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Accuracy and Precision
To many people, accuracy and precision mean the same thing: to
someone involved in measurement, the two terms should have very
different meanings.
One way in which this distinction is apparent is the
difference between a poll (a measurement) and a
vote (a count). The result obtained from counting
will be an exact result (barring blunder), while the
result obtained from measuring will only approach
the truth.
We are taught how to deal with exact
numbers, but are sometimes not aware of
the nature of results we obtain from
approximated or measured values.
The precision of an instrument
reflects the number of significant
digits in a reading;
Precision indicates
how close together or
how repeatable the
results are. A precise
measuring instrument
will give very nearly
the same result each
time it is used.
The accuracy of an instrument
reflects how close the reading is to
the 'true' value measured.
Accuracy indicates how close a
measurement is to the accepted
value. For example, we'd expect
a balance to read 100 grams if we
placed a standard 100 g weight on
the balance. If it does not, then
the balance is inaccurate
There is no such thing as a Perfect Measurement
As a consequence of the above fact, all measurements should include
an estimate of the accuracy conveyed by a given reading. The
accuracy estimate is reflected in the error term.
Result =
Every measurement should include the value, an error term and the units.
Precise and Accurate
Accurate, Not Precise
Neither Precise Nor Accurate
Precise, Not Accurate
Significant Figures
Significant figures are the
number of reliably known
digits used to locate a
decimal point reported in a
measurement. Proper use of
significant figures ensures
that you correctly represent
the uncertainty of your
measurements. For example,
scientists immediately realize
that a reported measurement
of 1.2345 m is much more
accurate than a reported
length of 1.2 m.
Guidelines for Significant Figures:
When a measurement is
properly stated in scientific
notation all of the digits
will be significant. For
example: 0.0035 has 2
significant figures which
can be easily seen when
written in scientific
notation as 3.5 x 10-3.
Fortunately, there are a
few general guidelines that
are used to determine
significant figures:
1. Whole Numbers
2. Integers and Defined Quantities
3. Multiplication and Division
4. Addition and Subtraction
Whole Numbers: The following numbers are all
represented by three significant digits. Note that
zeros are often place holders and are not
significant.
0.00123
0.123
1.23
12.3
123
12300 (The zeros here often cause confusion. As
written here, the zeros are not significant. If they
were, in fact, significant, then the use of scientific
notation would remove all ambiguity and the
number would be written 1.2300 x 104.)
The following numbers are all represented by one significant digit.
0.005
0.5
5
500
5,000,000
The following numbers are all represented by four significant figures.
0.004001
0.004000
40.01
40.00
4321
432.1
43,210,000
PERCENT ERROR
The significant figures rule in only approximate, and in some cases
may underestimate the precision of the answer.
Turn to page 7 in your physics book
Use the significant figures rule, but consider the % uncertainty
too, and add an extra digit if it gives a more realistic estimate
of uncertainty
Integers and Defined Quantities: Integers are assumed
to have an infinite number of significant figures. For
example, the 2 in C = 2pr, is exactly two and we can
assume that the number has an infinite number of
significant figures. However, the conversion factor 2.54 cm
which is used to convert inches to centimeters has three
significant figures.
Multiplication and Division: When multiplying or dividing
numbers, the result should have only as many significant figures
as the quantity with the smallest number of significant figures
being used in the calculation. For example, with your calculator
multiply 4.7 and 5.93. The calculator returns 27.871 as the
answer. A common mistake students make is to record what
comes out of the calculator as the correct answer. However, since
4.7 has only 2 significant figures, the result must be truncated to
2 significant figures as well. Taking all this into account and
remembering to round appropriately, the result should be
reported as 28.
Addition and Subtraction:
RULE: When adding or subtracting
your answer can only show as many
decimal places as the measurement
having the fewest number of
decimal places.
3.76 g + 14.83 g + 2.1 g = 20.69 g
= 20.7g
SCIENTIFIC NOTATION
Precision as to the significance of a value may be avoided
by expressing the number in standard exponential notation.
Addition and Subtraction
When adding or subtracting numbers in scientific notation,
their powers of 10 must be equal. If the powers are not equal,
then you must first convert the numbers so that they all have
the same power of 10.
(6.7 x 109) + (4.2 x 109) = (6.7 + 4.2) x 109 = 10.9 x 109 = 1.09 x 1010
(6.7 x 109) + (4.2 x 107) = (670 x 107) + (4.2 x 107)
So to make the exponent lower move the decimal of the
number to the right or to make the exponent larger move the
decimal of the number to the left. Then you can add or
subtract.
Multiplication and Division
It is very easy to multiply or divide just by rearranging
so that the powers of 10 are multiplied together
(6 x 102) x (4 x 10-5) = (6 x 4) x (102 x 10-5) = 24 x 102-5 = 24 x 10-3 = 2.4 x 10-2.
(8 x 102) = 8/4 x 102-5 = 2 x 10-3
(4 x 105)
ADDITIONAL PRINCIPLES
PRINCIPLE NUMBER ONE
If you are using exact constants, such as thirty-two ounces per quart
or one thousand milliliters per liter, they do not affect the number of
significant figures in you answer. For example, you might need to
calculate how many feet equal 26.1 yards. The conversion factor you
would need to use, 3 ft/yard, is an exact constant and does not affect the
number of significant figures in your answer. Therefore, 26.1 yards
multiplied by 3 feet per yard equals 78.3 feet which has 3 significant
figures.
PRINCIPLE NUMBER TWO
If you are using constants which are not exact (such as pi =
3.14 or 3.142 or 3.14159) select the constant that has at least
one or more significant figures than the smallest number of
significant figures in your original data. This way the number
of significant figures in the constant will not affect the number of
significant figures in your answer. For example, if you multiply
4.136 ft which has four significant figures times pi, you should use
3.1416 which has 5 significant figures for pi and your answer will
have 4 significant figures.
PRINCIPLE NUMBER THREE
When you are doing several calculations, carry out all of
the calculations to at least one more significant figure than
you need. When you get the final result, round off. For
example, you would like to know how many meters per second
equals 55 miles per hour. The conversion factors you would use
are: 1 mile equals 1.61 x 103 meter and 1 hour equals 3600
seconds. Your answer should have two significant figures. Your
result would be 88.55 divided by 3600 which equals 24.59 m/sec.
This rounds off to 25 m/sec. By carrying this calculation out to
at least one extra significant figure, we were able to round
off and give the correct answer of 25 m/sec rather than 24
m/sec.
How many significant figures are in each of the following numbers?
a) 1.234
a) 1.234: 4
b) 1.2340
b) 1.2340: 5
c) 1.234 x 10-3
c) 1.234 x 10-3: 4
d) 1.2340 x 10-3
d) 1.2340 x 10-3: 5
e) 1234
e) 1234: 4
f) 12340
f) 12340: 4 or 5
g) 0.012340
g) 0.012340: 5
Complete the following operations and express the answer
with the correct number of significant figures:
a) 1.421+ 0.4372 =
a) 1.421+ 0.4372 = 1.858
b) 0.0241 + 0.11 =
b) 0.0241 + 0.11 = 0.13
c) 0.14 + 1.2243 =
c) 0.14 + 1.2243 = 1.36
d) 760.0 + 0.011 =
d) 760.0 + 0.011 = 760.0
e) 1.0123 - 0.002 =
e) 1.0123 - 0.002 = 1.010
f) 123.69 - 20.1 =
f) 123.69 - 20.1 = 103.6
g) 463.231 - 14.0 =
g) 463.231 - 14.0 = 449.2
h) 47.2 - 0.01 =
h) 47.2 - 0.01 = 47.2
Perform the indicated operations. Express your answers with the
correct number of significant figures:
a) 42.3 x 2.61 = 110
a) 42.3 x 2.61 =
b) 0.61 x 42.1 = 26
b) 0.61 x 42.1 =
c) 46.1 / 1.21 = 38.1
c) 46.1 / 1.21 =
d) 23.2 / 4.1 = 5.7
d) 23.2 / 4.1 =
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