Virtually all processes exhibit variation, and it is our hope that such variation is constant and due only to common causes. However, identifying special cause variation and lack of process stability is critical to success, and the lower control limit is a fundamental part of identifying special cause variation.

## Overview: What is a lower control limit (LCL)?

On a control chart, the lower control limit is a line below the centerline that indicates the number below which any individual data point would be considered out of statistical control due to special cause variation. It is typically set to three standard deviations below the centerline for charts plotting central tendency, and some multiple of the centerline for charts plotting variation.

In the case of charts for central tendency, it is critical to utilize the correct estimate of standard deviation in calculating the lower control limit. A measure of short-term or “within subgroup” variation is utilized to identify the presence of special causes that appear over time, and these estimates are often adjusted by an unbiasing constant.

## 1 benefit and 1 drawback of a lower control limit

The lower control limit is a critical value as it allows for the most commonly used detection rule to be used on values below the center of the data.

### 1. It identifies unusually low values

The rule used to determine an unusually low value in a dataset is a point below the lower control limit. When this is observed, we typically determine it to be due to special cause variation and an indication that the process lacks stability.

### 2. It may fall outside of the possible data range

For certain datasets, and especially when utilizing Individuals control charts or charts for variation such a MR, R, and S, the calculation of the lower control limit may result in a value that is at or beyond the natural range of the process.

For example, a chart plotting cycle time may have a negative value for the lower control limit even though we cannot record negative times. In this case the lower control limit will not be useful in detecting special cause variation causing unusually low values, and alternative rules must be used.

## Why is a lower control limit important to understand?

Process stability is critical to improvement of the process, and the lower control limit is a fundamental part of assessing stability.

### It helps avoid tampering

By calculating a control limit based on statistical probability, we can identify whether a point lower than usual was to be expected given the common cause variability in the process or is an indication of something unusual happening. This, in turn, can help us avoid putting resources towards identifying the cause of points that were not actually indicative of a process shift.

### It identifies individual events

While other rules exist to identify more sustained shifts in a process, the control limits help us identify if a single point was likely due to special cause variation even if the process itself has not shifted.

## An industry example of a lower control limit

A manufacturer of food must take precautions to ensure that their products weigh, on average, at least as much as the declaration on the label. A control chart is made to identify whether the weights are stable by collecting packages five at a time each hour:

The lower control limit, labelled LCL on the graph, indicates that on this Xbar chart, any group of five packages averaging under 41.7503g is an indication that the process is unstable and special cause variation exists.

## 2 best practices when thinking about a lower control limit

While using the lower control limit may seem straightforward, there are some ways in which it is misused.

### 1. Use a logical estimate of standard deviation

When constructing a chart, we typically utilize only the variation within subgroups or short-term variation (such as the moving range) to eliminate as many possible special causes of variation as possible and only capture the common cause variation. The overall standard deviation of the entire dataset should not be used as it is more likely to contain special cause variation.

### 2. Verify the value is within the range of the dataset

On some charts and with some datasets, the lower control limit might fall outside of the range of values the process is able to produce. In this case, other rules should be utilized to detect special cause variation below the centerline.

## Frequently Asked Questions (FAQ) about lower control limits

### 1. How is the lower control limit calculated?

For charts that assess central tendency such as individuals and Xbar charts, the lower control limit is set to three standard deviations below the centerline. For charts that assess variation such as MR, R, and S charts, the lower control limit is set to a multiple of the centerline, which is specific to that chart.

### 2. How many points below the lower control limit indicate a process is out of control?

A single point falling below the lower control limit is an indication of special cause variation and should be investigated.

### 3. What is the difference between the lower control limit and the lower specification limit?

The lower control limit is calculated based on the data and indicates process stability. It does not indicate whether the process is performing well or poorly, only whether it is consistent. The lower specification limit, on the other hand, is the point below which the customer considers the process to have produced a defect. It is part of assessing process performance but does not assess stability.

## Defining what too low is

While processes produce a range of data, the lower control limit is a formal definition for the point in which we consider the data to be too low to have been due to common cause variation. Understanding the need for this formally and avoiding process tampering when there is no indication of special cause variation helps reduce noise and gain process stability.