Control Charts

Statistical Process Control

Control charts are a statistical process control (SPC) tool used to monitor and manage processes by tracking the performance of key variables over time.

Control charts help identify trends, shifts, or unusual patterns that may indicate potential problems within a process. As a result, they provide valuable insight into the process's stability over time.

The type of control chart you use depends on the format of your data. To help determine the most suitable chart, you can refer to a decision tree. We’ll explore this decision tree in more detail later.

But what exactly are control charts used for? To answer this, let’s start with an example, specifically, an Xbar-R chart.

Xbar-R Chart

Imagine you're working in quality management at a fulfillment center, where products are stored, packed, and shipped to customers. The main goal of the fulfillment center is to process orders efficiently and accurately.

Therefore, it's important to monitor the stability of this process. One way to do this is by measuring the time from order receipt to shipment. Our objective is to track the average processing time and ensure it remains within acceptable limits.

Of course, we need data to monitor the process. To collect this data, we take a random sample of five orders each day.

On the first day, we measure the processing time for five orders. For example, the first order took 12 minutes, the second took 14 minutes, and so on. We continue measuring processing times on the second day, third day, and so on. Let's say we collect data for a total of 25 days.

But how do we create an Xbar chart using this data? To do this, we first calculate the mean values of the five sampled orders for each of the 25 days.

Now we can create the Xbar chart. To do this, we plot the 25 days on the x-axis and the corresponding daily mean processing times on the y-axis. Resulting in one point for each day: the first, second, third, and so on.

We are almost finished. The last step is to add three important lines. The center line represents the overall mean of all the plotted points. The red lines above and below are the upper and lower control limits, which define the acceptable range of variation.

The Upper Control Limit (UCL) is the line above the center line, typically set at three standard deviations above the mean. It marks the upper boundary of expected process variation.

The Lower Control Limit (LCL) is the line below the center line, also set at three standard deviations from the mean. It defines the lower boundary of expected process variation.

It's important to note that there are different methods for calculating sigma. Some offer more accurate approximations, while others are simpler to apply. The most straightforward approach is to calculate the standard deviation of all the collected data.

Additionally, it's important to distinguish between control limits and specification limits. Control limits are determined by process variability and statistical calculations, while specification limits are set by customer requirements or engineering tolerances.

Now we have the so-called Xbar chart. In most cases, the Xbar chart is paired with the R chart, where R stands for range.

To create the R chart, we simply calculate the range for each day. The range is the difference between the largest and smallest values in the daily sample. For example, on day 1 the largest value is 15 and the smallest is 12, giving us a range of 3.

We then do the same for all other days.

If you want to create an Xbar-R control chart with DATAtab, simply copy and paste your data into the table and click on "Statistical Process Control". Then, select the variables and the Xbar-R chart will be produced automatically.

Xbar-R Chart vs Xbar-S Chart

Note: If you have continuous data with subgroups—as in our example, where several measurements are taken each day— an Xbar-R chart is used if the subgroup sample size is less than 10. If the sample size is larger, the Xbar-S chart is used.

Now that we know what an Xbar-R chart is, what about the other types? Let's start with the I-MR chart, which is easy to understand. Both chart types are used when we have continuous data. The only difference is that the Xbar-R chart uses multiple samples at each time point, while the I-MR chart uses only a single value per time point.

I-MR Chart

The I-MR chart stands for Individual-Moving Range chart. But how does it differ from the Xbar-R chart?

In an Xbar-R chart, multiple observations are collected at each time point. When only a single observation is available per time point, an I-MR chart is used instead.

To create the I chart, we simply plot the individual value at each time point. Since there is only one value per point, we cannot calculate a range like in the X̄-R chart. Instead, we use a moving range. In the Moving Range (MR) chart, we calculate and plot the absolute difference between consecutive data points.

For example, if the difference between the first and second points is 1, we plot a moving range of 1. Similarly, if the difference between the second and third points is 2, we plot a value of 2 on the MR chart.

To create an I-MR chart using DATAtab, simply paste your measured values into the table. Then, select a single variable, and the I-MR chart will be created automatically.

Control Charts for Discrete Data

Now that we have discussed control charts for continuous data, what if we have discrete data?

In control charts for discrete data, we focus on the number of defects in a process. Here, we differentiate between one defect per unit or several defects per unit. We’ll explain what that means in just a moment.

Additionally, in both cases, we need to consider whether the sample size remains constant or varies over time.

np Chart

Let's say you work for a company that manufactures light bulbs, and you want to monitor the proportion of defective bulbs produced each day.

To do this, you take a random sample of 10 light bulbs each day and record how many of them are defective.

For example, on the first day, 2 bulbs were defective; on the second day, 1 was defective; on the third day, 3 were defective, and so on. In practice, the number of defects would likely be much lower, and a larger sample size might be used for more accurate monitoring.

In this case, each unit can have only one defect (a bulb is either defective or not) and the sample size remains constant. Therefore, an np-chart is the appropriate choice.

The np-chart is used to plot the number of defective units over time, helping to identify trends or shifts in the defect rate. For example, if two bulbs were defective on the first day, one on the second, and so on, each of these counts would be plotted on the chart. Of course, you can also create an np-chart using DATAtab.

p Chart

Now you might be wondering, what is the difference between a constant sample size and a variable sample size? In our light bulb example, we used a constant sample size each day.

Alternatively, a machine might randomly sort out lamps throughout the day. For example, it might sort 155 lamps one day, 180 the next, then 121, and so on. In this case, the number of lamps in the sample changes each day, resulting in a variable sample size.

To calculate the error rate, we divide the number of defective lamps by the total number of lamps in the sample.

So, when there is only one possible defect per unit and the sample size varies, we use a p-chart. To create a p-chart, you need two columns: one for the sample size and one for the number of defects found in each sample.

With these values, we can calculate the proportions and plot the results.

c Chart

What if a single unit can have multiple defects? For example, suppose a car manufacturing plant wants to monitor the number of defects found on each car body to maintain high-quality standards.

Each day, one car body is inspected, and the total number of defects per car body is recorded. In this case, we use the c chart.

To draw the c chart, we simply record the number of defects found per car. For example, if 4 defects are found in the first car, we plot a point at 4.

u Chart

But what is an example of a u chart? In this case, there are several defects per unit and a variable sample size.

Imagine a software development team that wants to monitor the number of bugs per software release.

The individual releases, of course, vary in size. One way to measure the scope is by tracking the number of lines of code added. In this case, we have one column for the number of lines of code and another for the number of reported bugs.

This allows us to calculate the bugs per line of code. With this data, we can now create a u chart.

Of course, you can also create control charts online with DATAtab for discrete data. To do this, simply click on the Attributive option, select one or more defects, choose the measured values, and then either specify a constant sample size or provide the variable for the sample size. The correct control chart will then be displayed automatically. Below, you can see a list of the control charts along with instructions on how to create them.