  # Tukey Control Chart

There are many different types of charts.  P-charts are useful for analysis of mortality data but assume large number of observations.  X-bar charts are useful for analysis of satisfaction ratings but assume Normal data.  We present here an approach based on Tukey’s exploratory data analysis techniques.   The approach is robust and makes no assumptions regarding the distribution of outcomes plotted.  We refer to this approach as Tukey's Chart.

In a control chart, you monitor progress  over time.  You create a plot, where the X-axis is days or weeks since  start and the Y-axis is the outcome you are monitoring.  To decide if your  outcomes are different from historical patterns, the upper (UCL) and lower  control limits (LCL) are calculated.  These limits are organized in such a  way as to make sure that if your historical pattern has continued then 99% of  time data will fall within these limits.  This section shows you how to  calculate limits for Tukey charts - a type of chart useful for analyzing single  observations per time period. )

Figure 1 shows the structure of a typical control chart.   In this figure, all points, except for two, fall within the control limits.} Figure 1:  Components of a Control Chart

# How  to Read a Control Chart?

In a control chart, points outside the limits are unusual  and mark departure from historical patterns.  Two points in Figure 1 fall below  the LCL and therefore mark a real change.  All other points do not indicate any  real change, even though there are lots of them showing a rise or fall.   These fluctuations are random and not different from historical changes.

A control chart can be used to see if a process is stable,  meaning that observations are falling within anticipated limits.  If your  data falls within the control limits, despite day to day variations, despite  attempts to change the process, outcomes have not changed.

# Minimum Number of Observations

The more data you have, the more precision you have in  constructing the upper and lower control limits.  At a minimum, you need at  least 7 data points in the pre-intervention period to start a Tukey chart.   This is an absolute minimum and not a recommended number.  The actual  number you need depends on the consequence of waiting and collecting more data  versus using too little data and making an error in judgment.

Keep in mind that not all of the data you collect are used  for calculation of control limits.  Often, the limits are based on  pre-intervention period.  Then subsequent post-intervention observations  are compared to the pre-intervention limits.  When you make a change, you  want to see if the change has affected the outcomes.  In these  circumstances, you set the limits based on the pre-intervention data.  You  compare post-intervention findings to these limits.  If any points fall  outside the limits, you may then conclude that the intervention has changed the  outcomes.  See Figure 2 for an example of limits set based on  pre-intervention periods. The solid line shows the time period used to set the  limit, the dashed line is an extrapolation of the limit to other periods. Figure 2:  Post-intervention data compared to limits set based on pre-intervention data

Compare the chart in Figure 2 with the chart in Figure 1.   Both are based on the same data, but in Figure 2 the limits are based on the  first 7 days, before the intervention.  Figure 2 shows that post  intervention data are lower than LCL and therefore a significant change has  occurred.  When Figure 2 is compared to Figure 1, we see that more points  are lower than LCL in Figure 2.  By setting the limits to pre-intervention patterns, we were able to detect more accurately the improvements since the intervention.

Control limits can be calculated from either the pre or the post intervention period and projected over to the other period.  For example, control limits can be calculated from per-intervention period and extended to the post-intervention period.  Or the reverse: control limits can be calculated from the post-intervention period and extended to the pre-intervention period.  Either way, we are comparing the two periods against each other.  But since the results will radically differ, it is important to judiciously select the time periods from which the control limits are calculated.  The selection depends on the inherent variability in the pre- or post-intervention periods.  Control limits are calculated from the time period with least variability.  Typically, this is done by visually looking at the variability in the data or at the range of the data in pre- and post-intervention time periods.  But in Tukey chart, outliers may appear to cause variability but do not affect the control limit calculations.  Therefore, instead of visual tests or instead of looking at range of data, it is important to calculate the Fourth Spread (a concept explained in next section) and to select the time period with the smallest fourth spread.  This will produce control limits that are tighter and more likely to detect changes in underlying process.

Figure 3 shows the control limits derived from pre-intervention data.  The control limits are calculated from the pre-intervention period and shown as solid red line.  They are extended to the post-intervention period, shown as dashed red line.  The control chart compares the observations in the post intervention period to the control limits derived from the pre-intervention period. Figure 3:  Control Limits Derived from the Pre-Intervention Period

Figure 4 shows the control limits for the same data drawn from a post-intervention data.   Note that this time around the control limits are calculated from the post intervention period, shown as solid red line.  They are extended to the pre-intervention period, shown as dashed red line.  The control chart compares the pre-intervention observations to the control limits calculated from the post-intervention data. Figure 4:  Control Limits Derived from the Post-Intervention Period

Note that both analyses are based on the same data. Both analyses compare the pre- and post-intervention data by contrasting the observation in one period to control limits derived from the other period. In one case, the control limits are drawn from the pre-intervention period and in the other from the post-intervention period. Note the radical difference of the control limits derived from the two time periods. The Fourth Spread (a concept explained in the next section) for the pre-intervention period is 10 points while for the post-intervention period it is 18 points. Figure 3 is the correct way to analyze the data, because it is based on the control limits derived from the pre-intervention period, the time period with smallest Fourth Spread.  If control limits are based on the time period with the smallest Fourth Spread, they would be tighter and more likely to detect smaller changes in the underlying work process.

# Calculating Limits

We will use Tukey’s suggested limits for calculation of  confidence intervals. The procedure calculates control limits from  difference of upper Fourth and lower Fourth of data, a concept that Tukey named  Fourth Spread.

Most readers are familiar with median, a value where half  the data are below and half the data are above it. A lower Fourth is  similar to 25% quartile and is the median of the first half of the data.  At this point, 25% of the data are below this value.  An upper Fourth is  similar to 75% quartile and is the median of the upper half of the data; at this  point 75% of the data are below this value. The difference between the two Fourths is referred to as Fourth Spread. The Upper Control Limit is  calculated as the sum of the upper Fourth and 1.5 times the Fourth Spread.   The  Lower Control Limit is calculated as the difference of the lower Fourth and 1.5  times the Fourth Spread.

Here are the procedure for calculating Tukey’s control  limits:

1. List values from smallest  to largest.
2. Calculate median, where 50% of data are below if and & 50% above it.   If odd number of observations, take the value in the middle.  If  even number of observations, take average of the two middle ranked  numbers.
3. Divide the data set into  two halves using the Median.  Include the Median in both halves if  it Median is one of the observed data points.
4. Lower Fourth is the median of the lowest 50% of  the data, data from the smallest number till (or including) the median.
5. Upper Fourth is the median  of the top 50% of the data, data from (or including) median of the  full data set to the highest value.
6. Calculate Fourth Spread as  the difference between the two Fourths?
7. Calculate UCL and LCL  using the following two formulas:

LCL = Lower Fourth - 1.5 * Fourth Spread

UCL = Upper Fourth + 1.5 * Fourth Spread

## Example in Exercise Time & Weight Loss

Jane collected data in Table 1  regarding her exercise times.  She planned to exercise 3 times a week and  each time she exercised she recorded the time in minutes. When she did not  exercise, she recorded a 0 for the length of exercise. The first 7 days  recorded were pre-intervention. After this period, she and her spouse  joined a mixed group volleyball team. The question she wanted to know was  whether joining the team had made a difference in her exercise time.

 Table 1: Length of Exercise Day of observation Minutes  of exercise Sorted in order of length of exercise Rank Day of observation? Minutes of exercise 1 30 1 2 0 2 0 2 3 25 3 25 3 1 30 4 30 4 4 30 5 35 5 5 35 6 40 6 6 40 7 50 7 7 50 8 45 9 31 10 20 11 40 12 60 13 45 14 60 15 45 16 32 17 50 18 60

We can calculate the control limits from the pre-intervention or post-intervention period.  It so happens that the limits calculated from the pre-intervention period are tighter (meaning the difference of upper and lower control limit is smaller) and therefore the following shows how to calculate the control limits from the pre intervention period.  While we focus on setting the limits from the pre-intervention period, in reality you should do it from either period and select the time period that produces the tighter control limits.

The first step is to sort  pre-intervention data in order of length of exercise. This is shown in Table 1 in the last column of the Table. Next, we calculate the median,  this is the value where ½ the data (7 * .5 = 3.5, 3 points) are below it and ½ the data (3 points) are above it. The 4th data point with value of 30 is the median; 3 data points are below it and 3  above it.

Since median is an actual data  point, we include this point in the lower data set.  To calculate the Lower  Fourth, we calculate the half way point for the first half of the data. When we include the median, we have 4 points in the lower data set. The 25% quartile is halfway between the second and third point, in other words  between 25 and 30, which is 27.5.

To calculate upper Fourth, we calculate the half way point for the upper half of the data.  Again because  the median is an actual data point, we include this point in the upper dataset.   With the median, we have 4 data points from Median to the highest values. The upper Fourth is between the 5th and 6th data points  (between 35 and 40), and therefore its value is 37.5.

The Fourth Spread is the difference between the upper and lower Fourth, which is 37.5-27.5 = 10.  The Fourth Spread for the control limits calculated from post-intervention data is 18 points and this is why we have selected to calculate control limits from pre-intervention data.  The UCL is calculated as 37.5+1.5*10 = 52.5. The LCL is calculated as  27.5-1.5*10 = 12.5. A chart of the data is provided in Figure 5: Figure 5: Tukey's Control Chart for Data in Table 1

Examination of the chart shows  that in the first seven days, there was one very low point of no exercise, a  statistical abnormality.  After the first 7 days (used for setting the  limits), on 3 occasions the total exercise time exceeded the UCL.  In these  three days, there was a real increase in exercise time compared to the first 7  days. If these days correspond to joining the volleyball team, then the  intervention seems to have worked.

Let us look at another  example, this time on weight loss. A male, 48 year old man measured his weight for 8 weeks. Then he and his spouse changed food shopping habits. They excluded all sweets from their shopping (they stopped buying pops,  sweetened cereals, and chocolates for the house). The data for this person  is provided in Table 2.  Weight was recorded once a week.

 Table 2:  Recorded weight values Week Pounds over ideal  weight Sorted values Rank Pounds over ideal  weight 1 9 1 3 2 11 2 5 3 7 3 7 4 7 4 7 5 10 5 8 6 5 6 9 7 3 7 10 8 8 8 11 9 6 10 6 11 3 12 0 13 4 14 -1 15 -5 16 -2

As before, we need to calculate the control limits for pre and post-intervention periods and select the limits with the smallest difference.  It so happens that that the pre-intervention period has the smallest Fourth Spread and therefore we show the calculation of control limits from these data.  The first step is to sort pre-intervention data from least amount of pounds over weight to the  highest value. This is shown in Table 2 in the last column of the Table.   Next, we calculate the median, this is the value where ½ the data (8 * .5 = 4  points) are below it and ½ the data (4 points) are above it.  The value  should be between 4th and 5th data points, or between 7  and 8, so the median is 7.5.

Since median is not an actual  data point, we do not include this point in the calculations of Fourths.   To calculate the lower Fourth, we pick the half way point for the first half of  the data. We have 4 points in the lower data set. The Lower Fourth  is halfway between the 2nd and 3rd point, in other words  between 5 and 7, thus it is 6.

To calculate the Upper Fourth,  we calculate the halfway point for the upper half of the data.  Again  because the median was not an actual data point, we do not include this point in  the upper data set.  We have 4 data points for the highest values.  The Upper Fourth is between the 6th and 7th data points  (between 9 and 10), and therefore it is 9.5.

The Fourth Spread is the  difference between the Upper and Lower Fourth, which is 9.5-6 = 3.5. The  UCL is calculated as 9.5+1.5*3.5 = 14.75. The LCL is calculated as  6-1.5*3.5 = 0.75.   A chart of the data is provided in Figure 6: Figure 6:  Control chart for the weight data

Examination of the chart shows that in the first eight  weeks, all data points were within the limit. No weight was lost in the  pre-intervention period, even though there was considerable amount of  fluctuations.  Over the remaining 8 weeks and compared to the first 8  weeks, on 4 occasions the weight was lower than the LCL.  Therefore, there  was a real decrease in weight in the post intervention period.

# Example in Medication Errors

The following data show the error in PYXIS refills in a hospital.

 Name of Drug Refilled Date of Error Type of Error Promethazine inj 25 mg 05/13/07 Wrong strength Lidocaine 1% w/ Epi 06/13/07 Wrong strength Ofloxocin Opth 05/14/07 Wrong dose form Sensorcaine 0.25% 05/07/07 Wrong strength Lidocain 1% w/ Epi 06/20/07 Wrong strength Tramadol 50 mg tab 05/21/07 Wrong drug Morphine Sulfate 30 mg 07/01/07 Wrong dosage Table 3:  Recent PYXIS Refill Errors

Note that there are no pre- or post intervention time periods and therefore the entire data can be used for calculation of control limits:  To analyze this data we first calculate time between errors:

 Date of Ending Incident Days Between 5/13 6 5/14 1 5/21 7 6/13 23 6/20 7 7/1 11 Table 4:  Days to Errors

Next we reorder the observations from lowest days between errors to highest dates:  1, 6, 7, 7, 11, and 23 days.  The median for the data is 7.  The upper fourth is the median of 7, 11 and 23 and thus it is 11.  The lower fourth is the median of 1, 6, and 7 and thus it is 6.  The fourth spread is 5 days.  The upper control limit is the upper fourth plus 1.5 times the fourth spread which is 18.5.  The lower control limit is a negative number and thus it is re-set to zero.  Figure 7 shows the control chart: Figure 7:  PYXIS Refill Errors Improved from 5/21 to 6/13

# Example in Budget Variation

Suppose that we are looking at 12 month of data regarding our  clinic's budget. The question is whether the expenditures at any  particular month are higher than the general pattern across the 12 months.  The table below shows the budget deviation (expenditure minus budget amount) for  each of the months in thousands of dollars:

 Month Budget deviation in 1000su 1 23 2 -5 3 -70 4 -7 5 -8 6 9 7 12 8 30 9 24 10 25 11 -4 12 -2

Note that there are no pre- and post intervention time periods and therefore the entire data are used for estimation of control limits.  The first step is to sort the data:

 Rank Budget deviation in 1000 1 -70 2 -8 3 -7 4 -5 5 -4 6 -2 7 9 8 12 9 23 10 24 11 25 12 30

There are 12 data points, so the median is halfway between the  6th and 7th ranked data points. Therefore, the median is not included in the  lower and upper data sets because it is not an actual value in the data.  The Lower Fourth is halfway in between the 6 data points with lowest ranks, it  is between 3rd and 4th ranked data points and has the value of -6.  The  upper data set is the points ranked 7 through 12.  Median of this data set  is halfway in between the 9th and 10th ranked data items.  It is 23.5.  The Fourth spread is 29.5.

The UCL is 23.5+1.5*29.5 and the LCL is -6-1.5*29.5.  Figure 8 shows the control chart: Figure 8:  Tukey Chart for budget information (in 1000 of dollars)

The chart shows that all months are within control limits except  for March, where in there has been a large deviation from the budgeted amount.

# Which Chart is Right? Figure 9:  Tukey chart is best when there are single observations per time  period, outcomes are measured on an interval scale, and outliers are likely

When tracking data over time, you  have a number of options.  You could use a P-chart, designed specifically  to track mortality or adverse health events over time.     You could use a moving average chart to help  you construct control chart for an individual patient's data over time.   This section helps you decide which of these various charts are appropriate for  your application.  If you do not have a specific application in mind or if  you wish to learn more about each of the various different charts, skip this  section.  In the following, we ask you 4-7 questions and based on your  answers advise you which chart is right for the application that you have in  mind.!

Have you collected observations over different  time periods?

For more details on comparison of Tukey and Moving Average Control charts (XmR), click here

# Analyze Data

Advanced learners like you, often need different ways of understanding a  topic. Reading is just one way of understanding. Another way is through doing  and practicing the concepts learned in this section.  The following questions are designed to get you to think more about the concepts taught in  this session.

1. Assume that following data regarding one patient's rating of his/her health status over eight time period:
 Time period 1 2 3 4 5 6 7 8 Observations 90 85 92 67 98 83 94 90

Has the patient's health changed?  Note that there are no interventions and therefore the entire data can be used to estimate the control limits.

1. Analyze the following data using Tukey, XmR and Time-In-Between (more than 30 minutes of exercise considered a successful day) charts. Produce 3 charts and discuss if the findings from the 3 charts are similar.  To decide if the exercise time has changed, rely on the control chart with the smallest difference between upper and control limit.
 Day Minutes of exercise 1 25 2 30 3 32 4 0 5 15 6 17 7 15 8 40 9 15 10 28 11 0 12 60 13 20 14 24

1. Correct answers are needed for each of the four charts.  An error in any chart will constitute 25% of the grade.
2. All values must be computed as a formula except the data provided above.  It should be possible to change the data and the graph should change.  Submitting a file that does not do this will be counted as error in analysis.  This means that you may have to use if, count, average, median and other functions.  Do not under any circumstance enter the result of the calculation by hand.
3. Inappropriate displays will lose 10% of the grade for the chart, this means the following:
1. All lines must have a legend or have a label.  No lines should be labeled "Series 1".
2. The UCL and LCL should be drawn as lines with no markers, preferably in red.
3. Observations must be drawn with markers connected with lines.
4. There must be a title to the chart and x and y axis should be labeled.
5. There should not be any spelling errors anywhere in the chart.
6. All charts must be shown in the same file.  Do not email multiple files.  Label worksheets.

Please submit one file containing answers to all questions.  Please note that all cell values must be calculated using a formula from the data.  Do not enter values in any calculated cells.  Calculate each cell using Excel formulas.  Email►

# Presentations

To assist you in reviewing the material in this lecture, please see enclosed resources:

1. Tukey chart lecture  Slides
2. See an example in Excel  Video SWF►  Excel 2003

# More

• Introduction to Tukey chart  More►
• An empirical evaluation of Tukey chart   Counter Point►
• Economic evaluation of Tukey chart  More►
• Performance of Tukey chart  More►
• Wheeler argues against Tukey charts More►
• The collected work of John Tukey More►
• Tukey's writing in health care PubMed►
• Comparison to Individual Moving Range Control chart   Read