Bob Luttman, Robert Luttman & Associates
Histograms
Histograms provide a compressed picture of the variation and center of a data set. They are the perfect complement tcontrol charts. In fact, the combination provides a complete picture of a data set.
Basic Histogram
- Shows the location of the mean and modes of the data.
- Shows the variation in the data, especially the range.
- Identifies patterns in the data not ascertainable with statistics alone.
- Is a good aggregate graph of one variable.
- Can display data for quantitative (i.e., attributes data) variables.
Construction
- Categories gon "X" (horizontal) axis
- Values gon "Y" axis (vertical) axis
- Draw bar (or column) from X axis up tappropriate value line
- Calculating values:
Counts: Count all data points fitting the category or range
Percents: Divide individual counts by total number of data points
Do's and Don'ts
- Make the ranges equal when using quantitative data, especially time based data.
- Use "other", "greater than", or "less than" categories tindicate the presence of outliers. This makes the graph more readable.
- Sort qualitative categories in order of importance.
- Always start the "Y" axis at zero. Tstart anyplace else distorts the graphs proportions.
- Use percents rather than counts. This gives a consistent scaling tall histograms and allows you tcompare graphs with different sample sizes.
Interpretation
Skew
Positively skewed histogram of procedure times
Negatively skewed histogram of patient arrival relative to appointment time
Analyzing Skew
Centering
- Where are the mean and median?
- Relative teach other
- Relative tthe distribution
Tail
+ Is it long and thin, or, short and fat?
+ Is it a smooth tail or "bumpy"?
Possible Causes
+ Mathematical
+ Ex.: All data is greater than zer(time data)+ Ex.: Percents that must sum to 100%
+ Process
+ Limits (ex: size of a waiting room)
+ Deadlines
+ Appointment times
Mode(s)
Multimodal distribution of OR cancellations/day
Notice:
Modes at 5,7,9,12Outlier at 18
How many modes?
Location
+ Relative teach other+ Relative tmean and median
Shape
+ Spike or bell?
Outliers Analysis
Location
+ Relative tmean, median, or mode(s)+ Neighboring points
Relevance
+ Bad data or a Real Problem?
Most important point: Do Not Dismiss Them! Do Not Assume They Are a Fluke!
Histogram Enhancements
Grouping
A histogram enhanced to show both clinical service and admission type
Stacking
Same data, different format. The grouped histogram shown as a stacked histogram
+ Shows distribution within a category
Ex.: About 50% of Angiography patients are Outpatients
Note: If sample sizes are unequal, make that clear!
Layering
Again, the same data. This time three histograms are generated and layered. Data is percent of all patients so that different sample sizes are obvious.
Summary
- Histograms clearly picture the data showing peaks and valleys, clusters, and outliers.
- Histograms provide a graphical interpretation of the six basic statistical measures from above.
- Enhanced histograms detect and analyze stratification and/or multivariate data.
- Histograms, as part of a Paretanalysis, separate the "significant few" from the "trivial many"; thus focusing continuous improvement efforts.
