Reading and Building Control Charts

A control chart plots a process over time, so you can tell normal scatter from a real change. That matters because the right response to each is opposite. Real changes need action now, often while every part is still good. Normal scatter should be left alone, because reacting to it makes things worse. Statistical process control (SPC) is built on that one decision.

What You Will Be Able to Do

• Tell common cause from special cause variation.
• Choose the right chart for your data.
• Calculate limits for X-bar and R charts using the standard constants.
• Set up rational subgroups and starting limits.
• Read the common signals and explain why adjusting a stable process backfires.

Two Kinds of Variation

No process makes every part the same. Material varies, temperature changes, tools wear.

A process with only common cause variation is stable, or in statistical control. The main purpose of a control chart is to tell these two apart over time.

Control Limits Are Not Spec Limits

Control limits describe what the process actually does, and they come from process data. Specification limits describe what the customer requires, and they come from the drawing. Never draw spec limits on an averages chart, and never treat the two as the same. A process can be stable and still make bad parts, if it is too wide for the spec. A process can also make good parts while being out of control, so it is unpredictable and may soon go bad.

The Statistics You Need

• Mean (X-bar): add the readings and divide by how many there are. For 1.002, 1.001, 1.003, 1.002 and 1.001, the mean is 1.0018.
• Range (R): the largest minus the smallest. Here, 1.003 - 1.001 = 0.002.
• Standard deviation (sigma): how spread out the data is around the mean.
• Normal distribution: the bell curve many stable processes follow.

In a bell curve, about 68.27 percent of values fall within 1 sigma of the mean, 95.45 percent within 2 sigma, and 99.73 percent within 3 sigma. Control limits sit 3 standard deviations above and below the process mean. A stable process puts only about 3 points in 1,000 outside those limits by chance, so a point out there is worth investigating.

Choosing a Chart

First decide what kind of data you have. Variable data comes from a measurement that can land anywhere on a scale, like a diameter or a weight. Attribute data comes from counting or sorting, like pass or fail, or scratches on a panel.

A defective is a whole unit that fails. A defect is one flaw, and a unit can have several. Defectives go on p and np charts, and defects go on c and u charts.

Building an X-bar and R Chart

The X-bar chart shows whether the center is moving. The R chart shows whether consistency is changing. They are always used together.

1. Pick the characteristic. Chart what is critical to function or safety, or what has caused trouble. You cannot chart everything.
2. Prove the gauge. A Gage R&R study analyzes measurement system variation to see if it is small enough relative to the process. A noisy gauge can hide real signals.
3. Set up rational subgroups. See the section below.
4. Collect starting data. Take 20 to 25 subgroups while the process runs normally. For subgroups of 5, that is 100 to 125 readings.
5. Calculate the limits. Use the constants below.
6. Check the starting data. Look for signals. Investigate and remove any special cause you find, drop those points, and recalculate.
7. Run it live. Plot each new subgroup as it is measured.
8. Recalculate only when the process changes. New tooling, new material or an improvement is a reason. One bad point is not.

D3, the constant for the lower limit of the R chart, is 0 for every subgroup size up to 6. So for n = 5, the R chart's lower limit is 0.

Worked example. You collect 25 subgroups of 5 parts. The average of all the subgroup means (X-double-bar) is 1.0020. The average range (R-bar) is 0.0022.

X-bar chart limits:

• UCL = X-double-bar + A2 x R-bar = 1.0020 + (0.577 x 0.0022) = 1.0033
• LCL = X-double-bar - A2 x R-bar = 1.0020 - (0.577 x 0.0022) = 1.0007

R chart limits:

• UCL = D4 x R-bar = 2.114 x 0.0022 = 0.0047
• LCL = D3 x R-bar = 0 x 0.0022 = 0

To estimate sigma, use sigma = R-bar / d2 = 0.0022 / 2.326 = 0.00095. The constant d2 depends on subgroup size. It converts an average range into a standard deviation, because a bigger subgroup naturally shows a bigger range.

The limits on the X-bar chart are tighter than 3 sigma of individual parts, because averages vary less than single readings. A2 builds that in, so individual parts can sit outside these limits and the chart can still be right.

Limits for the Other Charts

MR-bar is the average moving range, the average gap between one reading and the next. If a lower limit comes out negative, use zero. Two quick examples:

• I chart: a mean of 12.40 and an MR-bar of 0.18 give limits of 12.40 +/- 0.48, so 11.92 to 12.88.
• c chart: an average of 4 defects per unit gives an upper limit of 4 + (3 x 2) = 10, and a lower limit of 0.

I-MR charts are less sensitive to small shifts than X-bar charts, because single readings vary more than averages.

Rational Subgroups

A rational subgroup is a snapshot of the process at one moment. Parts in a subgroup should be made close together, under the same conditions, so the range inside it shows only short-term variation. Changes between subgroups then show up as shifts on the X-bar chart.

• Take 3 to 5 consecutive parts, at intervals such as every 30 to 60 minutes or every 25 to 50 parts. Base the interval on production rate and risk.
• Do not mix sources in a subgroup, such as two machines, two cavities, two spindles or two shifts.
• Give each source its own chart, unless you have shown they have the same mean and spread.

Mix sources and the range gets inflated, the limits get too wide, and the chart stops seeing real changes.

Reading the Chart

Read the R chart first. The X-bar limits are calculated from R-bar, so if the R chart is out of control, those limits are not valid. Fix what is wrong with consistency, then read the X-bar chart.

The classic Western Electric rules are four zone tests. They look for patterns that are very unlikely in a stable process, and they go beyond simple out-of-limits points.

Many shops add run rules. Your control plan says which apply.

The last one looks good and is not. A chart that is too calm usually means the limits are too wide, because mixed streams inflated them.

Rule numbers differ between textbooks and software, so learn the patterns, not only the numbers. A point past the limit often comes from a broken tool, a slipped fixture, the wrong material, or a measuring error. Check the measurement before you touch the process.

Why You Leave a Stable Process Alone

Tampering means adjusting a process that is in control. If a stable part reads a little high and you offset it, you have moved a process that had not moved. The next part then carries that offset plus its own random error. If you correct the full error after every reading, the variance doubles, and the spread grows by about 40 percent.

So adjust only for a real signal, and follow your reaction plan. A planned offset that the control plan calls for, such as a set tool wear adjustment, is not tampering.

Habits That Keep Charts Useful

• Use the same gauge, method and spot on the part every time.
• Record each reading right away, not from memory at the end of the shift.
• Plot in real time. A chart updated once a day cannot catch anything early.
• When you find a special cause, write down what it was, when you found it and what you did.
• Keep inspecting. A chart supports inspection and does not replace it.
• Use software if you have it, and still understand what the signal means.

Common Mistakes

Check Your Understanding

1. You check one casting per batch, because each test destroys the part. Which chart fits?

I-MR. There is one reading at a time, so subgroups are not practical.

2. You count paint runs on each cabinet, and cabinet sizes vary. Which chart fits?

A u chart. You are counting defects, and the unit size changes, so you need defects per unit.

3. Subgroups of 5 give an R-bar of 0.0040, and the mean is 2.5000. What are the X-bar limits?

0.577 x 0.0040 = 0.0023, so the limits are 2.4977 and 2.5023.

4. Every reading is in spec, and the last 8 points sit above the center line. Your lead says to ignore it, because the parts are good. What do you say?

The process has likely moved, even though nothing is bad yet. Being in spec does not cancel a signal. Investigate now, while there is still time to act.

5. A stable process reads slightly low, with no signals. A new operator offsets the tool each time. What happens?

Variation goes up. They are chasing random scatter, and each change adds to the spread.

Key Takeaways

• A control chart separates common cause from special cause variation.
• Control limits come from process data, and spec limits come from the customer.
• Match the chart to your data: measured or counted, subgroup or single reading, defects or defectives.
• Start with 20 to 25 rational subgroups, and do not mix streams.
• Read the R chart first, then look for points, shifts, trends and patterns.
• Do not adjust a stable process, because tampering increases variation.