Home / Statistical Tools / Control Charts / Xbar-R and Xbar-S
Xbar-R and Xbar-S¶
From Excel click...
QXL Stat Tools Tab > Control Charts > Variables > Charts in Subgroups > Xbar-R or Xbar-S
These are the same chart with a different lower panel
Both are two stacked panels built from measurements collected in subgroups. The upper panel is identical on both: Xbar Chart, plotting the mean of each subgroup. Only the lower panel differs, and with it the estimator that feeds the limits on both panels.
| Xbar-R | Xbar-S | |
|---|---|---|
| Lower panel | Range Chart, plotting each subgroup's range | S Chart, plotting each subgroup's standard deviation |
| Within-subgroup sigma, default | Rbar, from the subgroup ranges | Sbar, from the subgroup standard deviations |
| Other estimator offered | Pooled standard deviation | Pooled standard deviation |
| Estimator not offered | Sbar | Rbar |
| Single-panel variants | Xbar only, R only | Xbar only, S only |
| Worksheet written | Xbar-R Chart |
Xbar-S Chart |
A range uses only the largest and smallest value in a subgroup. A standard deviation uses every value in it. Everything else on this page, and nearly every option on the Options page, behaves the same way on both.
Want to build one first?
How-To works through a finished chart step by step, from sample data you copy off the page straight into Excel. The data has a point that is out of control, so the result has something to read rather than twenty points in a row.
How-To with GroupBy does the same thing with a GroupBy column, so one run produces a separate chart per group.
Read the lower panel first on either chart: the limits on the means are computed from the within-subgroup spread, so the spread has to be stable before the means are worth judging.
Each chart has a dialog of its own, titled Xbar-R Chart or Xbar-S Chart, and each writes a worksheet of the same name.
When to use it¶
- Your measurements are continuous, and you collect two or more of them per point in time: five parts an hour, three samples per batch, four cavities per shot.
- You want the limits on the means to reflect only the variation inside a subgroup, so that a shift between subgroups shows up as a point outside the limits rather than being absorbed into wider limits.
Choosing between the two:
- Xbar-R when the subgroups are small. A range is an easy statistic to compute by hand and behaves well for a handful of observations per subgroup.
- Xbar-S when the subgroups are large enough that you want every observation in each one to contribute to the spread estimate, rather than only the two extremes.
If your process gives you one measurement at a time, use an I-MR chart instead. If you want only one of the two panels, both dialogs offer the single-panel variants listed above.
How it works¶
Quantum XL divides your rows into subgroups: a constant number of consecutive rows, an identifier column whose repeated value marks a subgroup, or one subgroup per row built across several measurement columns. Each subgroup contributes its mean and its spread, and those are what the two panels plot.
The center line of the Xbar panel is the grand mean of the individual observations. With unequal subgroup sizes that is not the same as the average of the subgroup means, because a larger subgroup carries more weight.
The limits come from a single within-subgroup standard deviation estimated once for the whole chart. The Xbar limits sit a number of standard errors either side of the center line, and the standard error uses each subgroup's own size: equal sizes give straight lines, unequal sizes give limits that step in where a subgroup is large and out where it is small.
The estimator is either Rbar (average range), selected when the dialog opens, or the pooled standard deviation. Sbar is not offered, because the companion panel on this chart plots ranges.
The Range panel uses the same estimate. Its center line is the range expected from a subgroup of that size, so it steps with subgroup size too, and its limits are fixed multiples of it. The lower limit is held at zero. A subgroup with fewer than two values has no range, so that position is empty on the Range panel while its mean is still plotted above.
Use unbiased constants (within sigma) is greyed out under Rbar, because the range estimator always applies its own constant. It becomes live only under the pooled standard deviation.
The estimator is either Sbar (average standard deviation), selected when the dialog opens, or the pooled standard deviation. Rbar is not offered, because the companion panel on this chart plots standard deviations.
The S panel uses the same estimate. Its center line is the standard deviation expected from a subgroup of that size, so it steps with subgroup size too, and its limits are fixed multiples of it. The lower limit is held at zero. A subgroup with fewer than two values has no standard deviation, so that position is empty on the S panel while its mean is still plotted above.
Use unbiased constants (within sigma) is live on this chart, because both estimators offered here apply an unbiasing constant. Clearing it flattens the S panel's center line to a single value at every point while the limits keep stepping, so the band becomes visibly asymmetric about a straight center line.
Two settings change what the limits describe rather than how they are computed. Estimate Limits From computes the limits from the first few subgroups only and extends them across the whole chart. Split Control Limits divides the chart into phases, each with its own center line and limits.
The exact formulas for both charts are on the Math Details page.
Output¶
The analysis writes one worksheet, named after the chart you ran: Xbar-R Chart or Xbar-S Chart. Under GroupBy there is one worksheet per group, each named with the group appended, for example Xbar-R Chart - {GroupName}.
Worksheets are divided by GroupBy group, not by chart. Every chart belonging to a group is stacked down that one worksheet, one below the other, so selecting three measurement columns without GroupBy produces three charts on a single sheet rather than three sheets. The User Input block is written once at the top of each worksheet; everything after it repeats for each chart on the sheet.
On the sheet, from the top:
- A User Input block in the first two columns, holding the analysis title, the columns you selected, and whether the calculation type was Excel or GroupBy. Modify Existing Analysis reads it back.
- The Xbar Chart panel, plotting each subgroup mean with its center line and control limits, plus the zone lines, reference lines, and phase dividers if you asked for them. Points that broke an enabled out-of-control test are drawn in a different color.
- The lower panel below it, Range Chart or S Chart, aligned so its points line up vertically with the panel above.
- A summary table to the right, listing for each panel the center line, the control limits where they exist, the count of out-of-control points, and the count of points you marked as outliers. A transformation adds rows naming the type and its parameters; a split chart adds a per-phase breakdown.
- A capability and histogram block below the panels, but only when you entered a specification limit or a target on the Capability tab.
When the analysis finishes, the Control Chart task pane opens on the right. See Task Pane for what it can and cannot change.