48
Reliability of Measurements
a hard and fast rule. The best situation would be to have a range of error for
each number; then the calculation could be performed with all the largest or all
the smallest values to see which digits in the result are reliable.
A case can be made for rounding off all numbers involved in a calculation
before the calculation is actually made. However, in this age of hand calculators this procedure adds little except extra effort. The simplest approach
probably is to enter all the numbers of a calculation through the keyboard without regard to significant figures. Then, by inspection at the end of the calculation, determine how many decimal places should be used (for addition and
subtraction) or how many significant figures should be used (for multiplication
and division) in the final result. The calculator cannot make the decision about the
proper number of decimal places or significant figures to use; only the operator can
do this. One of the most common errors in the use of calculators is to write
down all of the digits that appear in the display as a result of a calculation,
regardless of their significance. You must learn to think about your answers.
A pure number such as 3 or 4 has an unlimited number of significant figures
(4.000000
), as does a defined quantity such as TT (3.14159...) or e
(2.7182818 ....). Do not fall into the trap of excessively rounding off results that
come from equations using pure or derived numbers. For example, if you want
to find the volume of a sphere whose radius has been measured as 15.13 cm,
you shouldn't round off the answer to 1 x 10
4 cm
3 just because you are going to
use the formula V = %m3 , in which 4, 3, and rr each appears to have only one
significant figure. It is the measured values that determine the number of significant figures. In this case, the volume should be expressed to four significant
figures as 1.451 x 10
4 cm
3 . Your calculator probably has a TT key that will give
with one stroke the value of TT to 8 or 10 decimals.
DISTRIBUTION OF ERRORS
We have talked about significant figures and the general unreliability of the
"last figure" of a measurement. Now we shall talk about just how unreliable
these last figures are. Your experience has shown that really gross errors rarely
occur in a series of measurements. Suppose you were able to make an infinite
number of measurements on the same quantity (call it x). You would not be
surprised if, on plotting each observed value of x against the frequency with
which it occurred, you obtained a symmetrical curve similar to that shown in
Figure 5-1. One of the advantages of making an infinite number of measurements is that the average (x) of the values will be equal to the "true value" (/A),
represented by the dotted vertical line drawn from the peak of the curve. As
expected, the more a value of x deviates from p, the less frequently it occurs.
This curve is symmetrical because there is equal probability for + and - errors;
it is called a normal distribution curve. If you made your measurements in a more
careless manner or with a less sensitive measuring device, you would obtain a
distribution curve more like that in Figure 5-2, shorter and broader but with the
Reliability of Measurements
a hard and fast rule. The best situation would be to have a range of error for
each number; then the calculation could be performed with all the largest or all
the smallest values to see which digits in the result are reliable.
A case can be made for rounding off all numbers involved in a calculation
before the calculation is actually made. However, in this age of hand calculators this procedure adds little except extra effort. The simplest approach
probably is to enter all the numbers of a calculation through the keyboard without regard to significant figures. Then, by inspection at the end of the calculation, determine how many decimal places should be used (for addition and
subtraction) or how many significant figures should be used (for multiplication
and division) in the final result. The calculator cannot make the decision about the
proper number of decimal places or significant figures to use; only the operator can
do this. One of the most common errors in the use of calculators is to write
down all of the digits that appear in the display as a result of a calculation,
regardless of their significance. You must learn to think about your answers.
A pure number such as 3 or 4 has an unlimited number of significant figures
(4.000000
), as does a defined quantity such as TT (3.14159...) or e
(2.7182818 ....). Do not fall into the trap of excessively rounding off results that
come from equations using pure or derived numbers. For example, if you want
to find the volume of a sphere whose radius has been measured as 15.13 cm,
you shouldn't round off the answer to 1 x 10
4 cm
3 just because you are going to
use the formula V = %m3 , in which 4, 3, and rr each appears to have only one
significant figure. It is the measured values that determine the number of significant figures. In this case, the volume should be expressed to four significant
figures as 1.451 x 10
4 cm
3 . Your calculator probably has a TT key that will give
with one stroke the value of TT to 8 or 10 decimals.
DISTRIBUTION OF ERRORS
We have talked about significant figures and the general unreliability of the
"last figure" of a measurement. Now we shall talk about just how unreliable
these last figures are. Your experience has shown that really gross errors rarely
occur in a series of measurements. Suppose you were able to make an infinite
number of measurements on the same quantity (call it x). You would not be
surprised if, on plotting each observed value of x against the frequency with
which it occurred, you obtained a symmetrical curve similar to that shown in
Figure 5-1. One of the advantages of making an infinite number of measurements is that the average (x) of the values will be equal to the "true value" (/A),
represented by the dotted vertical line drawn from the peak of the curve. As
expected, the more a value of x deviates from p, the less frequently it occurs.
This curve is symmetrical because there is equal probability for + and - errors;
it is called a normal distribution curve. If you made your measurements in a more
careless manner or with a less sensitive measuring device, you would obtain a
distribution curve more like that in Figure 5-2, shorter and broader but with the