in a million. This being so, the prominent question to beheld
before the mind, when starting a series of measurements, should
be: Under all the circumstances of the case, what degree of pre-
cision is one justified in seeking? Assuming the accuracy of the
standards the possible precision attainable is generally related very
closely to the time spent upon the measurements, and the time
which one is justified in using is governed by circumstances which
should not be ignored. Suppose, for example, the object of a
measurement is to determine the specific resistance of a sample
of commercial graphite, With accurate standards and the expen-
diture of much time this might be determined to an accuracy,
perhaps, of a twenty-fifth of one per cent. But there would be
no justification in seeking such a high precision in this case
because it would be without value in view of the variability of
graphite. It is probable that different samples of graphite from
the same supply would differ in resistivity by one per cent or
more, and the same sample would certainly vary in resistivity from
day to day by much more than a twenty-fifth of one per cent. On
the other hand, if the object of a series of measurements is to
determine the minute variations over a period of time which take
ART. 102] EXTENT OF ELECTRICAL MEASUREMENT 5
place in a manganin-resistance standard, the most painstaking
care should be exercised to secure the necessary precision, and this
care and the time spent would be entirely justified, if circumstances
justified the research itself.
Again, great accuracy and painstaking care in finding how many
times the magnitude of the standard available is contained in the
quantity under measurement is not justified, if the uncertainty
must remain great regarding the true value of the standard itself.
Few lines of work require, as does electrical measurement, such
discriminating judgment as to the relative importance of things.
But no admonition or instruction can give this balanced judgment,
so necessary to successful performance, as does practice. In this,
as in most matters requiring skill, to measure well, one must
practice measurement. When, by experience, a discriminating
judgment has been acquired, it will be applied, not consciously, but
In making electrical measurements much more mischief is likely
to result from a careless commitment of gross errors than from
failure to give attention to details and to deduce the most prob-
able value from a set of observations. The gross errors may
result from a misconception of the nature of the problem, from an
entirely incorrect reading of the larger indications of an instrument,
from mistakes made in ordinary arithmetic, or from an improper
interpretation of units in calculating the results. The strained
attention often required to read small decimals, frequently causes
an entire loss of mental perspective as to the main features of the
problem. Important points are overlooked while minutiae are
carefully observed. In no line of effort is a novice more apt "to
strain at a gnat and swallow a camel," than when trying to make
a refined electrical measurement.
It is generally wise, as a precautionary measure, after the
apparatus is in adjustment and all the connections are completed,
to make what might be termed a survey measurement, and then
to deduce the results with a rough calculation, and consider well
if these results look reasonable. If they do, and the outfit is
seen to be in a proper, balanced, working condition, painstaking
observations may then be undertaken and the data worked up.
Nothing so assists in proving or disproving the reasonableness of
the observations as plotting the data in a curve, and this, in
almost every case, is to be recommended.
6 MEASURING ELECTRICAL RESISTANCE [ART. 102
In nearly every extended series of observations there will be,
in addition to accidental errors which are as likely to be plus
as minus, certain systematic errors which escape observation and
are not eliminated by taking the mean value of a number of read-
ings. The novice, noting a fine agreement among his observa-
tions, is often deceived into supposing he has attained an accuracy
far greater than the measurements really justify. The surest
way to obtain enlightenment is to remeasure the same quantity
by an entirely different method. The lack of agreement that is
apt to result is often a disagreeable surprise and gives one, as
nothing else can, a just estimate of how grudgingly Nature per-
mits the real truth to be extracted. A close agreement -in the
results of measurements made by two or three entirely different
methods gives, on the other hand, the highest assurance that a
real precision has been attained.
When a quantity is to be measured with precision, it is generally
wise to seek a method of measurement which will give the result
as directly as possible and without the necessity of making a
number of corrections. So-called "null" methods, in which the
quantity being measured is balanced against some other quantity,
are less rapid, as a rule, than deflection methods, in which the
quantity is measured in terms of the deflection of some instru-
ment, but ordinarily the former are much more accurate and there
is less necessity of applying corrections. Null methods are gener-
ally to be preferred for all precision work. In the descriptions
which follow, of the various methods of measuring electrical re-
sistance, the relative advantages of the two methods will become
The question as to what sensibility the indicating or measuring
instrument should have is an important one that must receive
careful consideration. It may be said, however, that, in general,
increased sensibility in the indicating instrument leads to a re-
duction in size and cost of all the rest of the equipment required.
On the other hand, more care, time, and skill are required to
work with sensitive instruments, and judgment must be con-
stantly exercised to choose a sensibility best adapted to the particu-
lar problem in hand.
It should be borne in mind that the only electrical quantity
which can be sold or exchanged for money is electrical energy,
or electrical power expended for a given time. Electrical power
ART. 103] EXTENT OF ELECTRICAL MEASUREMENT 7
is composite, being the product of E.M.F. and the current which
is in phase with the E.M.F. When the average power developed
in a certain time is multiplied by the time, the total amount of
electrical energy developed is obtained. Now energy has a real
existence and a market value. Consequently, from the industrial
standpoint, most electrical measurements have as their final
object the precise determination of that which has money value,
namely, electrical energy or watt-hours. While there are both
scientific and practical considerations which make it necessary
to measure separately such quantities as E.M.F., current, phase
angles, resistances, etc., one should not lose sight of the industrial
object to be attained, which is the proper and just charge for a
quantity of electrical energy sold. Other considerations, of course,
apply with those classes of measurement required in telephony
and telegraphy, or those made solely for scientific investigation.
Finally, it is strongly recommended to all those engaged in
electrical measurements that they give a careful study to the
meanings and physical interpretations of the electrical and mag-
netic units in use. In this connection the writer would recom-
mend the use and study of a little book called " Conversion
Tables," by Dr. Carl Hering.
103. Elements of the Theory of Errors.* There are, in gen-
eral, two classes of errors; systematic errors and accidental errors.
The former class often result from an incorrect value being as-
signed to the standards employed, from a faulty calibration of
the scale of an instrument, or from the constant presence and
influence of an unrecognized force, as, for example, the unrecog-
nized presence of a thermoelectric force in the circuit of the indi-
cating galvanometer. Systematic errors do not eliminate when
the mean value of a series of observations is taken as the result.
The latter class generally result from inaccuracies in reading
the instruments and from fluctuations above and below a mean
value of some quantity which determines the readings of the
instruments while the observations are being taken; for example,
fluctuations in the E.M.F. employed when measuring resistances
with deflection instruments. As accidental errors are as likely
to be plus as minus, they tend to eliminate from the mean value
as the number of readings is increased. But it may be added,
* Some of what follows under this heading is taken from "Instruments et
Methods de Mesures electriques Industrielles," par H. Armagnat.
MEASURING ELECTRICAL RESISTANCE [ART. 103
that the theory of probability shows that the precision increases
not directly with the number, but proportionally with the square root
of the number of measurements.
The difference between the value found in a single measure-
ment and the mean of the entire series of measurements is called
the apparent error. If the sum of all the apparent errors, without
regard to sign, be taken, and this sum be divided by the number
of measurements, we obtain the mean error. The theory of
probability shows that the mean error, obtained in this way,
provided the errors are accidental and not systematic, is very
closely the same as would be obtained if the mean error were
found by taking the true value of the quantity instead of the
mean value. Hence, in estimating the true value from a set of
measurements, one can say that this true value is equal to the
mean value found by the measurements plus or minus the mean
error. For example, suppose one has made five measurements of
a resistance, with the same apparatus and method, and all pre-
cautions have been taken to avoid systematic errors; the results
would be expressed as follows:
No of Meas.
127.8 = sum
0.640 = sum
25 . 56 = mean
0. 128 = mean error
True value = 25.56 0.128, which one would call
25.56 =t 0.13.
It is rare in electrical measurements that there is any need to
apply the more exact methods for arriving at the most probably
true result, which the theory of probabilities teaches us to apply,
and the matter will not be further considered here.
The numerical difference between the result of a measurement
and the true value of a quantity measured is called the absolute
error. The ratio of the absolute error to the magnitude measured
is called the relative error. One must take, however, in place of
the true value, which is unknown, the mean value, for expressing
ART. 103] EXTENT OF ELECTRICAL MEASUREMENT 9
the relative error. Thus, in the example above, 0.128 is the
mean absolute error, and ^~r^ = 0.00500 is the mean relative
If the relative error is multiplied by 100, it is then called the
per cent error. Thus, in the above example, 0.00500 X 100
= =b 0.500 of 1 per cent. Namely, the value obtained is equal to
25.56 ohms with a probable error of plus or minus one-half of
one per cent.
The relative error or the per cent error is the error which is of
interest, because the absolute error must always be considered
in relation to the absolute value, if it is to have any physical
meaning. Thus to measure 1000 ohms with a plus or minus error
of 1 ohm is quite permissible but to measure 10 ohms with a plus
or minus error of 1 ohm would be but rough work. In the former
case the precision would be 0.1 of 1 per cent and in the latter case
but 10 per cent.
In all that follows, unless specifically stated to the contrary, we
shall, in speaking of errors, always refer to the relative error, or to
the per cent error.
The result of a measurement is often given so as to be dependent
upon several partial results, or separate measurements.
Let x represent the value sought, and y the phenomenon, as for
example the deflection of a voltmeter, upon which the value
depends, then x will be some function of y, or
x = F (y). (1)
If an absolute error A?/ is made in observing the phenomenon, the
result will be in error some amount A#, such that
x + Az = F (y + Ay), (2)
A* = F (y + Ay) - F (y). (3)
If the error made in y is small, we can treat the increment in y as a
differential, in which case we can write
dx = F' (y) dy, (4)
and the relative error will then be
dx _ F' (y) dy (
To illustrate Eq. (5); suppose it is required to determine, by meas-
uring a current, the rise in the temperature T of a conductor, caused
by the current 7 which it is made to carry. If we assume the
10 MEASURING ELECTRICAL RESISTANCE [ART. 103
rise of temperature of the conductor to vary as the square of the
current carried, we can write
T = KI 2 , where K is a constant.
We then have
x is equivalent to T, y is equivalent to 7,
F (y) is equivalent to KI 2 , dx is equivalent to dT,
F f (y) dy is equivalent to 2 KI dl.
Hence, by Eq. (5),
dT = 2KIdI ^2dl
T = KI 2 I
Eq. (6) shows that the relative error in the determination of the
rise in temperature will be twice as great as the relative error
made in measuring the current.
Eq. (5) may be extended to the case of several variables, and
so permit us to estimate in advance the relative precision of the
final result, when we know the magnitude of the errors that may
be made in each of the separate elements measured and upon
which the final result depends.
x = F (u, v, Wj etc.). (7)
Differentiate this function in respect to each of the variables u,
v, Wj etc. Then we derive
dx_F' u (u, v,w,...) du+F' v (u, v, w, . . . ) dv+F' w (u, v, w) dw-\-
x F (u,v,w, . . . )
Apply Eq. (8) to the case of a circuit in which we wish to measure
the power consumed. If P is this power, R the resistance of the
circuit, and 7 the current flowing, we have
P = RI 2 . (9)
If to determine P we have to measure R and 7, we shall have
P = F(R ) I)' )
hence, by Eq. (8),
dP _ R2IdI + I 2 dR _ 2dl dR
= ~~ 7 !"''
Now, -y- is the relative error in measuring the current, which call
EJ, and -pr is the relative error in measuring the resistance, which
ART. 103] EXTENT OF ELECTRICAL MEASUREMENT 11
call E R . Then
j?- = 2E I + E R . (11)
Eq. (11) shows that, if + 0.33 J is the per cent error made in
measuring the current and + 0.33J is the per cent error in measur-
ing the resistance, the per cent error in determining the power
W0~ = 2 X 0.33 J + 0.33 } = 1 per cent.
If the per cent error in measuring the resistance happened to be
- 0.33J then the per cent error in determining the power would be
only + 0.33 J. But, as it is unknown whether the accidental errors
are positive or negative, no greater precision can be assigned to
the result than would be deduced upon the assumption that all
the partial errors have a like sign.
Certain further conclusions can be drawn from Eqs. (5) and
1st. The quantity Q a being measured is the sum of two factors
x, y, or
Q 8 = x + y.
By Eq. (8) the relative error would be
dQ s _ dx + dy
QT = ^TF
It is to be noted here that the relative error in the result cannot
exceed the relative error committed in either factor; for assume
dQ s dx
y = 0, then -77- =
2d. The quantity Q d is the difference of two factors x, y, or
Q d = x - y.
In this case the relative error becomes
dQ d _dx- dy ( .
Q d = x-y '
Here it is seen that the relative error is as much greater as the
difference (x y) in the two quantities is smaller. This is why,
in measuring a quantity which involves the difference of two
factors, one will always obtain a result which is much less exact
than is obtained in measuring each of the elements.
12 MEASURING ELECTRICAL RESISTANCE [ART. 104
3d. The quantity Q p is the product of two factors, or
QP = xy.
In this case the relative error will be
dQr dx dy
Q P x y
Thus the relative error in the result is equal to the sum of the
relative errors committed in measuring each of the factors.
4th. The quantity Q q is the quotient of two factors, or
In this case
ydx _ xdy
dQ Q _ j f_ _ dx dy
07 s ~T~ 7 "7."
Hence, in this case, the final relative error is equal to the sum of
the relative errors made in each factor when these have opposite
signs. But as it cannot be told what the signs will be, one must
assume that the case may occur in which the signs are unlike.
5th. When Q' p is the power m of a factor x, as Q' p = x m , the
relative error will be
dW = mx^dx = m dx,
Q P x m x
Hence the relative error in the result will be m times as great as
the relative error made in the factor.
6th. When Q r is the mth root of the factor x, as
Qr = X m ,
the relative error will be
Q r ~ 1 ~ m x
Hence, the relative error in the result will be of that made in
104. Application of the Theory of Errors. Having in mind
the above principles, we can reach a just estimate of the precision
ART. 1041 EXTENT OF ELECTRICAL MEASUREMENT 13
which may be expected in the measurement of resistance by de-
In practice, the instruments most used are such as have a scale,
like that of a Weston voltmeter that is, a scale with a total of
150 divisions. While an attempt is often made to estimate the
readings to one-tenth of a division, it is thought that one-fifth
of a division is as close an estimation as can be relied upon.
Assuming then that the quantity being measured is given directly
by a full scale deflection, 'the per cent error can scarcely be less
than ^-^ = 100 = 0.13 + per cent, and this per cent error
150 X 5
steadily increases as the deflection read becomes smaller. With a
reading of ten divisions the probable error would be 2 per cent.
An additional source of error will be introduced if the scale is not
laid off so that the deflections indicated are proportional to the
current passing thru the instrument. But in measuring resistances
with Weston voltmeters and millivoltmeters this source of error
can usually be disregarded. Not so, however, when the deflection
instrument is a galvanometer with telescope and scale or lamp
and scale. In such case the scale will probably have 250 divisions,
each side of a central zero, and one might be led to expect higher
precision than may be obtained with a pointer instrument of only
150 divisions. The scales of galvanometers, however, have divi-
sions of uniform length, and as few galvanometers deflect exactly
proportional to the current thru them, the deflections indicated
are not apt to be accurately proportional. The advantage
therefore of a longer scale may be offset by lack of proportionality
in the deflections.
Further, in resistance measurements by deflection methods, the
results in most cases are not given directly in terms of a single
deflection but involve the difference of two deflections, and, as
appears under case 2 ( 103), the precision of the final result will
be much less than the precision with which the individual deflec-
tions are read.
Again, the two or three readings which must be taken are not
made simultaneously but in succession, and this procedure always
involves the assumption that all conditions influencing the pre-
cision remain constant while the various readings are being taken.
With a generator, subject to variations in speed, as the source
of current, this assumption is hardly tenable.
14 MEASURING ELECTRICAL RESISTANCE [ART. 104
The theoretical per cent error in any of the cases given in the
following chapter for measuring resistances is easily deduced by
making use of the principle expressed by Eq. (8) ( 103). We
proceed to apply the principle to the method given in par. 207,
where two unknown resistances x\ and x 2 in series are to be deter-
mined by three readings of a deflection instrument. Let one of
the resistances be given by the expression
If the resistance R is given and is not subject to variation, the
three quantities which may vary and which are not independent
are D, d\ and d 2 . If we apply to the above equation the operation
expressed by Eq. (8) ( 103), we obtain
dx = dD - ddi dd 2 (D - di)
X! D-di-dz d 2 (D-di- d 2 ) '
Here dxi, dD, ddi, dd 2 are absolute errors.
These errors may assume either a positive or. a negative sign,
hence the relative error - - will depend not only upon the magni-
tude of the errors dD, dd\, dd 2 but also upon their sign. If the
error ddi 'is equal to, and of the same sign as dD, the first term of
Eq. (2) disappears.
It is essential, however, in computing the value of the relative
error in the result, to assume that the errors made in the elements
(namely, in this case, in D, d\, d 2 ) have signs which are the least
favorable to precision. Assume, then, that the error in di is
Now, in reading a voltmeter, it may be assumed that the
conditions would be so arranged that the deflection D comes near
the top of the scale, and it may be assumed that the readings can
be made to - of the largest deflection, or to . Then the errors
dD, ddi, dd 2 may all be as large as and most unfavorable to
precision as far as sign is concerned. With these assumptions we
D + D D
&, p + p p (U a>> D 2d, + D-d,
ABT. 104] EXTENT OF ELECTRICAL MEASUREMENT 15
To illustrate Eq. (3) let us make use of the readings obtained in
the measurement given in par. 207.
r\ -I rjrv o
Assume that = -^~ , namely, that the readings can be made
to -5^ of the greatest reading. Take the values D = 100.8,
di = 72.2, dz = 14.4. Then we have, by Eq. (3), as the maximum
per cent error,
<fa, 100.8 2 X 14.4 + 100.8 - 72.2
^~ 500 X 14.4 > 100.8 - 72.2 - 14.4
= 5.6 per cent.
The per cent error, actually made, was only +0.88 per cent.
This may be due in part to a more accurate reading of the instru-
ment than was assumed, and also to the fact that the errors can-
celled each other; if the latter only were the case we should have
100 ^ = - n D ~ dl , 100 = 2.8 per cent.
Xi pdz D di d 2
This last result shows that in the particular case of this measure-
ment the readings must have been made within about 0.05 of a
Returning to Eq. (1) ( 104) we see from par. 207 that, if one of
the resistances x 2 is infinity, the reading di would be zero and we
This is similar to the case given in Eq. (8), par. 201.
If now in Eq. (2) we put di = 0, we shall also have dd\ 0,
dxi _ dD ddzD _ dDdz - dd 2 D
~zT ~~ D - d 2 ~ dz (D - dz) ~ dz (D - d 2 ) '
= ddz = , we
minus sign and dD with a plus sign,
If, as before, dD = ddz = , we have, by taking ddz with a
m * = 100. (4)
Xi pdz D dz
Eq. (4) expresses the theoretical maximum per cent error which
would be obtained in the method described in par. 201 if the
readings can be made to - of the larger deflection. If Eq. (4) be
16 MEASURING ELECTRICAL RESISTANCE [ART. 105
applied to the measurement of 100 ohms, as given in the table in
par. 205, and it be assumed that = > we obtain as the maxi-
mum per cent error 1.4 per cent, while the actual error was 0.76
Enough has now been given to show the low accuracy that may
be expected from deflection methods of measuring resistances.
These methods, nevertheless, are of value in many situations.
If it be remembered that all copper and aluminum conductors,
such as magnet coils in arc lamps, etc., change in resistance about
4 per cent for every change of 10 C. in temperature, it will be
recognized that the rough methods of measuring resistances with
deflection instruments are often quite as precise as the conditions
demand, and because of their simplicity often more satisfactory
than methods which are more refined.
The apparatus required is, furthermore, very generally avail-
able; the measurements may be made quickly and in places where
more delicate apparatus could not well be used, and altho the
precision attainable is low, it is sufficient to meet such require-
ments as arise in connection with the measurement of insulation
resistances and the resistances of copper conductors, such as
105. Comments on Ohmic Resistance. An ohmic resist-
ance, considered as a quantity to be measured, may be viewed in
two ways. If we call E the drop of potential between two points
in an electric circuit, and / the current flowing, we may write