process going out-of-control and you detecting it. If the trend analysis shows that the process has gone out-of-control then you just spent time making defective parts and without proper traceability it will be very difficult to find and separate those parts for rework or for scrapping.

With * FSWorks *we aim to offer many features to our customers but probably none more important.

*offers continuous evaluation of trends with an instant alarm system, which alerts you to your process going out-of-control as soon as possible, while also offering comprehensive tracking for all the parts being produced on a line.*

**FSWorks*** Figure.* Customizable alarms based on continuous trend analysis in

*FSWorks*by Factory Systems

**Trend Analysis**

The primary purpose of control charts is to determine if the variation in a process is due to some assignable cause or is it inherent to the process. There are many ways to interpret a control chart in order to come to that determination, a simple approach may be to simply react to any points that fall outside of the control limits, but another popular approach is known as the Western Electric Rules, because it was developed by Western Electric Company, and is as follows,

A control chart indicates an out-of-control process if any of the following rules are met,

**Rule 1:** Any point outside of the control limits

**Rule 2:** 2 or 3 consecutive points beyond two standard deviations

**Rule 3:** 4 or 5 consecutive points beyond one standard deviation, on the same side of centerline

**Rule 4:** 7 consecutive points above or below centerline

**Rule 5:** 5 consecutive steps upward or downward

These rules can be adopted whole-cloth or be bent according to your specific requirements, depending on how strict you want your quality control to be.

* X-bar chart is based on the value of overall average range, R-bar, so if the R chart indicates an out-of-control process then our estimate of the range isn’t reliable and therefore we can’t be confident in the limits of the X-bar chart either.

**Future Steps**

When a process is determined to be out-of-control, the team needs to go looking for problems, once a problem is found, it needs to be fixed and the sample that indicated the existence of the problem needs to be excluded and the control limits recalculated and redrawn, these limits are called test limits.

If the new limits still indicate that the process is not in control then the above procedure needs to be repeated. However, if the process is now in-control, then extra attention needs to be paid to this process over the next several days, probably with an increased sampling frequency, and longer the process stays stable, the more confident you can be in its stability.

* If the R chart indicates an out-of-control process, because the points are above upper control limit, but the X-bar chart is in control, this indicates variability of the process has increased.

Else if the R chart indicates that the process is in control but the data points on the X-bar chart are trending downward or outside lower control limit, it indicates that the process’ average/center has decreased significantly.

If both charts indicate that the process is out-of-control, then there may be several issues to explore and address.

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