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c Chart How-To

This walkthrough builds a complete c chart from twenty-two equal inspection windows. This is the one attribute chart whose limits are straight lines, and the reason is worth understanding before you read the numbers: with a constant area of opportunity there is only one standard deviation to compute, and the Poisson model fixes it from the center line alone.

The data is on this page rather than in a file to download. Press Copy for Excel, then paste it into a blank worksheet.

The data

Twenty-two consecutive shifts on a coating line. Every shift is the same length and coats the same number of panels, so every subgroup covers the same area of opportunity. Defects is the number of coating faults found in the shift.

Defects
7
11
2
6
11
12
10
11
9
7
5
4
4
5
4
9
22
8
3
8
8
4

Shift 17 is the one to watch: 22 faults where every other shift ran between 2 and 12.

Steps

  1. Put the data in Excel

    Press Copy for Excel above the table. In Excel, open a blank worksheet, click cell A1, and press Ctrl+V. The header lands in A1 and the twenty-two shifts in A2:A23.

  2. Start the chart

    From the Excel ribbon: QXL Stat Tools Tab > Control Charts > Attribute Charts > C, U, and Laney's U'.

  3. Give it the cells

    In the Data Selection window set Selected Range: to A1:A23. Leave Data in Columns selected and First Row/Column is Header checked. Press Next >.

  4. Select the C variant

    The dialog opens on the Data tab with Variant: on U. Change it to C.

    Watch the Inspection Size group grey out

    The moment you select C, the whole Inspection Size group is disabled: both radios and the n = box. That is the visible sign that this chart assumes every subgroup covered the same area of opportunity. If your subgroups are not all the same size, this is the wrong variant and the u Chart is the right one; nothing in the dialog can detect the difference for you.

  5. Check the data column

    Check Defects in Defects Column.

  6. Finish

    Press Finish. The chart is written to a new worksheet named C Chart and the Control Chart analysis pane opens on the right.

What you should see

One panel with twenty-two points, its Y axis titled Count of Defects.

Value
c-bar 7.7273
sigma_Z 1.23
UCL 16.0667
LCL 0

Both limits are straight, and every point shares them. This is the only attribute chart where that is true, and it is why the summary table's UCL and LCL rows describe the whole chart rather than one subgroup's steps.

The center line is the plain average of the twenty-two counts, 7.7273. On a p or u chart the center line is a pooled, size-weighted figure; here there are no sizes to weight, so the arithmetic mean is the answer.

One number sets the entire band. The Poisson model says the variance equals the mean, so the standard deviation is

\[ \sigma = \sqrt{\bar{c}} = \sqrt{7.7273} = 2.7798 \]

and the limits are three of those either side: \(7.7273 + 3 \times 2.7798 = 16.07\) above, and below, \(7.7273 - 3 \times 2.7798\) is negative, so the lower limit is clamped to 0. Nothing on this chart can be marked for being too low, so shift 3 with its 2 faults is not a signal.

There is no separate spread to estimate and no sigma method to choose, which is why the dialog offers neither. It is also why a historical mean changes the limits as well as the center line: with the Poisson model they are the same number.

Shift 17 is out of control. It plots at 22 against the upper limit of 16.07, and is drawn in a different color. It is the only marked point. The next highest shift is 12, comfortably inside.

And there is a diagnostic you did not ask for. The summary table reports sigma_Z of 1.23 with a three-row Overdispersion Interpretation legend under it, comparing the scatter present with the scatter Poisson variation predicts. It is reported here and applied on none of the plain charts: applying it is what a Laney U' Chart does.

Only one test is running

An attribute chart opens with Rule 1 only. Rules 2, 3 and 4 are present and unchecked, and rules 5 through 8 are not offered on this dialog at all. Note also that Rule 9, the run-of-zeros test, is not here either: it belongs to the G Chart, so a c chart with a stretch of zero-defect shifts will not flag that stretch however long it runs.

Things to try next

  • Switch the variant to U and watch the limits bend. Set Variant: to U, and the Inspection Size group comes back to life. Give it a constant of 1 and the chart is arithmetically identical to this one, which is worth seeing once: a c chart is the special case of a u chart where every area of opportunity is 1. Give it a column of real sizes instead and the limits start to step. See the u Chart.
  • Supply the mean instead of estimating it. Type a known long-run count into Historical uBar (optional):. On this chart that single number moves the center line and the limits, because the Poisson model derives the spread from the mean. On a variable chart those would be two separate boxes.
  • Turn on the zone lines. Tick Show 1-sigma / 2-sigma zone lines and you get four more straight lines, one and two standard deviations out, at 10.51, 13.29, 4.95 and 2.17. On an attribute chart a zone step is one full standard deviation rather than a third of the band.
  • Turn on the pattern tests. Tick rules 2, 3 and 4 and rebuild. Nothing new is marked on this data, so the single Rule 1 signal was not hiding a pattern.
  • Check the diagnostic before trusting the band. \(\sigma_Z\) of 1.23 is close enough to 1 that the Poisson model is a fair description here. If it had come back at 2 or 3, these straight limits would have been about two or three times too tight, and the Laney U' Chart How-To shows what that looks like.

See Also