u Charts, p Charts and i Charts
In Quality Indicators in Healthcare, much of
the data is attribute data; that is, counts
of things, events or outcomes. We know, for
example, how many patient falls we had, how
many AMI patients were given beta blockers at
arrival or how many medication errors we have
recorded. Often, the data can be represented
on an i chart. The i chart is almost always
available but it is seldom the best choice.
Let’s take a look at the charts that may
be better and why. For many examples of such
data, view the indicators at http://pimd.statit.com,
particularly the CHF indicators under the Quality
While the i chart was designed to analyze measurements
in a slow-moving process, one can plot either
a count of events or a rate or proportion. However,
this application of the i chart is likely to
hide important information. For example, if
the i chart is a plot of the number of medication
errors in a period, it would in no way indicate
the number of doses administered in the period.
This is critical information in deciding whether
a spike in the number is statistically important.
The counts of one error vs ten errors may take
on a different meaning if we know that in the
former, there were only two doses given while
in the latter, there were 1000 doses given.
In the same way, plotting the rates of medication
errors (errors / total doses) on an i chart
does not take into account the size of the subgroup
in determining the control limits. Some critical
information is hidden from the decision maker.
Statistically, in estimating the confidence
in a rate of .25, it makes a difference if that
rate is 250 events in 1000 trials or 1 in 4.
While there is some information of the subgroup
size in the rate, the i chart does not take
different sizes into account in calculating
the control limits.
A c or u chart or a p chart are usually better
choices for representing these data. The c and
u charts work with what is known as an “Area
of Opportunity” in which any number of
events can occur. The Area of Opportunity is
some unit that presents an opportunity for the
event to occur and is also commonly referred
to as "Subgroup Size". Patients being
in a hospital present opportunities for patient
falls. Administering medication presents opportunities
for medication errors. Often the choice of a
c or u chart over a p chart is determined by
the question, "can an event happen more
than once in the area of opportunity?"
Can a patient fall more than once? Can there
be more than one medication error in a dose
of medication? If the answer is yes, then your
data indicate a c or u chart. These charts allow
for those situations where it is possible to
have more events than areas of opportunity.
If your area of opportunity is patient days,
it is possible to have more than one fall per
patient day, since it is possible for each patient
to fall more than once on a particular day.
Once you have determined that a c or u chart
is applicable, the decision of which one to
use is dependent on whether the area of opportunity
data are available. A c chart simply plots counts
and assumes a constant size for the area of
opportunity or subgroup. A u chart needs the
size of the area of opportunity for its calculations
because it plots counts per area of opportunity.
If you choose the number of patient days as
the area of opportunity for patient falls, then
the c chart could be applicable only if the
number of patient days was fairly constant across
all periods charted. For some indicators, you
may not have the actual size of the area of
opportunity but can make an educated assumption
that it is fairly constant. This may be dangerous,
however, without data to back up your assumption.
If the area of opportunity varies and if you
have the size of the area of opportunity for
each count, the u chart is the better choice.
The u chart will actually contain more information
than the c chart (or the i chart). As Hart and
Hart (2002) says, “Note that the u chart
may always be used and that the c chart is never
In u chart data, there are a number of areas
of opportunities in each time period. The u
chart calculates the plotted point by dividing
the number of events by this number of the areas
of opportunities in that time period. Be sure
to note that the number of areas of opportunities
can be fractional.
The area of opportunity for a u chart can be
manipulated to a unit that may be more acceptable
or reasonable to the user community, such as
Patient Falls per 100 Patient Days or Medication
Errors per 1000 doses. This is reasonable since
these charts are based on the Poisson distribution
which is applicable to infrequent events. The
u chart divides the number of events by the
number of units of areas of opportunities. A
chart of the transformed data is what Grant
and Leavenworth (1996) calls the ku chart, simply
because scaling factor, k, is applied to the
count of areas of opportunity, which has the
effect of dividing the number of Areas of Opportunities
by k. Statistically this manipulation has no
affect on the analytical results but may scale
the numbers to make more sense to the user.
Thus, 6 falls in 635 patient days becomes 6
falls in 6.35 100 Patient Days or .94 falls
per 100 patient days. This is a simple transformation
of dividing the number of Patient Days for each
period by 100. In this example, the Area of
Opportunity is Patient Day and the Unit of Area
of Opportunity is 100 Patient Days.
The p chart is used when there are only two
possible outcomes to some event and the sum
of the counts of the two outcomes equal the
total area of opportunity (the total number
of events). The former indicates a binomial
distribution. The latter says that the count
of outcomes is a subset of the area of opportunity.
Several data in healthcare fall into this group.
The medication administered was in error or
it was not. Beta Blocker was administered at
arrival or it was not.
The area of opportunity for this chart may
actually be a recorded event such as number
of medication doses administered or number of
AMI patients admitted. In this case, the count
of an outcome (numerator) is a subset of the
count of events (denominator). The count of
events is the Area of Opportunity. For all AMI
patients admitted, how many were administered
beta blocker at arrival? The event is an AMI
patient was admitted and the two possible outcomes
are that the beta blocker was administered or
that it was not. The number of AMI patients
who received the beta blocker is a subset of
the total AMI patients. With this type of data,
the number of a particular outcome divided by
the number of events can never exceed 1.
In contrast, Patient Falls is not a subset
of Patient Days. A number of Patients fell,
but no Patient Days fell.
Another concept to help decide on a u chart
or p chart is whether the actual number of opportunities
for an event can be counted. If counting the
opportunities is very difficult or impossible,
it is necessary to use a u chart instead of
a p chart. Obviously, the number of opportunities
for a patient falling would be very hard to
count. This leads one to attempt to find an
area of opportunity that would be more manageable,
but which will have an indicative relationship
to the possibility of an occurrence of the event.
Patient days for patient falls works because
Patient Days contains an indication of the opportunity
for patient falls. How do you think such things
as “Nurses on duty” or “Wheelchairs
available” or “Miles driven by the
Ambulances” would work as Areas of Opportunity
for Patient Falls?
Let’s look at some actual data to compare
Here are the data for falls in a particular
department in a hospital. The Rate is calculated
as NumFall/NumDays*100. NumDays_100 is NumDays/100.
An i chart on NumFalls:
Figure 1. i chart - Number of Falls (mouse over
data point to understand underlying data)
Notice that this chart does not give us any
out of control points. We would decide that
there are no assignable causes in this process.
An i chart on Rate:
Figure 2. i chart - Rate (mouse over data point
to understand underlying data)
Similarly, this chart shows that we have no
assignable causes in the process although there
is some information from the size of the subgroup
or area of opportunity.
A c chart on NumFalls
Figure 3. c chart - Number of Falls (mouse over
data point to understand underlying data)
If we could assume that the number of patient
days was fairly constant, we could get by with
this chart. However our number of patient days
range from 492 to 1048. We are on very thin
ice if we make the assumption of equal area
A u chart on numfalls by numdays:
Figure 4. u chart - Number of Falls per Patient
Day (mouse over data point to understand underlying
The u chart gives us an alert for an assignable
cause. That is the function of control charts,
to find possible areas of improvement. In this
case, it appears that August of 2005 may have
had something change and we need to investigate
to see if we can find a cause.
u chart on numfall per 100 patientDays:
Figure 5. u chart - Number of Falls per 100
Patient Days (mouse over data point to understand
This chart gives the same information as the
chart in Figure 4. The transformation of data
makes it easier to use. It displays a mean of
.371 instead of .00371 rounded to .004. Notice
also that this chart has a calculated lower
control limit of 0 while the i charts in Figures
1 and 2 have negative control limits. Counts
cannot go negative, so Attribute Charts (u,
c, p, etc.) take this into account. Such an
assumption cannot be made for measurement data.
The i chart was designed for measurement data.
AMI 6 - Beta Blockers on Arrival
A similar situation is true for i charts and
p charts. Consider the following data on AMI
6 Beta Blockers at Arrival. Rate is Num/Den
and RateI is Rate*100.
First an i chart:
Figure 6. i chart - AMI 6 Rate (mouse over data
point to understand underlying data)
And the p chart
Figure 7. p chart - AMI 6 (mouse over data point
to understand underlying data)
Again we see that the i chart does not alert
us to possible assignable causes that the p
chart does. We are in the business of improving
our processes. If we were simply working to
show that our processes are in control, the
i chart might suffice. If we wish to find areas
for improvement, we might choose a different
chart more in line with the data.
In conclusion, if you can avoid an i chart,
do so. We have several i charts on our Statit
piMD demo site. For some, such as LOS charts,
the i chart works well. For others, the rate
is the only data we had. If you get into such
a situation, perhaps you can ask why this is
the only data available. After all, the rate
needed to be calculated somehow. To summarize:
- You are counting events (Patient Falls)
- The event occurs over some area of opportunity
- The event can occur more than once in the
Area of Opportunity (One patient falls twice
on one day)
- The number of opportunities for the event
to occur is difficult to count (Need an indicative
Area of Opportunity)
- Number of Areas of Opportunity may be fractional
(6.35 100 Patient Days)
Use the c chart in place of the u chart
- Only if you do not have the Area of Opportunity
- Only if you can make an educated assumption
that the Area of Opportunity is roughly equal
across all periods.
- You are counting outcomes of an event class
(Beta blockers on arrival for AMI patients
- An event has exactly two possible outcomes
(The sum of the counts of the outcomes equals
the count of events)
- Each event is countable (Number of AMI patients
- Each event is an area of opportunity for
the outcome under study (Was the beta blocker
given at arrival)
E.L., & Leavenworth, R.S. (1996). Statistical
Quality Control (5th ed.). Boston: McGraw-Hill.
M.K., & Hart, R.F. (2002). Statistical Process
Control for Health Care. University of Wisconsin,