Statistics
The X̄-S Chart watches both the centre and the spread of a process once your improvement is in place. This guide walks through the tool's built-in sample data, the limits it calculated, and the decisions the chart supports.
The X̄-S Chart monitors subgroup mean and standard deviation. It is built for subgroups larger than ten. At that size the sample standard deviation gives a tighter read on dispersion than a range does, so shifts in centre and in spread both show up earlier.
You reach for this chart in the Control phase of RDMAICS. The analysis is done. The improvement is in. What you need now is a way to tell whether the new level holds when the people who built it stop watching it every day.
Two charts sit together for a reason. The mean chart answers the question of where the process sits. The standard deviation chart answers the question of how consistent each subgroup is. A process can drift in either direction without the other moving, so you read both or you read neither.
The tool opens with sample data already loaded. That data is fictional. It describes a food service operation tracking an order measure across thirty days, split between a lunch shift and a dinner shift, under a single series called Ten-order subgroups.
Picture the situation. A project team has just finished work on order handling. The daily figures have settled into the high twenties and low thirties. The team is about to roll off. Somebody has to decide whether the process is predictable enough to hand over, and what the owner should watch for afterwards.
The thirty daily values run from 27 to 31 for twenty nine days. The final value, Day 30 on the dinner shift, is 40. That single point is the interesting part of the case, and the rest of this guide comes back to it.
Four columns carry the information. Observation or period labels each row, here as Day 1 through Day 30. Measurements in time order holds the value itself. Group or stratum records the shift, either lunch or dinner. Series or scenario names the dataset, here Ten-order subgroups.
The subgroup size field is set to 10. That is the single most important input on the screen. The tool groups the measurements in the order they are supplied, so the first ten rows form subgroup one, the next ten form subgroup two, and the final ten form subgroup three. Thirty rows at a size of ten gives you three subgroups.
Time order is not a formality. The chart assumes the rows arrive in the sequence the process produced them. If you sort the table by value, or by shift, the subgroups stop meaning anything and the limits become decoration.
The group column lets you keep the shift label attached to each reading. It does not split the chart. In this sample the lunch and dinner readings alternate, so every subgroup mixes both shifts.
The result panel reports three points, which matches three subgroups of ten. The centre line for the mean chart sits at 29.5333. The upper limit is 31.4394 and the lower limit is 27.6272. Sigma is reported as 2.0099. Out of control is 0.
Below that, the signals note says that no rule was triggered. The tool states the conclusion carefully. The process is in statistical control over this period, which means it is predictable, not that it is good enough. That distinction is the whole point of the Control phase.
Look at the subgroup mean chart and the three plotted points sit inside a narrow band. Look at the subgroup standard deviation chart and you see a different shape. The third point climbs towards the upper control limit, because the value of 40 on Day 30 sits in the third subgroup and pulls its spread up while barely moving its mean.
That is the behaviour the S chart exists to expose. An unusual single reading inside a subgroup of ten dilutes into the average. It does not dilute into the standard deviation. Read the mean chart alone and you would call this period quiet.
The first decision is whether the improvement can be handed over. With zero out of control points and no rule triggered, the answer for this period is that the process is predictable. You can write a control plan around a centre of 29.5333 and limits of 31.4394 and 27.6272, and the owner has something concrete to check against.
The second decision is what the owner acts on. A point outside the limits on the mean chart means the centre moved. A point outside the limits on the standard deviation chart means consistency changed. Those two signals usually point to different causes and different responses, so the control plan should name both.
The third decision is baseline setting. The tool notes that the limits came from all three points, and invites you to set a baseline above so later data is judged against an earlier period. That is how you keep the gain honest. Freeze the limits from the period when the improvement was working, then plot next month against those limits rather than letting the limits drift with the data.
The fourth decision is attention. The rising third point on the standard deviation chart is a prompt to go and look at Day 30 before the team disbands, while the people who understand the process are still in the room.
Most problems with this chart come from setup rather than interpretation. The arithmetic is handled for you. The judgements about grouping and timing are not.
The sample case shows two of these clearly. Three subgroups is a thin basis for limits, and one unusual value inside a subgroup of ten can move the standard deviation chart while leaving the mean chart calm.
The X̄-S Chart describes behaviour over time. It does not explain it. The chart can show you that the third subgroup has more spread than the first two. It cannot tell you why Day 30 read 40, whether a dinner shift was short staffed, whether a measurement was recorded wrongly, or whether something changed upstream. That work stays with you and with the people who run the process.
It does not set a target. The centre of 29.5333 is where the process currently sits. It is not a goal and it is not a specification. Nothing on this screen compares the process to what the customer needs, so a process can sit happily inside its control limits and still miss every requirement.
It does not separate the strata. The group column holds the lunch and dinner labels, but the chart treats the mixed subgroups as single points. If you suspect the two shifts behave differently, this chart will not answer that question, and the mixing may even mask it.
It does not judge whether three subgroups is sufficient, decide which period deserves to be the baseline, or tell you when to review the limits again. The tool does offer an AI review that explains the result in plain language, which is useful when you hand the chart to an owner who did not sit through the project. The decisions about grouping, baselines and action remain yours.
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