PRINCIPLES OF MEASUREMENT
Computation with decimals frequently involves the addition or subtraction of
numbers which do not have the same number of
decimal places. For example, we may be asked
to add such numbers as 4.1 and 32.31582. How
should they be added? Should zeros be annexed
to 4.1 until it is of the same order as the
other decimal (to the same number of places)? Or, should .31582
be rounded off to tenths? Would the sum be
accurate to tenths or hundredthousandths? The
answers to these questions depend on how the
numbers originally arise.
Some decimals are finite or are considered as
such because of their use. For instance, the decimal
that represents 1/2, that is 0.5, is as accurate
at 0.5 as it is at 0.5000. Likewise, the
decimal that represents 1/8 has the value 0.125
and could be written just as accurately with
additional end zeros. Such numbers are said
to be finite. Counting numbers are finite. Dollars
and cents are examples of finite values. Thus,
$10.25 and $5.00 are finite values.
To add the decimals that represent 1/8 and 1/2, it
is not necessary to round off 0.125 to tenths. Thus,
0.5 + 0.125 is added as follows:
Notice that the end zeros were added to 0.5 to carry
it out the same number of places as 0.125. It
is not necessary to write such placeholding zeros
if the figures are kept in the correct columns and decimal points are aligned.
Decimals that have a definite fixed value may
be added or subtracted although they are of
different order. On the other hand, if the
numbers result from measurement of some kind,
then the question of how
much to round off must be decided in terms of
the precision and accuracy of the measurements.
ESTIMATION
Suppose that two numbers to be added resulted from measurement. Let us say
that one number was measured with a ruler marked off in
tenths of an inch and was found, to the nearest tenth of an inch, to be 2.3
inches. The other number measured with a
precision rule was found, to the nearest
thousandth of an inch, to be 1.426 inches.
Each of these measurements requires estimation between marks on the rule, and
estimation between marks on any measuring instrument is subject to human error.
Experience has shown that the best the
average person can do with consistency is to
decide whether a measurement is more or less
than halfway between marks. The correct way to state this fact
mathematically is to say that a measurement made with an instrument marked off
in tenths of an inch involves a maximum
probable error of 0.05 inch (five hundredths
is onehalf of one tenth). By the same
reasoning, the probable error in a measurement made with an instrument marked in
thousandths of an inch is 0.0005 inch.
PRECISION
In general, the probable error in any measurement is onehalf the size of the
smallest division on the measuring ‘instrument.
Thus the precision of a measurement depends
upon how precisely the instrument is marked.
It is important to realize that precision
refers to the size of the smallest division
on the scale; it has nothing to do with the
correctness of the markings. In other words,
to say that one instrument is more precise than another does
not imply that the less precise instrument is poorly manufactured. In fact, it
would be possible to make an instrument with very
high apparent precision, and yet mark it carelessly
so that measurements taken with it would be
inaccurate.
From the mathematical standpoint, the precision of a number resulting from
measurement depends upon the number of
decimal places; that is, a larger number of
decimal places means a smaller probable
error. In 2.3 inches the probable error is
0.05 inch, since 2.3 actually lies somewhere between 2.25 and 2.35. In 1.426
inches there is a much smaller probable error
of 0.0005 inch. If we add 2.300 + 1.426 and
get an answer in thousandths, the answer, 3.726
inches, would appear to be precise to thousandths;
but this is not true since there was a
probable error of .05 in one of the addends. Also 2.300 appears to be precise to
thousandths but in this example it is precise
only to tenths. It is evident that the precision
of a sum is no greater than the precision of the
least precise addend. It can also be shown that the
precision of a difference is no greater than the
less precise number compared.
To add or subtract numbers of different orders,
all numbers should first be rounded off to the
order of the least precise number. In the foregoing
example, 1.426 should be rounded to tenthsthat
is, 1.4.
This rule also applies to repeating decimals. Since
it is possible to round off a repeating decimal
at any desired point, the degree of precision desired should be determined and
all repeating decimals to be added should be rounded to
this level. Thus, to add the decimals generated by 1/3, 2/3, and 5/12 correct to
thousandths, first round off each decimal to
thousandths, and then add, as follows:
When a common fraction is used in recording the
results of measurement, the denominator of the
fraction indicates the degree of precision. For
example, a ruler marked in sixtyfourths of
an inch has smaller divisions than one marked
in sixteenths of an inch. Therefore a measurement
of 3 4/64 inches is more precise than a
measure of 3 1/16 inches, even though the 16 two fractions are numerically
equal. Remember that a measurement of 3 4/64
inches contains a probable error of only
onehalf of one sixtyfourth of an inch. On the other hand, if the smallest
division on the ruler is onesixteenth of an
inch, then a measurement of 3 1/16 inches contains a probable error of one
thirtysecond of an inch.
ACCURACY
Even though a number may be very precise, which
indicates that it was measured with an instrument
having closely spaced divisions, it may not
be very accurate. The accuracy of a measurement
depends upon the relative size of the
probable error when compared with the quantity
being measured. For example, a distance of 25 yards on a pistol range may be
measured carefully enough to be correct to the
nearest inch. Since there are 900 inches in 25
yards, this measurement is between 899.5 inches
and 900.5 inches. When compared with the
total of 900 inches, the 0.5inch probable error
is not very great.
On the other hand, a length of pipe may be measured
rather precisely and found to be 3.2 inches
long. The probable error here is 0.05 inch,
and this measurement is thus more precise than that of the pistol range
mentioned before. To compare the accuracy of the two measurements, we note that
0.05 inch out of a total of 3.2 inches is the
same as 0.5 inch out of 32 inches. Comparing
this with the figure obtained in the other
example (0.5 inch out of 900), we conclude
that the more precise measurement is actually
the less accurate of the two measurements considered.
It is important to realize that the location of the
decimal point has no bearing on the accuracy of the number. For example, 1.25
dollars represents exactly the same amount of
money as 125 cents. These are equally
accurate ways of representing the same
quantity, despite the fact that the decimal
point is placed differently. Practice
problems. In each of the following problems,
determine which number of each pair is more
accurate and which is more precise:
1. 3.72 inches or 2,417 feet
2. 2.5 inches or 17.5 inches
3. 5 3/4: inches or 12 7/8: inches
4. 34.2 seconds or 13 seconds
Answers:
1. 3.72 inches is more precise.
2,417 feet is more
accurate.
2. The numbers are equally precise.
17.5 inches is more
accurate.
3. l2 7/8; inches is more precise and more accurate.
4. 34.2 seconds is more precise and more accurate.
