#### Slide rule accuracy and precision

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# Slide Rule Precision, Accuracy and Significant Digits

Dave Hoyer, July 2012

# Contents

 Introduction Significant Digits Summary Fractional Significant Digits Precision and Accuracy Worked Examples Accuracy Limits Measuring the Accuracy of a Slide Rule

# Introduction

How accurate is a slide rule? What's the difference between accuracy and precision? Here are some suggested answers to these questions, specifically with respect to using slide rules for multiplying two numbers.

# Summary

• Precision and accuracy are not the same thing. Very roughly, precision is how many significant digits you can read off the scale, and accuracy is how close your result is to the true result.
• A damaged or poorly made slide rule can have high precision but low accuracy.
• The precision and accuracy of a well made slide rule both improve with slide rule scale length.
• A linear slide rule scale has the same precision at all points along its scale, whereas the number of significant digits at the low end is one greater than at the high end.
• A formula is presented for determining the number of significant digits of any slide rule. This can be used to compare the potential accuracy of different slide rules.
• To get one digit more accuracy a slide rule scale needs to be 10 times longer and have ten times as many division markings.

# Precision and Accuracy

The precision of any instrument is a measure of how finely it can be read repeatedly. For example, on my Faber Castell 52/82 slide rule the C scale is a single log scale 250 mm long. The first division after 1 is 1.01 and the division is 1.1 mm wide. You can sub-divide this by eye into 10 sub-divisions if you are careful. We can say that the precision near 1 is ±0.001 or 0.1%. Similarly, the last division before 10 is 9.95, this division being 0.55 mm wide. You could estimate by eye to ±0.01, which is still 0.1% (of 10).

The accuracy is how close your answer is to being “true”, or exactly right. If for example the slide rule was incorrectly manufactured, with some of the divisions slightly in the wrong place, then it would have poor accuracy at those places even though the precision remains the same.

So if the slide rule in the example above was perfectly manufactured, and if our eye can estimate 10 sub-divisions inside a 1.1 mm division (or 5 over a 0.55 mm division), then we can say it has a precision and relative accuracy of 0.1% over its whole range. The absolute accuracy would be ±0.001 at the low end near 1 and ±0.01 at the high end near 10.

# Accuracy Limits

What if the slide rule scale was made shorter but still had the same number of marked divisions? At some point the divisions would simply be too close together to reliably sub-divide by eye to the same degree. An 8” slide rule for example would have a C scale 200 mm long with a first division of 1.01 at a distance of 0.86 mm. I reckon I may be able to estimate 10 sub-divisions between 1 and 1.01 on a good day, but not very confidently. Anything smaller and I would find it difficult to estimate the division into tenths without extra tools such as a microscope and/or a finely divided reference scale. So for the sake of argument let’s say that 1mm is the smallest interval we can reliably sub-divide into 10 by eye.

We can use this observation to define a ground rule for the accuracy of a slide rule at any point along its length, by using the values represented by two adjacent divisions and the width between them.

Dave’s Ground Rules for determining the accuracy of a slide rule..

• You can’t sub-divide a division by eye into more than 10 sub-divisions
• A division must be at least 1 mm wide to sub-divide by eye into 10
• A division less than 1 mm wide can be divided into a proportionally smaller number of sub-divisions (ie. less than 10).

This can be put into a formula for the accuracy at any point along the scale of a slide rule. First define the following parameters in relation to a marked division on a single log scale from 1 to 10..

• A the absolute accuracy
• L the length of the scale, mm
• T  the maximum number of “by eye” sub-divisions
• V1 the value at a mark on the scale (i.e. the start of the division)
• V2 the value at the next mark after V1 (i.e. the end of the division)
• W the width of the division, mm The width of the division along the scale is…

• W  =  L * (log10(V2)- log10(V1))

The number of “by eye” sub-divisions allowed is…

• T  =  10, if W is 1 mm or greater, or
• T  =  10 * W,  if W is less than 1 mm (rounded to an integer; not less than 1)

The absolute accuracy at V1 is…

• A  =  ± (V2 - V1)/T

# Significant Digits

The number of significant digits is how many meaningful digits can be written down. With slide rules this is determined by the smallest interval we can estimate by eye, which is somewhat subjective but we can make some ground rules. In the example above with a 250 mm (10”) long single log scale we can meaningfully distinguish between 1.001 and 1.002, which we can call 4 significant digits. And we can distinguish between 9. 98 and 9.99, which we can call 3 significant digits.

On smaller slide rules the first division after 1 may be 1.02 rather than 1.01, and we should be able to estimate within this division to ±0.002. We could distinguish between 1.002 and 1.004 but no finer. This is not quite a full 4 significant digits, but is clearly better than 3 significant digits. In this case we need to have a way to use fractional significant digits to enable us to properly compare the accuracies of different slide rules.

# Fractional Significant Digits

The C scale on my Pickett N600-ES is 125 mm long and its first division 1.1 mm away is 1.02, which we can divide into ten by eye to get ±0.002, which in turn seems like “not quite 4 significant digits”. We need to give it a fractional number of significant digits somewhere between 3 and 4.

Let’s call the number of significant digits N. Then we could say..

• N = 1 + Log(1/A)  ……..  where “Log” is the base 10 logarithm.

Try it out:

 250 mm scale at 1.001 N = 1+Log(1/0.001) =  4.0 250 mm scale near 9.98 N = 1+Log(1/0.01)    =  3.0 125 mm scale near 1 N = 1+Log(1/0.002) =  3.7

So using this approach the number of significant digits of a slide rule is a function of both the physical distance between two divisions, and the difference in value represented by the two divisions.

# Worked examples

Some worked examples (table)..

 Description L, mm V1 V2 W, mm T A SigD Pickett 600, 5" 125 1 1.01 0.5 5 0.002 3.7 " 125 3 3.02 0.4 4 0.005 3.3 " 125 9.95 10 0.3 3 0.0167 2.8 Faber 52/82, 10" scale 250 1 1.01 1.1 10 0.001 4.0 " 250 3 3.02 0.7 7 0.0029 3.5 " 250 9.95 10 0.5 5 0.01 3 14" scale 357 1 1.01 1.5 10 0.001 4.0 " 357 3 3.02 1.0 10 0.002 3.7 " 357 9.95 10 0.8 8 0.0063 3.2

The table below has a summary of calculated significant digits (SigD) for various slide rules, from the 4” Sun-Hemmi to the giant Loga 15m and the Thacher 20m cylindrical slide rules.

 Description Type L, mm SigD @ 1 SigD @ 3 SigD @ 10 Sun-Hemmi, 4" Linear 100 3.6 3.2 2.6 Pickett 600, 5" Linear 125 3.7 3.3 2.7 K&E 4053-3, 8" Linear 200 4.0 3.5 3.0 Faber Castell 52/82, 10" Linear 250 4.0 3.5 3.0 14" Linear 357 4.0 3.7 3.0 20" Linear 500 4.0 3.7 3.0 Otis King, 66" Helical 1677 4.8 4.3 3.8 Fuller's Calculator, 41' Helical 12500 5.0 5.0 4.7 Loga 15m Cylindrical 15000 5.0 5.0 4.8 Thacher 20m Helical 20000 5.0 5.0 4.7

All the above also assumes that the slide rule is “true”, namely that its markings are correctly placed to within the measurement accuracy.

# Measuring the Accuracy of a slide rule

Without special tools the best way to get a quick feel for the accuracy is to do some multiplications and compare the result with the correct result. Try these..

 1.012345^2  =  1.0248424 to check out a result close to 1 3.1415956^2  =  9.8696041 to check out a result close to 10 9.876543^2  =  97.546101 to check out a result close to 100