1. Youden Analysis

2. Youden Analysis Introduction to W. J. Youden
Components of the Youden Graph
Calculations
Getting the “Circle”
What to do with the results.

3. W. J. Youden 1900-1971 Born in Australia
1921 – B.S. in Chemical Engineering
1924 – Ph.D. Analytical Chemistry
– Plant Research
– World War II
1948 – NBS Statistical Consultant
Interesting – Youden was a statistician for bombing accuracy during World War II.

4. Components of Youden Graph
Y Axis
2sd limit of the random components
Origin (0,0) Median(x,y) Known(x,y)
X Axis
Youden’s analysis is directed toward interlaboratory comparisons of chemical testing. These types of tests are more procedure dependent than that of mass measurements. Youden’s main objective was to determine the precision of a procedure and expect all labs to meet this level of precision. He also expected lab’s systematic errors to be small compared to the precision (random error). However this type of analysis can be used effectively to evaluate mass measurements. Remembering that in mass measurements it is usually the random errors that are small as compared to the systematic errors.
Also, note that the “origin” is not always (0,0). In fact, the origin is only at (0,0) when ‘differences’ are used. In most cases, the origin will be determined by the “best value” for the artifact, such as the NIST value (if available), the mean (average) value, or the median value. The median value is most commonly used because it is not as vulnerable to ‘outliers’.
The advantage of using Youden Analysis is its unique ability to separate random & systematic errors. An Error that is purely systematic will fall on the 45 degree line. For example, if a lab’s mass standard is off by 2 mg, both artifacts will have a 2 mg error from the true value and will fall on the 45 degree line.
45 degree

5. Line Graphs to Youden Graphs
Lne Graphs have been the most common method of graphing round robin results. However, when the data allows, Youden Graphs can provide valuable information to the participants concerning possible systematic errors.
This slide illustrates how data can be taken from line graphs and used to create a Youden Graph (must meet certain criteria first).
Data from one artifact is graphed on the X-axis and data from the another artifact is graphed on the Y-axis.

6. Systematic and Random Components
Total magnitude of Error = 7.28
Calculated by using the formula for the distance between two points (x1,y1) and (x2,y2):
Intercept Point
Draw a line from the Point to the 45 degree line (Perpendicular)
Youden’s method of determining vector lengths were approximations and were designed for ease of use. In the early 1960’s all calculations were done by hand with the use of a slide rule (slip-stick). Now, with the availability of calculators and computers the vectors can be easily defined. Modifications to Youden’s original methods have been made and the vectors and their associated intercept points are now calculated by using formulas for the “distance between two points” , the Law of Sines”, and Pythagorean formulas for right triangles.
Plot the Point (-2,-7) X-axis = Y-axis = -7

7. Systematic and Random Components
Systematic Distance from Origin to Intercept
Calculated by using a variation of the Pythagorean formula for 45o right triangles:
Origin=(x1,y1)
Point = (x2,y2)
Random Distance from Point to Intercept
Calculated using the formula for the distance between two points:
There are many different methods to calculate the vectors we need, the methods used here are just the ones that came to mind first.
***This portion of the training needs to be refined and redefined in future presentations to show the “actual” random & systematic components that comprise the total error as shown in the calculations and graph on the next two slides.

8. Fitting the Ratio of Systematic & Random Errors to the Total Error
Systematic Component = (negative or positive)
Random Component = (always positive)
Sum Random & Systematic = 9.900
Total Error = 7.280
The absolute value of the “systematic component” added to the “random component” must be fitted to the Error Vector.
Using the Law of Sines for triangles, it can be verified that this “ratio method” is valid.
The next slide shows the “actual” random error.

9. Where do we get the Circle?
Each Point will have a “Random Error”
A horizontal line drawn from the “45 degree intercept point” to the error vector shows the proper random and systematic components.
A Circle is defined by its origin (or center) and by its radius. The origin is determined by the “true” values of artifact#1 and artifact#2. The radius of the circle is based on the standard deviation of the lab random errors. This standard deviation of the random errors is then multiplied by a confidence factor that should make the circle contain 95% of all points IF the systematic errors were eliminated.
The next slide shows the calculations for determining the standard deviation of the random errors and the radius of the circle.
Random Error
=2.60

10. (Youden’s) Calculating the radius of the Circle
Each participant’s point provides a perpendicular.
Each perpendicular is squared.
These squares are then summed and divided by n-1.
The square root of this result is an indication of the standard deviation based only on the random components of each point.
Multiplying the standard deviation by 2.45 gives the value for the radius of the circle. (95% of the points should fall within this circle if all systematic errors could be eliminated.)
THIS SLIDE IS CURRENTLY HIDDEN.
FORMULA NEEDS TO BE CHECKED.
THE FORMULA LOOKS LIKE A ‘STANDARD DEVIATION’ FORMULA BUT IT DOES NOT USE RESIDUALS.

11. (modified) Calculating the radius of the Circle
Each participant’s point provides a random error (ran).
Each random error is squared.
These squares are then summed and divided by n-1.
The square root of this result is an indication of the standard deviation based only on the random components of each point.
Multiplying the standard deviation by 2.45 gives the value for the radius of the circle. (95% of the points should fall within this circle if all systematic errors could be eliminated.)
(This slide is self-explanatory)

12. Getting the Circle on the Graph
Formula of a Circle
Formula rewritten in terms of y
We use the “Formula for a Circle” to calculate points on the circle. X and Y will be offset by the value for artifact#1 and artifact#2.
(The bottom formula is a hyperlink to a spreadsheet to show the layout of data used to draw the circle on the graph)

13. Rules of Youden Analysis
Requires Two Artifacts
Must have two values to plot a point
Artifacts must be same Nominal Value
“Cannot compare Apples & Oranges”
Same procedure must be used to test both Artifacts
SOP - Restraint - Equipment - Metrologist
Artifacts should not be Tested at Same Time
Random errors appear to become more systematic when tested at the same time
Participants should be working at the same precision level
Don’t Over-Analyze
A point that lies outside the circle doesn’t necessarily mean that there is a problem (although it is never a good thing)
(Self-explanatory)
(Don’t over-analyze: This should be expanded when more training time is allowed.)

14. Let’s take a look at the Spreadsheet
We will now take a look at a Youden Spreadsheet. And we will attempt to perform a file transfer so each of you may have a copy. The Spreadsheet is still being developed, so if you have any suggestions (after you have time to explore) let me know and I will try to incorporate the changes and keep the spreadsheet updated. Maybe we could have it as a “downloadable” on the OWM website.
(“Spreadsheet” is a hyperlink to the Youden Spreadsheet)