What are upper and lower control limits

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UCL = Upper Control Limit

LCL = Lower Control Limit

Control Limits are calculated based on the amount of variation in the process you are measuring.  One measure of variation is standard deviation*.  A common method of calculating control limits is the mean +/- three standard deviations.  For example, if the average part width measurement is 0.50” and the variation in measurement has a standard deviation of 0.005”, then the control limits are UCL = 0.515” and LCL = 0.485” (based on 3 standard deviations).

A process is considered “in control” when measured data points fall within the three standard deviation control limits.   Sigma σ is the Greek symbol used in statistics to represent standard deviation. The term “Six Sigma” refers to the common practice of using +/- 3 standard deviations about the mean to calculate control limits.

Control limits are not the same as specification limits.   Specification Limits (or “spec” limits) are set by the customer and typically cannot be exceeded without consequences.  For example, if your customer requires the part width to be 0.50” +/- 0.03, then your specification limits are USL = 0.53” and LSL = 0.47”.  If the part exceeds these limits, then it must be scrapped or re-worked.  The customer will not accept a part outside their tolerances.

*There are several different methods available for calculating standard deviation.  Consult a statistician to determine which method is appropriate for your application.

Control limits distinguish control charts from a simple line graph or run chart. They are like traffic lanes that help you determine if your process is stable or not. Control limits are calculated from your data. Control limit formulas are complex and differ depending on the type of data you have.

You can try and calculate control limits yourself, but ...

  • It will suck up a bunch of your time and you've got better things to do.
  • You'll probably make mistakes and get the wrong answer.
  • You will find your homegrown template hard to maintain.

What are upper and lower control limits

In a stable process:
68.3% of the data points should fall between ± 1 sigma.
95.5% of the data points should fall between ± 2 sigma.
99.7% of the data points should fall between the UCL and LCL.

How do you calculate control limits?

  1. First calculate the Center Line. The Center Line equals either the average or median of your data.
  2. Second calculate sigma. The formula for sigma varies depending on the type of data you have.
  3. Third, calculate the sigma lines. These are simply ± 1 sigma, ± 2 sigma and ± 3 sigma from the center line.

    + 3 sigma = Upper Control Limit (UCL)

    - 3 sigma = Lower Control Limit (LCL)

Why are there so many formulas for sigma?

The formula for sigma depends on the type of data you have:

  • Is it continuous or discrete?
  • What is the sample size?
  • Is the sample size constant?

Each type of data has its own distinct formula for sigma and, therefore, its own type of control chart.

There are seven main types of control charts (c, p, u, np, individual moving range XmR, XbarR and XbarS.) Plus there are many more variations for special circumstances. As you might guess, this can get ugly. Here are some examples of control limit formulas:

p Chart formula

What are upper and lower control limits

Individual Moving Range Chart formula

What are upper and lower control limits

X bar R Chart formula

What are upper and lower control limits

* "Introduction to Statistical Quality Control," Douglas C. Montgomery *

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What are upper and lower control limits

Just select your data and QI Macros does all of the calculations and draws the control chart for you.

QI Macros calculations are tested and accurate.

QI Macros built in code is smart enough to:

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QI Macros Also Makes it Easy to Update Control Limit Calculations

Once you create a control chart using QI Macros, you can easily update the control limits using the QI Macros Chart Tools menu. To access the menu, you must be on a chart or on a chart embedded in a worksheet.

What are upper and lower control limits

Here's what you can do with the click of a button:

There are also options to easily re-run stability analysis after changing data or control limit calculations.

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What are upper and lower control limits

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What are upper and lower control limits

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What are upper and lower control limits


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Welcome to the Omni upper control limit calculator aka UCL calculator! A simple tool for when you want to calculate the upper control limit of your process dataset. The upper and lower control limits are critical indicators to help you determine whether variation in your process is stable and caused by an expected source.

If you read further, you can learn what control limits and control charts are, how to calculate the upper control limit and implement it in real life. To better understand the concept, we have prepared an example for you as well. Come along! 👩🏻‍🏫

Control limits are used to detect whether the variation in a process we are observing is located within the expected limits.

More specifically, control limits help us see whether the observed variation in the process of interest is due to random or special causes. Any variation detected inside the control limits probably occurred by chance. On the other hand, variation outside of the control limits likely occurred due to special causes. The upper control limit (UCL) and the lower control limit (LCL) serve as boundaries for expected deviation in data.

Sounds complicated? Here is an example:

  • Let’s say your bakery takes 40 minutes on average to bake bread 🍞. Due to random causes, sometimes the baking process takes 46 minutes, but sometimes 34 minutes is enough. Because this variation in time is due to common causes, it is within statistical control. However, if your oven breaks down and bread baking takes one hour, the variation in time is caused by a particular cause (e.g., oven malfunctioning).

If you're wondering how to calculate the control limits of your process dataset, here are the UCL and LCL formulas below:

  • The upper control limit formula:
    UCL = x - (-L * σ)
  • The lower control limit formula:
    LCL = x - (L * σ)


  • x – Control mean;
  • σ – Control standard deviation; and
  • L – Control limit you want to evaluate (dispersion of sigma lines from the control mean)

Note: although the control limit you wish to evaluate could be any number, we set our calculator's default control limit as three-sigmas since it is most commonly utilized. If you want to learn more about the three-sigma rule, check Omni empirical rule calculator, and if you wish to evaluate different control limits, use our calculator's Advanced mode 🧠 feature!

Now that you know how to calculate the upper control limit let's talk about the use of control limits. Control limits are usually utilized by Six Sigma practitioners as a statistical quality control for detecting whether variations in the production process of interest are out of control (not stable). To do such statistical process monitoring, we look at control charts. If the control chart indicates that the process is out of control and variation is above the upper and lower control limits, analyzing the chart can help determine the particular cause of this variation. If you're also interested in your process's capability to produce results relative to customer requirements, check process capability index calculator.

What are upper and lower control limits
Process control chart

Remember the bakery example? 👨🏻‍🍳

Suppose you used our control limit calculator and determined that the upper control limit for breaking bread is 46 minutes 🕐. If the oven is not working correctly and takes one hour to bake bread instead of 40 minutes (average time of baking), the control chart of the process will display unexpected variations. In this case, data at some point in time will appear well above the upper control limit; therefore, as a bakery owner, you can assume that process performance is degraded because of the particular cause, e.g., malfunctioning oven, rather than random causes.