Last Rev.

: 11 JUN 08 Kline-McClintock Uncertainty : MIME 3470 Page 1

KLINE-MCCLINTOCK1 METHOD and the uncertainty is calculated as:

(EXPERIMENTAL UNCERTAINTY)
1/ 2
~~~~~~~~~~~~~~   dV  2 2

 dV 
The first thing the student must be aware of is that this procedure V     L  2
  D  2

does not calculate an error; instead, it states what the possible error   dL   dD  
—the uncertainty—is in a final result based on experimental 1/ 2
measurements or a tolerance in fabrication.   2

2 2 
 D 2D 
  3L2     L  2
  L D  2

So what causes the uncertainty? Basically, it is due to the    2 
  4  
coarseness of measuring tools. A simple six-inch scale graduated  
in increments of 0.1 inches can only be read accurately to the 1/ 2
  2

2

2 
nearest 0.1 of an inch. The uncertainty is taken as one half of a  3  
   3 5   0.2     3 5   0.1 
2 2 2
  
graduation or 0.05 inches. An alternate dimensional uncertainty is    2 
  2  
a tolerance as used in drawing of machine parts. If a dimension is  
8mm  0.1mm, then the 0.1mm is the uncertainty in the dimension.
The Kline-McClintock Method determines the uncertainty of a

  67.9314   0.2    23.5619  0.1
2 2 2 2 1/ 2

 184.5870  5.5516
1/ 2
calculation given certain measurements and the tolerances on those  13.7891in . 3

measurements. To exemplify how this method works, assume one

is to determine the volume of a cube containing a cylindrical hole This value is not a percent!
(see the figure below).
D
The ideal volume calculation is
2
 D
V  L3    L
L3  2 
2
 3
  5 3    5
 2
L2
 125  35.3429  89.6571in3 .
L1 Thus, one could have a percent error as large as
To make this a simple explanation of the procedure, assume that L   13.7891 
   100  15.38%
= L1 = L2= L3. The volume, ideally, would then be:  89.6571 
2
 D even though the errors in individual measurements are less than 4%.
V  L3    L (A.1)
 2 
If one measures D and L with measurements
 0.2 
L  5"0.2"    100  4.000%
 5 
 0.1 
D  3"0.1"    100  3.333%
 3 
what is the anticipated possible error (either positive or negative)
when the volume is calculated using Equation A.1? In general, if n
measurements xn are being made, each with a measurement
tolerance of n, and a function F is calculated using the measured
values, then the uncertainty or tolerance in the calculation can be
determined as:
1/ 2
  2 2 
dF   dF 
 F      1  2     2  2
.
  dx1   dx 
 2  
 
For the proposed example,
2
 D
F  V  L3    L
 2 1  0.2
x1  L 1  0.1
x2  D

1
Kline, S. J., and F. A. McClintock. “Describing Uncertainties in
Single-Sample Experiments.” Mechanical Engineering, Vol. 75, No. 1, January
1953: 3-8.
Last Rev.: 11 JUN 08 Kline-McClintock Uncertainty : MIME 3470 Page 2