1. Tukey Control Chart Farrokh Alemi, Ph.D.
These slides were organized by Dr. Alemi. These slides discuss a control chart based on median’s distribution, as organized by Tukey and first proposed by Alemi.

2. Seven Steps 1. Assumptions
Making a Tukey chart is relatively easy. It can be done in seven steps. It start with checking assumptions

3. Assumption
1 per Period
Tukey charts assume there are 1 observation per time period.

4. Assumption
Outliers
Tukey charts are based on median distribution and are not affected by outliers as much as the X-bar control charts or other averaging control charts are affected.

5. Seven Steps 2. Median & Fourths
In step 2 we calculate medians and fourths

6. 2. Median & Fourths Median, ½ Below
You are familiar with median, a value where half the data are below and half the data are above it. 

7. 2. Median & Fourths Median, ½ Below Order Equal halves Odd or Even?
You are familiar with median, a value where half the data are below and half the data are above it.  To do this Order values from smallest to largest. Select median, where data are in two halves. If odd number of observations, take the data point in the middle, where 50% of data are below it. If even number of observations, take average of the two middle ranked numbers. If median is one of the data points, include it in both the upper and lower half of data. Excel has a function called median that calculates the median of data.

8. 2. Median & Fourths Lower Fourth, ¼ Below
A lower Fourth is similar to median but 25% of data fall below it. A quartile is the median of the first half of the data.  At this point, 25% of the data are below this value. 

9. 2. Median & Fourths Upper Fourth, ¾ Below
An upper Fourth is similar concept. It is 75% quartile and is the median of the upper half of the data; at this point 75% of the data are below this value. 

10. Seven Steps
3. Fourth Spread
Step 3 calculates fourth spread.

11. Upper Fourth-lower Fourth
3. Fourth Spread
Upper Fourth-lower Fourth
The difference between the upper and lower fourths is referred to as fourth spread.

12. Seven Steps
4. Control Limits
Step 4 calculates control limits

13. UCL = Upper Fourth + 1.5 * Fourth Spreads
LCL = Lower Fourth – 1.5 * Fourth Spreads
We will use Tukey’s suggested limits for calculation of confidence intervals.  The procedure calculates control limits from difference calculated fourths and 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.  

14. Seven Steps
5. Plot Chart
Next we plot observations and control limits

15. Seven Steps
6. Interpret Findings
Finally we interpret findings

16. Seven Steps 7. Distribute
And distribute the control chart and our interpretation of it.

17. Example: Exercise Data
Pre-intervention
Jane collected data in this Table 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.

18. Analysis of Exercise Data: Step 1
Lower half of data
Upper half of data
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.
Two halves both include Median because it is an observed data point
Median

19. Analysis of Exercise Data: Step 2
Lower Fourth is Median of lower data set, it is a value between 25 & 30, so it is 27.5
 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. 

20. Analysis of Exercise Data: Step 3
Upper Fourth is Median of upper data set, it is a value between 35 & 40, so it is 37.5
 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 data set.  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.

21. Fourth Spread = 37.5-27.5 = 10 Upper Fourth Lower Fourth
The Fourth Spread is the difference between the upper and lower Fourth, which is = 10.  The UCL is calculated as *10 = 62.5.  The LCL is calculated as *10 = 12.5.

22. UCL = 37.5+1.5*10 = 52.5 Upper Fourth Fourth Spread
The UCL is calculated as *10 = 62.5.  The LCL is calculated as *10 = 12.5.

23. UCL = 37.5-1.5*10 = 52.5 Lower Fourth Spread Fourth
.  The LCL is calculated as *10 = 12.5.

24. Control Chart for Exercise Data
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.

25. 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 shows the budget deviation (expenditure minus budget amount) for each of the months in thousands of dollars:

26. Analysis of Budget Data: Step 1
Sort the data from lowest to highest value
First we sort the data. Notice how the rank order of observations change.

27. Analysis of Budget Data Continued
Lower Fourth is Median of lower half and is -4.5
Lower half
Fourth spread is 23.5 minus -4.5 = 29
Median
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 -3.5.  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.
Upper Fourth is Median of upper half and is 23.5
Upper Half
UCL is * LCL is * 29

28. Control Chart for Budget data
The chart shows that all months are within control limits except for March, where there was a significant deviation from the budgeted amount.

29. Tukey control chart works with medians and is less sensitive to outliers than mean-based charts