Chapter 2 Understanding SPC Charts
Now that we’ve explored the concepts of common and special cause variation through the example of handwritten letters, the next step is understanding how to apply these ideas to healthcare processes. This application involves creating charts that represent the “voice” or behaviour of a process (usually) over time. By leveraging statistical theory, these charts help determine whether a process is exhibiting common or special cause variation. Such visualisations are known as SPC charts, control charts, or process behaviour charts.
SPC charts are point-and-line plots of data collected over time. They serve as operational definitions for identifying common and special cause variation. An operational definition is one that is designed to be practical, useful, and ensures consistency in interpretation across different individuals.
Figure 2.1: Control chart of systolic blood pressure
Figure 2.1 illustrates a control chart showing daily blood pressure measurements. One data point (#6), marked with a red dot, stands out from the others. This deviation triggers a signal, indicating that the data point is likely due to a special cause.
Special causes are signalled by unusual patterns in data. By “unusual” we mean something that is unlikely to have happened purely by chance, like a freak data point. Common cause variation is simply noise. The absence of signals of special cause variation is evidence that the behaviour of the process is consistent with common cause variation. The presence of signals is consistent with special cause variation.
There are many different types of control charts and the chart to be used is determined mostly by the type of data to be plotted. But regardless of the type of data, (most) SPC charts look the same and are interpreted the same way, by employing a set of statistical tests (or rules) to help identify unusual patterns.
In this chapter, we introduce seven of of the most common types of control charts (The Magnificent Seven) used in healthcare and their practical applications. In the following chapter (3), we delve into the rules and statistical tests for identifying unusual patterns in data. Later, in Part 2 of this book, we provide a detailed guide on performing the calculations necessary to construct and analyse control charts.
2.1 Anatomy and physiology of SPC charts
Shewhart’s groundbreaking innovation was the introduction of control limits, which define the boundaries of common cause variation in data. Data points falling within these limits (without unusual patterns) indicate common cause variation, while those outside indicate special cause variation.
Figure 2.2: Standardised control chart. SD = standard deviation; CL = centre line; LCL = lower control limit; UCL = upper control limit
Figure 2.2 demonstrates a standardized Shewhart control chart, created using random numbers from a normal distribution with a mean of 0 and a standard deviation of 1. The x-axis represents time or sequence order, while the y-axis displays the indicator values. Each dot, connected by a line, represents a data point sampled sequentially. The data included in a data point is called a subgroup representing a set of observations (a sample) collected together at a specific point in time and under similar conditions.
The chart features a Centre Line (CL), which represents the mean of the data values, and a Lower Control Limit (LCL) and an Upper Control Limit (UCL), defining the range of common cause variation. Because the control limits are (usually) positioned three standard deviations (sigma) on each side of the centre line they are often referred to as 3-sigma limits.
The data points cluster symmetrically around the centre line, with most near the mean. As the distance from the centre line increases, data points become increasingly sparse. The control limits are positioned to encompass nearly all data points arising from common cause variation. Conversely, points outside the limits are likely to represent special causes.
2.2 Why 3-sigma limits?
Shewhart’s choice of 3-sigma limits was based on empirical observations from real-world experiments. For data following a normal distribution, like in Figure 2.2, these limits contain about 99.7% of all data points, meaning the likelihood of a point falling outside the limits purely by chance is approximately 3 in 1000 (0.3%). In a control chart with a total of 20 data points the chance of all data points being within the control limits is about 95% (1 - 0.99720).
But before we go down that rabbit hole we note that these theoretical considerations were rejected by Shewhart who found that most real-life data are not normally distributed. Shewhart argued that the choice of 3-sigma limits was made simply because they “work” (Shewhart (1931), p. 18), because this practical choice balances sensitivity (detecting special causes) with specificity (avoiding false alarms), regardless of the data type or distribution. More on this will be discussed in Appendix C.
2.3 Some common types of control charts: The Magnificent Seven
As described above, the control limits are positioned at three standard deviations (SDs) above and below the centre line. Thus, all types of control charts follow this general formula for control lines:
CL±3SD
To construct a control chart, the essential components are the process mean and the process standard deviation. However, it is important to use the right standard deviation. Rather than using the overall SD of all data – which could include variation from special causes and artificially inflate the control limits – we rely on a pooled average of the within-subgroup SDs. This approach ensures that the calculated control limits reflect only the common cause variation inherent to the process.
The exact method for determining the within-subgroup SD depends on the type of data being analysed. For readers eager to dive into the specifics, the formulas for constructing control limits can be found in Table 6.1 later in the book.
Generally, data come in two flavours: count data and measurement data.
2.3.1 Counts
Counts are positive integers representing the number of events or cases. While the distinction between events and cases may seem subtle, it is crucial for accurate analysis:
Events are occurrences that happen in time and space, such as patient falls.
Cases refer to individual units possessing (or lacking) a particular attribute, such as patients who have fallen.
When counting events, every occurrence is recorded. For example, if a patient falls multiple times, each fall is counted as a separate event. In contrast, when counting cases, each unit (e.g. the patient) is counted only once, regardless of how many times the event (e.g. the fall) occurred for that individual.
Both events and cases can be expressed as ratios, which are counts divided by a relevant denominator, also known as the area of opportunity. However, the way these ratios are constructed depends on whether we are analysing events or cases.
Rates: Events are often expressed as rates, calculated as the number of events per unit of time. For instance, the rate of falls might be expressed as the number of falls per 1,000 patient-days. The denominator is a continuous variable (e.g., time) and is inherently different from the numerator, which counts discrete events.
Proportions: Cases are expressed as proportions or percentages, calculated as the number of cases divided by the total number of cases and non-cases. For example, the proportion of patients who fell might represent the percentage of patients who experienced one or more falls. The numerator and denominator represent the same category of count (e.g., patients). The numerator is always a subset of the denominator, meaning it cannot exceed the denominator.
The most common chart types for count data are:
- C chart: count of events, e.g. number of patient falls.
- U chart: event rates, e.g. number of patient falls per unit of time.
- P chart: case proportions, e.g. proportion of patients who fell.
2.3.2 Measurements
Measurements refer to data collected on continuous scales, often including decimals. Examples include blood pressure, height, weight, or waiting times.
As an example, consider waiting or “door-to-needle” times. These may be plotted as either individual times where each data point represents one patient or as an average time for all patients in a certain period of time, for example an hour, day, week, or month. When producing SPC charts for measurement data, the choice of control chart depends on how the measurement data are grouped.
For individual measurements, we use an I chart (also called an X chart) where each subgroup represents one individual measurement. The I chart is often used in combination with a Moving Range (MR) chart to plot the variability between consecutive individual measurements.
For subgroup averages, we use an X-bar chart where subgroups consist of multiple measurements, such as the average of several patients’ waiting times within a given period. the X-bar chart is often used in combination with an S chart to plot the within-subgroup standard deviations, which helps visualize the variability within each subgroup.
- I and MR charts: individual measurement, subgroup size = 1.
- X-bar and S charts: multiple measurements, subgroup size > 1.
2.4 Summary of common SPC charts
A Shewhart control chart is a point-and-line plot of data over time, augmented by three horizontal lines:
- Centre line (CL): Represents the overall mean of the data.
- Lower control limit (LCL) and upper control limit (UCL): Define the boundaries of natural process variation.
While there are various types of SPC charts tailored to different types of data, all share a consistent structure and behaviour. When choosing the right chart we must first decide what type of data we have, count data or measurement data:
- For count data:
- C chart: For event counts.
- U chart: For event rates (events per unit of time or opportunity).
- P chart: For proportions (cases as a subset of the total).
- For measurement data:
- I chart with MR chart: For individual measurements and moving ranges.
- X-bar chart with S chart: For subgroup averages and within-subgroup variability.
In the next chapter, we will explore the rules and techniques used to detect signals of special causes. We will present a validated set of rules that achieve a high sensitivity to special cause variation while maintaining a reasonable false alarm rate.