1. Xbar Chart
Farrokh Alemi, Ph.D.

2. Why Chart Data? To discipline intuitions To tell a story
To discipline intuitions. Data on human judgment show that we, meaning all of us including you, have a tendency to attribute system improvements to our own effort and skill and system failure to chance events. In essence, we tend to fool ourselves. Control charts help see through these gimmicks. It helps us see if the improvement is real or we have just been lucky. An example may help you understand my point. In an air force, a program was put in place to motivate more accurate bombing of targets during practice run. The pilot who most accurately bombed a target site, was given a $2,000 bonus for that month. After a year of continuing the program, we found an unusual relationship. Every pilot who received a reward did worse the month after. How is this possible? Rewards are supposed to encourage and not discourage positive behavior. Why would pilots who did well before do worse now just because they received a reward? The explanation is simple. Each month, the pilot who won did so not because he/she was better than the others but because he/she was lucky. We were rewarding luck; thus, it was difficult to repeat the performance next month. Control chart helps us focus on patterns of changes and go beyond a focus on luck.
In a field like medicine, over time a poor outcome occurs the natural tendency is to think of it as poor care. But such rash judgments are misleading. In an uncertain field such as medicine, from time to time there will be unusual outcomes. If you focus on these occasional outcomes, you would be punishing good clinicians whose efforts have failed by chance. Instead, focus on patterns of good or bad outcomes. Then you know that the outcomes are associated with the skills of the people and the underlying features of the process and not just chance events. Control charts help you see if there is a pattern in the data and move you away from making judgments about quality through case by case review.
To tell a story. P-charts display the change over time. These charts tell how the system was performing before and after we changed it. They are testimonials to the success of our improvement efforts. Telling these types of stories helps the organization to:
celebrate small scale successes, an important step in keeping the momentum for continuous improvement.
communicate to others not part of the cross functional team. Such communications help pave the way for eventual organization wide implementation of the change.
You can of course present the data without plotting it and showing it. But without charts and plots, people not be able to see the data. Numbers makes people understand the data but plots and charts, especially those drawn over time, make people connect a story with the data, they end up feeling what they have understood. For many people seeing the data is believing it. When these charts are posted publicly in the organization, control charts prepare the organization for change. They transfer and explain the experience of one unit of the organization to other units.

3. Steps to create an X-bar chart?
Check assumptions
Calculate grand average
Calculate standard deviation
Calculate standard deviation for each time period
Calculate control limits
Plot chart
Interpret and distribute

4. Example Data
Over several months, we tracked the satisfaction with our unit.
The question is whether the unit has improved over time.

5. Are variations due to chance?
How would you answer this question? Look at the data.
There are wide variations in the data. Could these variations be due to chance?
The first step is to calculate the average for each time period.
The x-y plot already tells you a lot. But it will tell you more, if you add to the plot the upper and lower control limits, between which one expects 95% of the data. The upper and lower control limits in an X-bar chart is based on the assumption that data are normally distributed. So before we calculate these limits, we need to check and see if the assumptions are met.

6. 1. Check Assumptions Continuous Interval Scale Independent events
Normal distribution
Constant variance
Continuous Interval Scale. The variable being averaged must be a continuous variable on an interval scale, where the differences between the scores are meaningful. An ordinal scale cannot be averaged. Satisfaction rating and health status ratings are generally assumed to be interval scales.
Independent events. The observations over each period of time are not affected by the previous observations. In our example, the satisfaction ratings in time period two should not be affected by ratings in the first time period. This assumption will be violated in an example where the same patient is rating the unit in every time period. It is likely that this patient's first impression affects subsequent evaluations. The assumption seems reasonable when different patients are rating in different time periods.
Normal distribution. If we were to stack all the ratings, most will fall on the average rating, some on each side. A normal distribution suggests that the stack will peek on the average, slowly decline on both sides of the average, and the shape of the curve will be symmetrical. The law of large numbers says that no matter what the distribution of a variable is, the average of the variable will tend to have a Normal distribution. As the number of cases for calculation of the average increases, the average is more likely to be Normal. A minimum of four cases is needed for applying the law of large numbers. The law of large numbers tells us that the average of any distribution, no matter how strange, has a Normal distribution.
Constant variance. This assumption can be verified on a control chart. It states that deviations from the average should not consistently increase or decrease over time.

7. 2. Calculate Grand Average
Average values for all time periods and all cases, plot as a line
Calculate and plot the grand average. The grand average is the average of all ratings across all time periods. Do not calculate this by averaging the mean of each time period. The correct way to do this is to sum all the ratings for all time periods and divide the sum by the number of ratings. In the example provided here, it makes no difference how you calculate the grand average, but in situations where the number of cases in each time period is changing, it does make a difference.
With the central tendency line in, we have a visual line to compare the data to. It already tells us a lot, it gives us a sense of which time periods are closer to our central tendency line.

8. 3. Calculate standard deviation
Calculate standard deviation for all observations S = [i=1,…,nj j = 1, …m (Xij -GA )2 / (n-1)]0.5 GA is grand average n is number of observations across time periods
If you notice, in some time periods you have larger differences from the central tendency line than in others. The question arises whether these differences are so large that they are beyond what could be expected from chance events. To answer this question we need to add two other lines to our plot. These are upper and lower control limits. Then points beyond these limits are differences that could not be due to chance. They indicate real changes in the patients' satisfaction with our services. To calculate the upper control limit first calculate the standard deviation of the observations.
Estimate the standard deviation for observations within each time period by dividing the standard deviation of all of the observations by the square root of the number of cases in the time period.

9. 4. Calculate standard deviation for each time period
Estimate the standard deviation for each time period St = S / (nt)0.5
Estimate the standard deviation for observations within each time period by dividing the standard deviation of all of the observations by the square root of the number of cases in the time period.

10. 5. Calculate Control Limits
UCL = GA * St
LCL = GA – 1.96 * St
1.96 sets the limits so that 95% data fall within the two limits
GA is the grand average
St is the standard deviation for the time period
Calculate the upper lower limit for each time period as the grand average plus 1.96 times the standard deviations of the time. For our first time period this will be *2.05
Calculate the lower limit for each time period as the grand average minus 1.96 times the standard deviation of that time period.

11. Calculations in Excel
Standard deviation function over range of all observations
Dividing by square root of number of observations in the time period
Grand average
Constant for 95% confidence limits
=Average ($B$2:$E$5)
=Average(B2:E2)
You can use the standard deviation function in Excel to calculate the standard deviation of observations. The function is =stdev and the arguments of the function are the range of the data.

12. 6. Plot Control Chart
The x-axis is time, the Y-axis is the observed values in this case satisfaction. Observations are show by single markers. Control limits are shown as lines with no markers
Note that the control limits are straight lines in this example because in every time period we sampled the same numbers of cases. If this were not the case, the control limits would be tighter when the sample size was larger. If observations fall within the control limits, then the change in the observed rates may be due to chance.

13. 7. Interpret findings & distribute
First time period is above UCL
Second time period below LCL
Note also that the second time period is lower than the lower control limit. Therefore, patients rated our services worst in this time period and the change in the ratings were not due to chance events. It marks a real change in satisfaction with our services. We would not have known this until we created a control chart. +
The first time period is just also above the upper limit, so we did well in this time period.