Percent of Error
The accuracy of a measurement is determined by the RELATIVE ERROR. The
relative error is the ratio between the
probable error and the quantity being
measured. This ratio is simply the fraction
formed by using the probable error as the numerator and the measurement itself
as the denominator. For example, suppose that
a metal plate is found to be 5.4 inches long,
correct to the nearest tenth of an inch. The
maximum probable error is five hundredths of
an inch (onehalf of one tenth of an inch)
and the relative error is found as follows:
Thus the relative error is 5 parts out of 540. Relative
error is usually expressed as PERCENT OF ERROR, When the denominator of the
fraction expressing the error ratio is divided into the numerator, a decimal is
obtained. This decimal, converted to percent,
gives the percent of error. For example, the
error in the foregoing problem could be
stated as 0.93 percent, since the ratio 5/540
reduces to 0.0093 (rounded off) in decimal
form.
Significant Digits
The accuracy of a measurement is often described in terms of the number of
significant digits used in expressing it. If
the digits of a number resulting from
measurement are examined one by one, beginning with the lefthand digit,
the first digit that is not 0 is the first significant
digit. For example, 2345 has four significant
digits and 0.023 has only two significant digits.
The digits 2 and 3 in a measurement such as 0.023
inch signify how many thousandths of an inch
comprise the measurement. The Os are of no
significance in specifying the number of thousandths
in the measurement; their presence is
required only as "place holders" in placing the
decimal point.
A rule that is often used states that the significant digits in a number
begin with the first nonzero digit (counting
from left to right) and end with the last
digit. This implies that 0 can be a
significant digit if it is not the first digit in
the number. For example, 0.205 inch is a measurement
having three significant digits. The 0
between the 2 and the 5 is significant because
it is a part of the number specifying how
many hundredths are in the measurement.
The rule stated in the foregoing paragraph fails
to classify final 0s on the right. For example, in a number such as 4,700,
the number of significant digits might be
two, three, or four. lf the 0s merely
locate the decimal point (that is, if they
show the number to be approximately fortyseven hundred rather than forty
seven), then the number of significant digits is
two. However, if the number 4,700 represents a
number such as 4,730 rounded off to the nearest hundred, there are three
significant digits. The last 0 merely locates
the decimal point. If the number 4,700
represents a number such as 4,700.4 rounded
off, then the number of significant digits is four.
Unless we know how a particular number was
measured, it is sometimes impossible to determine
whether righthand Os are the result of
rounding off. However, in a practical situation it is normally possible to
obtain information concerning the instruments used and the degree
of precision of the original data before any
rounding was done.
In a number such as 49.30 inches, it is reasonable to assume that the 0 in
the hundredths place
would not have been recorded at all if it were
not significant. In other words, the instrument used for the measurement can be
read to the nearest hundredth of an inch. The
0 on the right is thus significant. This
conclusion can be reached another way by
observing that the 0 in 49.30 is not needed
as a place holder in placing the decimal
point. Therefore its presence must have some other significance. The
facts concerning significant digits may be
summarized as follows:
1. Digits other than 0 are always significant.
2. Zero is significant when it falls between significant
digits.
3. Any final 0 to the right of the decimal point
is significant.
4. When a 0 is present only as a place holder
for locating the decimal point, it is not significant.
5. The following categories comprise the significant
digits of any measurement number:
a. The first nonzero lefthand digit is significant.
b. The digit which indicates the precision of
the number is significant. This is the digit farthest
to the right, except when the righthand digit
is 0. If it is 0, it may be only a place holder
when the number is an integer.
c. All digits between significant digits are
significant.
Practice problems. Determine the percent of
error and the number of significant digits in each
of the following measurements:
1. 5.4 feet
2. 0.00042 inch
3. 4.17 set
4. 147.50 miles
Answers:
1. Percent of error: 0.93%
Significant digits: 2
2. Percent of error: 1.19%
Significant digits: 2
3. Percent of error: 0.12%
Significant digits: 3
4. Percent of error: 0.0034%
Significant digits: 5
