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Article Contributed By: Dr. Alan Morris, reproduced by permission
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Accuracy vs. Precision Accuracy vs. Precision

* Many people add to the content of the Slide Rule Universe, sometimes with pictures, sometimes with research, other times with original technical content or articles. We are fortunate to have this article contributed by Dr. Alan Morris on a topic that will interest many visitors to this site.


* The discussion of accuracy and precision is especially relevant to slide rules, because both of the terms are often misused, and frequently misunderstood where slide rules are concerned. Some of the recent discussion threads on the mail list clearly illustrate this. While this article is by no means the final resolution of this issue, it does help clarify the situation, and gives a concrete method of evaluating precision and accuracy in your own rules. Both very useful accomplishments.


* One thing to remember is that both the accuracy and precision of any computing device (whether a slide rule, calculator or computer) need to be suitable for the intended purpose. If you are working with components of 5% accuracy, for example, in an electronic design, precision of 0.01% is not really meaningful. For computing complex orbital trajectories, you will no doubt want all the precision and accuracy (not to mention attention to small details like UNITS, as NASA learned recently on the Mars project) you can possibly obtain.


* In addition, no matter what precision is required, you have probably ASSUMED that the method used for calculation is always intrinsically accurate, no matter what level of precision it is capable of. This is not true, and after reading this article, you will understand much better why all these things are so.


Walter Shawlee 2, Webmaster of SRU


The article will appear below shortly.

Do you have useful material to contribute to the SRU site?

Email Walter HERE to discuss it.

Slide Rule Accuracy vs. Precision
Alan Morris, Dr.Eng., P.E. Version, 8/10/99
	

Introduction

Patently, the most widely published slide rule instruction 
manual was the book by Kells, Kern and Bland which 
accompanied every K+E Log Log Duplex slide rule ever 
manufactured.  This book, in its beginning pages, usually 
"Section 3," has a paragraph entitled "Accuracy of the Slide 
Rule."  That this very paragraph discusses slide rule 
precision, not slide rule accuracy, shows that a gross error 
was propagated for decades by the authors engaged by K+E 
to write their manuals and, further, that no editor or stylist 
ever corrected the paragraph.  Because there is a world of 
difference between the meanings of the two words 
"accuracy" and "precision" this present paper is written to 
clarify the meanings of the contructs of accuracy and 
precision, and will do so by means of examples which are 
related to scales usage.

This paper will also present a discussion of both initial 
manufacture and longevity effects  on slide rule accuracy; a 
discussion of accuracy and precision characteristics of slide 
rules shorter and longer than ten inches as compared with 
the accuracy and precision of ten inch rules; and, in an 
Appendix, the paper will present a twelve-level slide rule 
accuracy evaluation sequence for 10" LogLog Duplex slide 
rules, a sequence developed through an extensive program 
of slide rule evaluations.

 
The Laboratory Scale Experiment

In an Elementary Physics course laboratory, a class of 30 
students, arranged in teams of two, are given the following 
materials and assignment:

Each team is handed a 12" steel scale, a piece of  white 
bond paper, a sharpened pencil, and a 10x magnifying 
glass; the steel scales are engine divided in 1/100 inch 
intervals.

The student teams are instructed to make two marks an 
arbitrary distance apart of their sheets of paper.

The student teams are then instructed to use their steel 
scales to measure the distance between their two marks 100 
times, using the magnifying glass,
recording each measurement made in tabular form on their 
data sheet,  and   to alternate measuring between the two 
team members.  The distance measurements are to be 
made to the nearest 1/100 of an inch, with interpolations to 
be made should the pencil mark lie between two adjacent
l/100 inch graduations of the steel measuring scale.

When the series of 100 measurements is complete, the 
student teams are to compute the average and the standard 
deviation of their individual sets of data, and then hand in 
their results.

When, at the following session of the Elementary Physics 
laboratory, the student teams are handed back their 
measurement results papers from the previous  laboratory 
session, the students find that each team's results, i.e., the 
average and the standard deviation computed, are graded 
with a big red "F" for Fail.  Naturally, the students want to 
know why their very careful work  had universally been 
graded Fail.

To answer the students' questions, the teacher handed back 
to each student team one of the steel scales that they had 
used at the previous laboratory session.  The students
were directed to study the fine writing at the left end of the 
rules, which writing stated:
 
	"Linear shrink: steel, puddled, l:50"

The teacher explained that, although the scales looked  like 
fine, accurate, engraved steel rules, in fact these steel rules, 
marked "12", were actually 12.24" long!  In other words, a l2" 
measurement made with the rule would be about a l/4" 
longer than 12", that a 6" measurement made with the rule 
would be about an l/8" longer than 6", and so forth.  The 
teacher further explained that these rules were used to size 
patterns for sand molding of puddled steel alloy, and that 
castings made with that alloy shrink one part in 50 in every 
direction upon cooling.  Thus the pattern created for the
casting using this alloy would correspondingly have to be 
made one part in 50 larger in every direction in order to 
assure that the cooled casting would be of the desired 
dimensions.

Then to drive home hard the pivotal point of the entire 
exercise, the teacher lectured the students thus:

Use of the shrink rules to measure distances in actual inches 
and fractions of inches, down to l/l00" and, further, down to 
an estimated l/l000 of an inch by interpolation, was totally 
erroneous, since the shrink rules could be counted on to 
make measurements, say, of a 12" distance to only " scale 
intervals, not l/100" or, more ridiculously, to l/l000" by 
interpolative estimates.

Thus the shrink rules were precise, because measurements 
made with the rules could be determined to within 1/l00 inch, 
and interpolations could be made to an estimated l/l000 inch.  
But for making true measurements, e.g., the measured 
distance between the marks made with sharpened pencil 
during the experiment, the rules were not accurate.

In conclusion, the teacher stated, the rules appeared to be 
accurately made, but the rules were not accurate for 
measuring actual distances, the rules were only precise.



Other Accuracy vs. Precision Examples

Having presented the Laboratory Scale Experiment findings, 
the following examples will serve to further demonstrate the 
total, and absolute, difference in the meanings of the two 
distinct constructs:  accuracy, and precision:

A gas tank gage in an automobile has a finely divided scale 
which can be used to read to the nearest 1/10 gallon.  
However, unbeknownst to the operator of the vehicle, a 
miscreant has secretly bent the needle of the gage at a point 
near the needle's pivot, a point that is hidden by the fascia of
the instrument panel. The miscreant who bent the gage 
needle arranged the bend so that when the needle showed 
the gas tank as being "Full," the tank would actually be half-
full.  The gage then becomes an instrument that is precise, 
but that is woefully inaccurate.
	

A watch dial is graduated in l/5th of a second intervals 
between each minute mark.  Thus the watch is precise.  But  
unbeknownst to the person using the watch to observe the 
time, the watch is five minutes slow; reading a time to the 
nearest 1/5th second with this watch, while being precise, is 
ridiculous, because the time reading is five whole minutes 
away from the true time - the watch is inaccurate.


The Constructs of Accuracy and Precision as Applied to 
Log Log Duplex Rules     
	      
Contrary, then, to what Kells, Kern and Bland stated in every 
edition of the K+E instruction book, the readings the authors 
describe relate only to precision, i.e., the scale intervals that 
permit a user to read or set the rule to three or more places.  
Having the scale properties of precision states nothing  
about, and has no relationship whatever to, the properties of  
accuracy of the rule.

The accuracy of a slide rule has, at the time of manufacture, 
everything to do with how the engraving or printing of  all of 
the scale graduations correspond with the true 
mathematically-computed positions of every graduation on 
the rule.  

Assuming then for the moment that a particular slide rule 
was accurately laid down at the time of manufacture, nothing 
specific can be said about the effects on that rule's accuracy 
down through time; those effects can include not only 
damage and abuse, but also in the case of a wood or paper 
rule, shrinkage or expansion,  non-uniformly in a single 
direction or differentially in numerous directions throughout 
the entire volume of the rule.  

In the case of the Log Log Duplex rule there is of  course the 
all-important consideration of transfer of calculations from 
front to rear and from rear to front sides of the rule. Thus, in 
a Duplex  rule the accuracy-damaging effects of time are 
potentially  greatly enhanced because of the two-sided 
referencing that must be done with that style of rule, even if it 
is assumed that the Duplex rule was laid down accurately, 
both sides, and both sides in registry, at the time  of 
manufacture.

Additional Accuracy-Limiting Factors in LogLog Duplex 
Rules

When a rule is manufactured, the wood may not have been 
properly aged, and so the body or slide or both may warp, 
either in a single curve, or in a wavy curve, or the slide 
portion may warp differently than the body portions.  In  the 
latter two cases, the slide at various points along the mating 
edges, will lie either above or below the adjacent surface of 
the body, leading to parallax errors on reading and on 
setting, even if all of the rule's graduations were accurately 
laid down during manufacture.  A rule can also become 
curved, one wave, multiple waves, differential waves, 
through bad storage or careless handling, or from warping 
that occurs over time; the limiting parallax effects above-
described also apply under these circumstances.

Either at the time of manufacture, or through aging, some of 
the body or slide edges may lose planar flatness, and flare 
out at some or all points along the body mating edges, or at 
the slide edges, or at all four mating edges, body and slide.  
This flaring-of-edges effect introduces parallax errors on 
reading and on setting.

It may prove impossible to bring front and rear cursor 
hairlines into perfect coincidence while at the same time 
bringing the pair of  hairlines into perfect registration with the
front and rear sides of the rule.  The usual import of this 
impossibility, should it arise, is that the front and rear sides 
were either not in registration at the time of manufacture
or the front and rear sides through time have proceeded out 
of overall registration.

Another accuracy-limiting cursor effect is related to the fit of 
the cursor to the slide rule body.  Even if the cursor hairlines 
are in perfect coincidence front-to-rear, if the cursor is 
slightly loose on the body in the transverse direction, having 
some slack in that direction, then it is possible that:

	a. The cursor can become angled with respect to the 
surface of the body, causing inaccurate readings from front-
to-rear because the front-to-rear axis of the hairlines is not 
perpendicular to the body of the rule

	b. The cursor may shift position when the rule is 
flipped over to utilize the other side of the rule body.

	c. The cursor may lie fully flat on one side of the 
rule, causing the hairline of the cursor on the other side of 
the rule to be too high above the surface of the other side of 
the rule, leading to parallax errors in accuracy of reading and 
setting.

The cursor cannot be so tightly fitted on the body of the rule 
so as to be capable of being moved only with difficulty, yet 
the optimum free play of the cursor in the transverse position 
can be afforded with only a few thousandth's of a inch of 
transverse movement.  However, having this near-perfect fit 
of the cursor means that any dirt that gets under the cursor 
windows must be removed; this removal of dirt can be 
accomplished easily by use of triangularly shaped, slightly 
moistened, slips of 20 lb. white paper, where the tip of
the paper triangle is introduced under the cursor window, 
and then the cursor is slid  back and forth atop the wider 
portions of the paper triangle.

There must be minimum gap widths between the mating 
scale edges, for if these gaps are too wide, there will be 
accuracy errors on reading and setting the rule.  Gap width 
can be a function of maladjustment of the adjustable stator, 
but due to possible differential shrinkage and expansion of a 
rule through time, it may be impossible to reduce the gap by 
binding down with the adjustable slider without locking the 
slider in place.
	
Some have suggested to this writer that a slide rule might 
expand or contract along its length direction in such a way 
that it, if originally accurately laid down, will remain
accurate.  Leaving out Pickett metal rules, K+E, Dietzgen 
and Hemmi rules are all made of wood.  Hemmi rules are 
made of a superior and more stable wood, bamboo, than the 
wood, mahogany, of which K+E and Dietzgen rules are 
made.  Wood is an non-homogenous material and there is 
no reason why wood, on expansion or on contraction, would 
do so in an absolutely linear and uniform manner. If such 
were indeed possible, the rule at every point along its length, 
body and slide, both sides, would have to expand or contract 
with a uniformity of l/l000 of an inch, a certain impossibility.  
To make realistic, but at the same time totally impracticable, 
the linear expansion and/or contraction suggestion, the rule 
would have to be constructed from heavy bars of platinum-
iridium alloy, an alloy having an exceedingly low coefficient 
of linear expansion or contraction.  Until October 1960, the 
international meter was defined as the distance between  
two marks on a platinum-iridium bar housed in Paris.  To 
present an idea of the level of accuracy involved with the
standard of length, in October 1960, by international 
agreement, the meter was redefined to be l,650,763.73 
wavelengths in vacuo of the orange-red spectral line of 
krypton 86.

Another effect that contributes to the potential masking of  
rule inaccuracies lies in the thickness of the graduations.  
Pickett rules in general have thicker graduations than 
Hemmi, K+E and Dietzgen rules.  When the writer has 
conducted evaluations of  Pickett rules, many have been 
noted to be inaccurately printed, but some Pickett rules that 
have been found to be accurate do meet the higher Levels 
(see Appendix) of accuracy  through the agency of  
somewhat too-thick graduation lines. The writer has 
observed that the graduations of 1945 Hemmi rules are 
thinner than the graduations found on the last-produced 
1975 Hemmi rules.

Visual Acuity and the Slide Rule

The resolving power of the human eye is related to the 
visual angle subtended by the finest detail that the eye 
can distinguish.  However, it is a property of the human
eye, a property long made use of in optical devices such 
as split-image rangefinders, that the eye can distinguish 
line objects and the coincidence of or the lack of  
coincidence of line objects, at visual angles far less than 
those at the limit of the resolving power of the eye. For 
example, one can easily see a distant telephone or power 
line although the visual angle subtended by the distant 
line object is much smaller than the visual angle 
subtended at the resolving power limit of the eye.  One 
can distinguish at a distance if one line is close to, but not 
touching, another line.

The line-distinguishing property of the human eye makes 
facile the reading and setting of a slide rule, since the eye 
can work well with critical line alignment or line non-
alignment; these being the visual tasks involved with slide 
rule calculations.

Slide Rules Shorter Than 10"

"Pocket" slide rules of the LogLog Duplex design have 5" 
scale lengths.  Even if such a rule is accurately laid down, 
there are two effects which serve to limit the accuracy
of a 5" rule as compared with a l0" rule:

	a. The thickness of the graduation lines on the 5" 
rule cannot be less than the thickness of the graduation 
lines found on the 10" rule, while, logically, the 
graduations on a 5" rule should be  the thickness of the 
graduations on a l0" rule.  As discussed above in this 
paper, too-thick graduation lines serve to mask 
inaccuracies, and thus lead to errors on setting and on	    
reading.

	b. The 5" rule is less precise than a l0" rule, since 
the 5" rule is not as finely divided as a 10" rule, 
necessarily so, as otherwise, the 5" rule's scales would 
be rendered useless through overcrowding.

Slide Rules Longer Than 10"

Examples of rules longer than 10" include the 20" Log 
Log Duplex rule, certain cylindrical rules, large circular 
rules, classroom wall demonstration rules, and the 
multiply-staved Thacher rule.  It should be clear from this 
paper that rules longer than 10" can certainly provide 
more precision of setting and of reading, but again it
will here be reemphasized that the scale properties of 
precision has no relationship whatever to the properties of 
accuracy of the rule.

If, for example, on a 10" rule, 1.01 and 9.95 can each be 
set on a graduation line, and if, for example, on 20" rule, 
1.005 and 9.975 can each be set on a graduation line, 
there is absolutely no warranty that the increase of  
precision afforded by the 20" rule due to the increased 
fineness of the graduations on the 20" rule will result in 
more accurate calculations with the 20" rule than can be 
made with the 10" rule.  This is because longer rules 
made of wood or paper could not be manufactured with 
greater accuracy than a 10" rule, and rules longer than 
10" cannot withstand the effects of longevity, namely,  
differential expansion and contraction, warping, edge-
flaring, as well as can a 10" rule.  The costs of 
manufacturing an accurate rule longer than 10" would far 
exceed the costs of manufacturing an accurate 10" rule.  
It is well known in one of the scale procedures of highest 
accuracy, the field of ruling the lines of diffraction 
gratings, that the lengthy engraving machine lead screw 
is the single most extraordinarily costly element of the 
entire machine.  Correspondingly, if a manufacturer set 
out to make accurate slide rules longer than 10", he 
doubtless would not utilize a wooden base for the slide 
rule body; also, his engraving machines would
have to be crafted to be accurate over a distance of a 
least twice the length of a l0" rule, radically increasing  the 
costs of the machines and correspondingly the sales 
prices of long accurate rules.

All this is not to say that all 10" rules were all accurately 
made, in fact, there is no proof that  very  many 10" slide 
rules at all were accurately made.  What we do have in 
the way of proof, however, is the converse proof, made 
inferentially, through  this writer's exhaustive and 
continuing program of slide rule evaluations,  being as 
follows:


The fact that the writer has identified  several near perfect 
and several fully perfectly accurate  10" Log Log Duplex 
slide rules, albeit many, many years having passed since 
these identified slide rules were manufactured, means
that at one time or another the machinery for producing 
the 10" scale length duplex slide rule was capable of 
producing a totally accurate rule.                 

Obviously, it is only through unknown circumstances that 
a slide rule that originally was accurately  manufactured 
would present itself today as still being an accurate slide 
rule.

Definitions

For scale readings and/or settings, as obtain in using 
slide rules, then:

	a. To how many places can the scale be read or 
set is the measure of  precision of the scale.

	b. How close a reading or setting is to the true 
value of the number is the measure of the accuracy of the 
scale.


In Conclusion

The 10" Log Log Duplex slide rule is the optimum design for 
a slide rule, for that design is the single slide rule design that 
at the same time provides:

	a. Potential for overall accuracy.

	b. Two-sided design that allows for openness of 
scales layout, front and rear, and that makes complex 
calculations involving trigonometric, log, ln, reciprocal, 
square root, cube root, and exponential functions easy to 
accomplish, all in a compact package that can be easily 
grasped and that balances well in the hand.  

As the single function of a slide rule is to enable the user to 
make accurate calculations, and as accurate calculations 
can only be made using a slide rule that is accurate, then it 
is this writer's opinion that an accurate 10" Log Log Duplex 
slide rule is the premier slide rule and that any slide rule that 
is less than accurate, or is of any design or configuration 
other than 10" Log Duplex, is merely a curiosity.

	


Appendix:  Accuracy Evaluation Sequence For 10" Log 
Log Duplex Slide Rules

This accuracy evaluation presented below increases greatly 
in difficulty as a candidate rule is sequenced through each 
succeeding Level.

Tools Required

8X optical loupe
Set of screwdrivers of proper tip width and tip sharpness
Wooden or plastic small mallet for tapping slider and 
adjustable stator (can be  smooth wooden handle of old-style 
small screwdriver)

Preliminary Evaluation

Warping - examine rule, end-on, for warping; if excessive 
(see present paper), reject the rule.

Flaring - examine mating edges of scales to note the extent 
of any flaring; if excessive (see present paper), reject the 
rule.

Cursor Fit - examine fit of cursor; if loose transversely (see 
present paper) reject the rule.


Slider/Stator Gaps - check the gap widths between mating 
edges (see present paper); if gaps cannot be uniformly 
minimized, reject the rule.


Level Rating Evaluation

Level l:	 Check C vs. D on front side.

Level 2: Check A vs. B on rear side.

Level 3: Check C and D vs. A and B.

Level 4: Check CF vs. DF.

Level 5: Check D vs. Sin.

Level 6: Check CF and DF vs. D and  Sin.

Setting A:  Align stators to slider.

Level 7: Check C vs. D vs. CF vs. DF.

Level 8: Check A vs. B vs. Sin vs. D.

Level 9: Check C vs. D vs. CF vs. DF vs. A vs. B vs. Sin vs. 
D.

Cleaning:	Clean the slide rule following the methods of 
Bruce Babcock (Oughtred Journal, Vol. 2, No. 2, October 
1993, p. 18);  clean the underside of the 	cursor windows 
(see present paper).

Setting B:	Align cursor hairline to body on front side.

Level 10:	Check all front side scales, bottom to top of 
rule.

Setting C:	Align cursor hairline to body on rear side.

Level 11:	Check all rear side scales, bottom to top of 
rule.

Setting D:	Align cursor to bring front side hairline into 
coincidence with rear side hairline, while at the same time 
taking care to maintain the front side alignment of Setting B 
as well as the rear side alignment of Setting C.

Level 12:	Check all scales, front side to rear side, 
bottom to top of rule.
	

Article copyright 2000 by Dr. Alan Morris.

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