Can a Control Chart Be Drawn with Only One Data Point per Shift? —— Five Steps to Implement the Individual-Moving Range (I-MR) Chart

A quality engineer in a heat treatment workshop has been staring at a "control chart that doesn't look like a control chart" for two months. The critical components are inspected by furnace batch, and only one batch is produced per shift. The destructive testing can only take one sample, so the data collected over a month, when arranged by time, amounts to just 20 to 30 points. The training materials for the X̄-R chart require 4 to 5 consecutive samples per subgroup and a total of 20 to 25 subgroups, but this scenario can't even form a "subgroup."

Thus, three makeshift methods have emerged: some directly copy the tolerance upper and lower limits onto the chart as control limits, which never trigger an alarm; others force different furnace batches and material lots into a single subgroup to calculate the range, leading to fluctuating control limits; and still others simply don't draw the chart, waiting for batch issues to arise before tracing back the data. As a result, the process variation of the critical characteristics remains a black box.

Having a small amount of data is not a reason to abandon the control chart; choosing the right type of chart is the key. The Individual-Moving Range (I-MR) chart is specifically designed for processes where "only one data point is available at a time."

1. Why Can the I-MR Chart Be Built Without Subgroups?

The sources of variation in a conventional X̄-R chart are two: the differences among several products within the same subgroup (within-subgroup variation, reflecting common causes) and the differences between subgroup means. The first item is the basis for estimating process variation, but in single-piece production, destructive testing, and extremely slow processes, there is no "within-subgroup" variation. Each shift or furnace batch produces only one data point, making it impossible to use within-subgroup variation.

The I-MR chart uses a different approach: it estimates process variation using the difference between adjacent measurements—the moving range (MR). Adjacent points are close in time, with minimal changes in process conditions, material lots, and personnel. Their differences primarily reflect common causes, effectively substituting for the range within subgroups. This is the statistical basis for building a control chart in a single-value scenario.

The I-MR chart consists of two paired charts that must be drawn and interpreted together:

• The Individual Chart (I Chart), which monitors whether the process level has shifted;
• The Moving Range Chart (MR Chart), which monitors whether the process variation has increased.

Drawing only the top chart shows the level but not the variation; drawing only the bottom chart does the opposite. Both charts are necessary for a complete picture.

It is suitable for specific scenarios: single-piece or extremely slow processes like heat treatment by furnace batch, large castings, electroplating solutions, and periodic sampling in continuous processes; characteristics subject to destructive testing (where only one sample can be tested each time); and key characteristics measured by automated online inspection, but only one value is recorded per time interval.

2. Five Steps to Build the Chart

Step 1: Determine Measurement Points and Frequency. The I-MR chart has the same requirements for sampling representativeness as the X̄-R chart. At least 20 to 25 individual data points are needed to establish control limits; fewer points make the limits less reliable. For key characteristics, it is recommended to collect 50 to 100 points. The frequency should cover a complete process cycle (including shift changes, tool changes, and material lot changes), and the time points for machine stops, model changes, and material lot changes should be marked on the records. Otherwise, it will be impossible to explain anomalies later.

Step 2: Calculate the Moving Range and Estimate the Process Standard Deviation. Arrange the individual values x₁, x₂, ... in time order. The i-th moving range MRᵢ = |xᵢ − xᵢ₋₁|. The first point has no preceding data, so n individual values yield n−1 moving ranges. The moving range is the difference between adjacent points, essentially the range of a subgroup of size 2. Dividing the average moving range MR̄ by the constant d₂ (d₂ = 1.128 when n = 2) gives the process standard deviation estimate: σ̂ = MR̄ ÷ 1.128.

Step 3: Calculate the Control Limits for Both Charts.

• Individual Chart: The center line is x̄, and the control limits are x̄ ± 3 × σ̂ = x̄ ± 2.66 × MR̄ (since 3 ÷ 1.128 ≈ 2.66);
• Moving Range Chart: The center line is MR̄, the upper control limit is 3.267 × MR̄ (D₄ = 3.267 when n = 2), and the lower control limit is 0.

It is crucial to clarify one point: the center line is not the target value, and the control limits are not the tolerance limits. Using the specification upper and lower limits or the "defect rate target" as control limits is the most common mistake with I-MR charts—either the chart remains quiet as if there are no issues, or it triggers false alarms daily, neither of which reflects the process.

Step 4: Evaluate Stability, Remove Outliers, Recalculate, and Fix Limits. The limits calculated from the first batch of data are used as "trial limits." Identify and remove points influenced by special causes, investigate the reasons, and then recalculate the center lines and control limits using the remaining data. Continue this process until there are no obvious anomalies on the chart, then fix the limits and transition to monitoring. Fixed limits are not permanent: if the process, tooling, measuring instruments, material lots, or product specifications change, new samples must be taken and the limits recalculated.

Step 5: Tailor the Out-of-Control Rules. On the individual chart, common rules such as "points outside the limits, 7 consecutive points on the same side, continuous increase or decrease, and 2 out of 3 points beyond 2σ" can still be applied. However, on the moving range chart, typically only the "points outside the limits" rule is used, without continuous trend rules. The reason is that adjacent moving ranges share the same data point (MRᵢ and MRᵢ₊₁ both include xᵢ), leading to autocorrelation. Applying continuous rules designed for independent data would generate many false alarms.

3. A Numerical Example

The hardness of a critical component in a heat treatment process is recorded by furnace batch. The individual values (HRC) for 20 consecutive batches are: 54.2, 55.1, 54.5, 55.8, 54.9, 55.3, 54.1, 55.6, 54.7, 56.2, 54.4, 55.0, 55.4, 54.6, 55.2, 54.8, 55.1, 54.3, 55.5, 54.9.

• Center line x̄ = 55.0 HRC;
• Average moving range MR̄ = 0.88;
• σ̂ = 0.88 ÷ 1.128 ≈ 0.78 HRC;
• Control limits for the individual chart = 55.0 ± 2.66 × 0.88, i.e., 52.6 to 57.3 HRC;
• Upper control limit for the moving range chart = 3.267 × 0.88 ≈ 2.9 HRC.

Assume the 21st batch reports 57.6 HRC, which falls outside the control limits of the individual chart, triggering an alarm. The difference from the previous batch is 2.7, which is still within the 2.9 limit of the MR chart. This illustrates why both charts must be used together: the process level has shifted (possibly due to changes in furnace temperature settings or loading quantity), but the variation between adjacent points hasn't increased—detecting the level shift early with the individual chart allows for investigation before several more batches are affected.

Conversely, if the individual chart shows no level shift but the MR chart has consecutive points outside the limits, the focus should be on the source of variation: material lot consistency, clamping methods, and measurement techniques, rather than adjusting the set values. The same data, when interpreted with both charts, points to two different sets of causes.

4. Four Typical Misuses

1. Drawing Only the Individual Chart, Not the MR Chart. This discards half the information about variation. When the process is gradually drifting, the MR chart often shows the first signs of change, and drawing only one chart is like closing one eye.
2. Using Constants from the X̄-R Chart. Applying A₂ (0.577), D₄ (2.114), and d₂ (2.326) for n = 5 to the individual chart will result in overall misalignment of the control limits. The I-MR chart only uses the constants d₂ = 1.128 and D₄ = 3.267.
3. Substituting "Reasonable-Looking" Limits for Calculated Limits. Specification limits, last month's average, or the experienced line of a senior technician cannot replace the control limits calculated as three times the standard deviation from the data.
4. Ignoring Measurement System Resolution. The individual chart records only one measurement per point, so any fluctuation in the measuring instrument will be entirely reflected in the point-to-point differences. Instruments with insufficient resolution or without a Measurement System Analysis (MSA) will produce charts that reflect the instrument, not the process—confirm the measurement system's usability before building the chart.

5. Three Questions for Implementation

Before building the chart, ask yourself three questions: Are the data points arranged in time order? Does each point represent an independent measurement? Have there been significant changes in process conditions during the data collection period, and are these change points marked? If you can answer all three questions, follow the five steps, and the chart will be reliable. If you can't answer any one of them, address the conditions first before building the chart; otherwise, the chart will be just for decoration.

Without subgroups, use adjacent differences to estimate variation—draw the individual chart to monitor the level and the moving range chart to monitor the variation. Both charts together form a control chart.

Knowledge code: 6.3.1

Version: v20261003

Author: QTank QTank is dedicated to providing systematic professional knowledge, methodologies, and practical tools for quality management practitioners, helping companies continuously improve their quality capabilities.