C chart

C chart (also written as c-chart) is an attribute control chart used in statistical process control to monitor the total number of nonconformities (defects) per unit when the sample size remains constant. Walter A. Shewhart developed this chart at Bell Telephone Laboratories during the 1920s as part of his broader work on quality control methodology^[1]. The chart assumes that defects follow a Poisson distribution and is particularly useful when multiple defects can occur within a single inspection unit.

Historical background

The c chart emerged from Shewhart's pioneering research at Western Electric and Bell Labs between 1918 and 1931. On May 16, 1924, Shewhart wrote his famous internal memorandum introducing the concept of control charts to distinguish between common-cause and special-cause variation^[2]. His boss George D. Edwards later recalled that this single-page document "set forth all of the essential principles and considerations which are involved in what we know today as process quality control."

The p-chart came first, followed by the np-chart, c-chart, and u-chart. Shewhart published his theoretical foundations in two landmark books: Economic Control of Quality of Manufactured Product (1931) and Statistical Methods from the Viewpoint of Quality Control (1939). During World War II, the War Department mandated these methods for wartime production, spreading their use across American industry.

Structure and components

A c chart consists of three horizontal lines plotted against time:

Center line (CL) - represents the average number of defects per unit, calculated as the mean of historical data
Upper control limit (UCL) - set at three standard deviations above the center line
Lower control limit (LCL) - set at three standard deviations below the center line (often zero if calculation yields a negative value)

The control limits are calculated using the Poisson distribution. For a process with average defect count c-bar:

UCL = c-bar + 3 * sqrt(c-bar)
LCL = c-bar - 3 * sqrt(c-bar)

Application requirements

Several conditions must be met before implementing a c chart^[3]:

The sample size must remain constant across all subgroups
The opportunity for defects to occur should be large, while actual occurrences remain small
Defects should occur independently and randomly
Each unit has equal probability of containing a defect

Manufacturing environments commonly use c charts to track scratches on painted surfaces, soldering defects on circuit boards, or documentation errors in batch records. Service industries apply them to monitor complaint counts per time period or errors per transaction batch.

Comparison with other attribute charts

The c chart belongs to a family of four attribute control charts:

Chart type	What it measures	Sample size requirement
p chart	Proportion of defective units	Variable or constant
np chart	Number of defective units	Constant
c chart	Number of defects per unit	Constant
u chart	Average defects per unit	Variable or constant

A key distinction exists between "defective" and "defect." A defective item fails to meet specifications entirely. A defect is a single nonconformity - one item may contain multiple defects without being classified as defective.

Interpretation guidelines

Process analysts examine c charts for signals indicating special-cause variation^[4]:

Any point falling outside the control limits
Seven or more consecutive points on one side of the center line
Seven or more consecutive points trending upward or downward
Fourteen or more points alternating up and down
Two out of three consecutive points beyond two standard deviations from the center line

When a process remains in statistical control, only common-cause variation affects the defect count. This stability allows managers to predict future performance with reasonable confidence. Out-of-control signals demand investigation to identify and eliminate assignable causes.

Practical example

A textile manufacturer inspects 100-meter fabric rolls for weaving defects. Weekly inspection data over 20 weeks yields an average of 4.2 defects per roll. The control limits would be:

UCL = 4.2 + 3 * sqrt(4.2) = 4.2 + 6.15 = 10.35
LCL = 4.2 - 3 * sqrt(4.2) = 4.2 - 6.15 = -1.95 (set to 0)

Any roll with more than 10 defects triggers investigation. The lower limit of zero means extremely low defect counts, while unusual, do not signal process problems.

Limitations

The c chart has several constraints that practitioners should recognize. It requires constant sample sizes - varying inspection units necessitate the u chart instead. The Poisson assumption may not hold for processes where defects cluster or occur in bursts. Small average defect counts can produce asymmetric control limits, reducing the chart's sensitivity to process improvements. Overdispersion, where variance exceeds the mean, violates model assumptions and requires alternative approaches such as generalized control charts.

Infobox5 — recommended articles

Control chart

Statistical process control Quality control Quality management

References

Montgomery, D.C. (2019). Introduction to Statistical Quality Control, 8th Edition. Wiley.
Shewhart, W.A. (1931). Economic Control of Quality of Manufactured Product. Van Nostrand.
Wheeler, D.J. (2010). Understanding Statistical Process Control, 3rd Edition. SPC Press.
ASQ Quality Press (2018). The Certified Quality Engineer Handbook, 4th Edition.

Footnotes

<references> <ref name="shewhart">Shewhart developed control charts while working at Bell Labs to improve telephone transmission reliability.</ref> <ref name="memo">The May 16, 1924 memorandum is considered the birth of statistical quality control.</ref> <ref name="requirements">ASQ Quality Resources provides detailed guidance on c chart implementation requirements.</ref> <ref name="interpretation">Western Electric rules and Nelson rules guide control chart interpretation.</ref> </references>