Control chart for individual measurement (I-chart)

A tool for monitoring quality improvement processes

Many measures are made on a continuous scale using some kind of device (laboratory instrument, oxygen saturation, bedside monitors of heart rate and blood pressure) and patient-specific measure (such as inpatient falls). Often the frequency of data collection cannot be controlled for these situations and types of data. The I-chart is appropriate for any of these, and may generally represent all types of data distributions. However, in contrast to attribute control charts, it is very sensitive to the underlying data distribution. This should be investigated by probability plots and frequency histograms, and where necessary, data transformation performed to achieve a normal distribution as much as possible.

Example: Hospital infectious waste

Hospital management monitors the amount of infectious waste produced by each ward on a monthly basis, making sure that it is disposed of safely, and investigating unusual fluctations in the amount. Figure 1 shows the data for infectious waste (column B) per patient day over a period of 21 months (column A).

How to calculate an individual measurements I-chart

  • Calculate the (k-1) moving ranges (MR) in column C as the absolute difference between each line and the previous line (regardless of whether there was an increase or decrease). For example:
    C3 = ABS(B3 - B2) = ABS(-0.10) = 0.10
  • Calculate the average of the moving ranges (MRbar) nothing that there is one less MR (n=19) than the original data (n=20).
    MRbar = AVERAGE(C3:C22) = 0.4005
  • Calculate the provisional upper limit of the moving ranges (ULMR)
    ULMR = 3.27 * MRbar = 1.3096
  • Remove any moving range bigger than the ULMR. For each row of data, test this by putting the forumla in each row of column E. For example,
    E3 = IF(C3 > ULMR, "", C3)
    Tip: putting invalid values equal to "" makes the Excel functions for COUNT, AVERAGE etc ignore them, giving the correct calculations in the next step.
    Result: E21 is blank because C21 (=1.53) is greater than ULMR (=1.3096)
  • Recalculate the average moving range (MRbarnew).
    [Note: this recalculation should be done only once]
    ULbarnew = AVERAGE(E3:E22) = 0.3411
  • Calculate the average of the individual data (Ibar). This is the centerline (CL) on the I-chart.
    Ibar = AVERAGE(B2:B22) = 7.077
  • Calculate lower control limit:
    LCL = Ibar - (2.66 * MRbarnew) = 7.077 - (2.66*0.3411) = 6.17
  • Calculate upper control limit:
    UCL = Ibar + (2.66 * MRbarnew) = 7.077 + (2.66*0.3411) = 7.98
  • Draw the control chart using data (column B), CL, UCL, and LCL. The result is in Figure 2.
The values 2.66 and 3.27 used in the above calculations are the sample size-specific anti-biasing constants (A3 = 2.66, B4 = 3.27) for n = 2, as given in most textbooks on statistical process control.
[Factors in Variables Control Chart]
Figure 1. Infectious waste per patient day (data and calculations) (Provost p.156)
Provost p.156 (data)
Figure 2. Infectious waste per patient day (I-chart)
Provost p.156 (graph)