X-bar and R Chart: Control Limits, Example and Interpretation
Open the X-bar and R Control Chart (opens in a new tab)
What is a control chart?
A control chart is a graph for checking, in time order, whether a process is stable. Even when products are made by the same process, their dimensions and weights always vary. This variation is of the following two kinds.
- Chance variation: small variation that cannot be avoided, such as slight differences in material
- Out-of-control variation: variation with a cause, such as equipment trouble, tool wear, or a change in material lot, that should be identified and removed
A control chart has a center line and upper and lower control limit lines, and from the position and arrangement of the points it tells whether out-of-control variation is occurring. The state in which there is no out-of-control variation and only chance variation remains is called “in control” (a state in which the process is stable).
What is an X-bar and R chart?
An X-bar and R chart takes several data points at a time from the process and, for each such set (a subgroup), shows the following two charts, one above the other.
- chart (upper): Shows changes in the subgroup means. Detects shifts in the process mean
- R chart (lower): Shows changes in the subgroup range (maximum − minimum). Detects changes in the process variation
Subgroup size n is used from 2 to 10, and 4 or 5 is common.
Formulas
Let k be the number of subgroups and n the subgroup size.
| Symbol | Meaning |
|---|---|
| Subgroup mean | |
| R | Subgroup range (maximum − minimum) |
| Mean of the subgroup means (the overall mean) | |
| Mean of the subgroup ranges |
chart
Center line =
Upper control limit (UCL) = + A2 ×
Lower control limit (LCL) = − A2 ×
R chart
Center line =
Upper control limit (UCL) = D4 ×
Lower control limit (LCL) = D3 × (not considered when n is 6 or less)
A2, D3 and D4 are coefficients determined by the subgroup size n.
| n | A2 | D3 | D4 |
|---|---|---|---|
| 2 | 1.880 | — | 3.267 |
| 3 | 1.023 | — | 2.574 |
| 4 | 0.729 | — | 2.282 |
| 5 | 0.577 | — | 2.114 |
| 6 | 0.483 | — | 2.004 |
| 7 | 0.419 | 0.076 | 1.924 |
| 8 | 0.373 | 0.136 | 1.864 |
| 9 | 0.337 | 0.184 | 1.816 |
| 10 | 0.308 | 0.223 | 1.777 |
The control limits are lines drawn above and below the center line at a distance of “3 times the standard deviation of the subgroup mean.” The coefficients are used to obtain this value from the subgroup range.
Worked example
Suppose the subgroup size is n = 5 and the measured values of one subgroup are 10.1, 9.9, 10.2, 10.0 and 9.8.
- Subgroup mean = (10.1 + 9.9 + 10.2 + 10.0 + 9.8) ÷ 5 = 10.0
- Subgroup range R = 10.2 − 9.8 = 0.4
Suppose that calculating this for 25 subgroups gives = 10.00 and = 0.40. Since n = 5, A2 = 0.577 and D4 = 2.114.
chart
- Center line = 10.00
- UCL = 10.00 + 0.577 × 0.40 = 10.00 + 0.23 = 10.23
- LCL = 10.00 − 0.23 = 9.77
R chart
- Center line = 0.40
- UCL = 2.114 × 0.40 = 0.85
- LCL: not considered, since n = 5
How to read a control chart
On a control chart, you check whether any points fall outside the control limits and whether the arrangement of the points shows any pattern.
Point Point flagged by a rule
- A: Stable: The points are scattered randomly above and below the center line, within the control limits. Only chance variation is present
- B: A point outside the control limits: Even one point outside the limits indicates a high likelihood of an out-of-control condition. Investigate what happened at that time
- C: Run: The points continue on one side of the center line. The process mean may have shifted
- D: Trend: The points keep rising (or keep falling). The process may be changing gradually, for example because of tool wear
Typical rules for detecting out-of-control signals
The eight rules below, given in standards such as ISO 7870-2 (JIS Z 9020-2 in Japan), are commonly used to judge out-of-control signals on a control chart. They are similar to what are known as the Nelson rules (and the Western Electric rules) in English-language references. The distance from the center line to the control limit is divided into three equal parts, and the parts are called zones C, B and A, in order from the center line.
| Rule | What it may indicate | |
|---|---|---|
| 1 | 1 point is beyond a control limit | A sudden out-of-control event |
| 2 | 9 consecutive points are on the same side of the center line | A shift in the mean |
| 3 | 6 consecutive points keep increasing, or keep decreasing | A gradual change, such as tool wear |
| 4 | 14 consecutive points alternate up and down | Two machines, operators or similar sources are alternating |
| 5 | 2 of 3 consecutive points are in zone A or beyond, on the same side of the center line | The start of a shift in the mean |
| 6 | 4 of 5 consecutive points are in zone B or beyond, on the same side of the center line | The start of a shift in the mean |
| 7 | 15 consecutive points are within zone C | The variation is too small (there may be a problem with how subgroups are formed or how data are recorded) |
| 8 | 8 consecutive points are on both sides of the center line, and none of them is within zone C | Two different conditions are mixed together |
In Japan, rules such as “7 consecutive points on one side of the center line” and “7 consecutive points rising or falling” have also long been widely used to flag out-of-control conditions.
In the newer edition of JIS Z 9020-2, these rules are positioned as guidelines rather than fixed criteria.
Which rules to use varies from workplace to workplace. If your company or customer has its own rules, follow them.
Control limits and specification limits are different
Control limits and the upper and lower specification limits are completely different lines.
- Control limits: lines calculated from the process data. Used to judge whether the process is stable
- Specification limits: the lines the product is required to meet; set by the design or the customer
Even if the process is stable (in control), out-of-specification product will occur if the variation is large relative to the specification limits. Conversely, even if everything is within the specification limits, a point outside the control limits is a sign that an out-of-control condition has occurred in the process. The two are easy to confuse, so be careful when drawing specification limit lines on a control chart.
Collecting data
- Forming subgroups: Put items made under nearly the same conditions into the same subgroup (for example, items made consecutively in the same time period). If items made under different conditions are mixed in a subgroup, R becomes large and out-of-control signals are easier to miss
- Number of subgroups: When finding control limits, a guideline is to collect about 20 to 25 subgroups
- Using the control limits: Once you have confirmed that the process is stable, fix those control limits and plot the subsequent data against them
Relationship with Cpk
The process capability index Cpk presupposes that the process is stable. Even if you calculate Cpk for a process with out-of-control conditions, you cannot tell the process capability. The basic order is to first confirm with an X-bar and R chart that the process is stable, and then calculate Cpk.
How to calculate Cpk is explained in What Is Cpk?
Making one in Excel
This is an example where subgroups of 5 values are in columns B to F, with 25 subgroups in rows 2 to 26.
| What to calculate | Example formula |
|---|---|
| Subgroup mean (column G) | =AVERAGE(B2:F2) |
| Subgroup range R (column H) | =MAX(B2:F2)-MIN(B2:F2) |
=AVERAGE(G2:G26) | |
=AVERAGE(H2:H26) | |
| UCL of the chart | + 0.577 × (when n = 5) |
| LCL of the chart | − 0.577 × |
| UCL of the R chart | 2.114 × |
Make columns G and H into line charts and add the center line and control limit lines, and you have an X-bar and R chart.
Calculating with the tool on this site
With the X-bar and R Control Chart, just paste in your data and it calculates the control limits and draws an chart and an R chart. It also flags points outside the control limits. You can also fix control limits calculated from past data and use them. No registration is required, and it is free to use.