58
Reliability of Measurements
45.725 ± (3.182X0.0160) = 45.725 ± 0.0509 g
There is a 95% probability that any weight value picked at random has a value
that lies within 0.0509 g of the average. The precision of the mean is given by
45.725 ± (3.182)
( °'°
160) = 45.725 ± 0.0255 g
V4
That is, there is a 95% probability that the true value of the weight of 50 ml of this
liquid lies within the interval 45.725 ± 0.0255 g.
Relative Error
It frequently is convenient to express the degree of error on a relative basis,
rather than on an absolute basis as above. A relative basis has the advantage of
making a statement independent of the size of the measurements that were
made. For example, the statement that a solid contains 10% silver is a relative
statement; it says that one-tenth of the solid is silver, and it is understood that a
large sample of the solid would contain more grams of silver than a small one.
Percentage is "parts per hundred," and it is found for this example by multiplying the fraction of the sample that is silver (in this case, 0.1) by 100. It would be
as correct to multiply the fraction by 1000 and call it " 100 parts per thousand,"
or to multiply the fraction by 1,000,000 and call it "100,000 parts per million."
The choice of parts per hundred (percentage) or parts per thousand or million is
determined by convenience. If the fraction were very small, say 0.00005, it
would be more convenient to call it 50 parts per million than to call it 0.005 parts
per hundred or 0.005 per cent.
PROBLEM:
Express the 95% confidence level of the standard deviation of the mean obtained
in the previous problem as percentage, as parts per thousand, and as parts per
million.
SOLUTION:
The 95% confidence level of the standard deviation of the mean was found to be
±0.0255 g, where the weight itself was (on the average) 45.725 g. Infraction of
the total weight that might be error is
= 0.000558
45.725 g
The relative error thus can be written as
(0.000558X100) = 0.0558 parts per hundred = 0.0558 percent
(0.000558X10') = 0.558 parts per thousand (ppt.)
(0.000558X10
11
) = 558 parts per million (ppm.)
Reliability of Measurements
45.725 ± (3.182X0.0160) = 45.725 ± 0.0509 g
There is a 95% probability that any weight value picked at random has a value
that lies within 0.0509 g of the average. The precision of the mean is given by
45.725 ± (3.182)
( °'°
160) = 45.725 ± 0.0255 g
V4
That is, there is a 95% probability that the true value of the weight of 50 ml of this
liquid lies within the interval 45.725 ± 0.0255 g.
Relative Error
It frequently is convenient to express the degree of error on a relative basis,
rather than on an absolute basis as above. A relative basis has the advantage of
making a statement independent of the size of the measurements that were
made. For example, the statement that a solid contains 10% silver is a relative
statement; it says that one-tenth of the solid is silver, and it is understood that a
large sample of the solid would contain more grams of silver than a small one.
Percentage is "parts per hundred," and it is found for this example by multiplying the fraction of the sample that is silver (in this case, 0.1) by 100. It would be
as correct to multiply the fraction by 1000 and call it " 100 parts per thousand,"
or to multiply the fraction by 1,000,000 and call it "100,000 parts per million."
The choice of parts per hundred (percentage) or parts per thousand or million is
determined by convenience. If the fraction were very small, say 0.00005, it
would be more convenient to call it 50 parts per million than to call it 0.005 parts
per hundred or 0.005 per cent.
PROBLEM:
Express the 95% confidence level of the standard deviation of the mean obtained
in the previous problem as percentage, as parts per thousand, and as parts per
million.
SOLUTION:
The 95% confidence level of the standard deviation of the mean was found to be
±0.0255 g, where the weight itself was (on the average) 45.725 g. Infraction of
the total weight that might be error is
= 0.000558
45.725 g
The relative error thus can be written as
(0.000558X100) = 0.0558 parts per hundred = 0.0558 percent
(0.000558X10') = 0.558 parts per thousand (ppt.)
(0.000558X10
11
) = 558 parts per million (ppm.)