E

A

B

C

D

F

G

H

-15

-10

-5

0

5

10

15

-15 -10 -5 0 5 10 15

10 kg #1

10 k

g #2

Youden Analysis

Youden Analysis

• Introduction to W. J. Youden

• Components of the Youden Graph

• Calculations

• Getting the “Circle”

• What to do with the results.

W. J. Youden 1900-1971

• Born in Australia

• 1921 – B.S. in Chemical Engineering

• 1924 – Ph.D. Analytical Chemistry

• 1924-1948 – Plant Research

• 1942-1945 – World War II

• 1948 – NBS Statistical Consultant

Components of Youden Graph

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10

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

X Axis

Y Axis

2sd limit of the random components

45 degree

Origin (0,0) Median(x,y) Known(x,y)

RMAP 10 kg (Jan 1, 00 thru Oct 1, 00) 10 kg #1 (analyzed versus NIST)

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

10.0

12.0

mill

igra

ms

10 kg #1 NIST NIST Unc NIST Unc

RMAP 10 kg (Jan 1, 00 thru Oct 1, 00) 10 kg #2 (analyzed versus NIST)

-20

-15

-10

-5

0

5

10

15

mill

igra

ms

10 kg #2 NIST NIST Unc NIST Unc

Line Graphs to Youden Graphs

E

A

B

C

D

F

G

H

-15

-10

-5

0

5

10

15

-15 -10 -5 0 5 10 15

10 kg #1

10 k

g #2

Systematic and Random Components

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10

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

Plot the Point (-2,-7) X-axis = -2 Y-axis = -7

Total magnitude of Error = 7.28

Calculated by using the formula for the distance between two points (x1,y1) and (x2,y2):

28.7

53494

)07()02(

)()(

22

212

212

d

d

d

yyxxd

Draw a line from the Point to the 45 degree line (Perpendicular)

Intercept Point

)5.4,5.4(

5.4

2/)72(

2/)(

yx

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10

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

Systematic and Random Components

364.6

2/]9[

2/]5.45.4[

2/)]05.4()05.4[(

2/)]()[( 1212

d

d

d

d

yyxxd

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:

536.3

25.625.6

2]5.2[2]5.2[

)]5.4()7[()]5.4()2[(

)()(

22

212

212

d

d

d

d

yyxxd

Fitting the Ratio of Systematic & Random Errors to the Total Error

Systematic Component = -6.364 (negative or positive)

Random Component = 3.536 (always positive)

Sum Random & Systematic = 9.900

Total Error = 7.280

600.2)280.7(900.9

536.3

680.4)280.7(900.9

364.6

Random

Systematic

Where do we get the Circle?

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10

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

Random Error=2.60

Each Point will have a “Random

Error”

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.)

1

2

n

rans

(modified) Calculating the radius of the Circle

Getting the Circle on the Graph

• Formula of a Circle

circle) of radius(r 222 yxr

22 rxy

•Formula rewritten in terms of y

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)

Let’s take a look at the

Spreadsheet