52
Reliability of Measurements
In order to make the value of s useful for other calculations, it is customary to
write it to one more decimal place than the least significant figure of the measurements; we shall do the same for x when it is used in the calculation ofs, but at
no other time. If we did not adopt such a convention, we would frequently find,
for a good set of data, that a standard deviation rounded off to the decimal place
corresponding to the least significant figure of the measurements would have a
value of 0. This would be a useless and misleading result because variations in
the measurements actually exist. A good compromise is to write the average to
the correct number of significant figures as discussed on p 46, but to write the
error statement to one more decimal place than corresponds to the least significant figure in the average. An error statement is the average deviation, the
standard deviation, or one of the various confidence intervals to be discussed on
pp 56-57. Thus, in the example just shown, you would write
x = 10.320 ± 0.0048 m (for average deviation)
x = 10.320 ± 0.0061 m (for standard deviation)
For a very poor set of data, with a very large standard deviation, it is foolish to
show the error statements to one more decimal place than the least significant
figure. If the room measurements just cited actually had a standard deviation of
±0.0756 m, it means that you were silly to think of measuring to the nearest
0.001 m, and in any case you should round the standard deviation to ±0.076 m.
The decision as to when you do, or do not, write the error statement to one
more decimal place than the least significant figure is arbitrary. (Some professors say that you should use the extra decimal place if the standard deviation is
less than 0.4% of the average of the measurements.)
Many calculators permit you to determine x and j directly without the need of
setting out the calculations as described in the last problem. After entering each
measurement through the keyboard, you press the S+ key. After entering all of
the numbers in this fashion, you press the x key (or keys) to get the average and
then the ^ key (or keys) to get the standard deviation. The average—which may
be displayed to, say, eight decimal places—must of course be rounded off to the
proper number of significant figures.
A small complication may arise from the use of calculators that determine x
and s directly. With these calculators, the standard deviation is calculated from
the average deviation that includes all of the digits in the display. The resulting
value probably will differ slightly from the one you would obtain if you previously rounded the average to one more decimal place than the least significant
figure. This difference usually is small and can be ignored. Thus you need not
worry about somewhat different answers obtained with different types of
calculators.
Probability Distribution for Large Numbers of Measurements
In order to appreciate the usefulness of the standard deviation as a measure of
reliability, we must take a closer look at the curves in Figures 5-1 and 5-2. The
Reliability of Measurements
In order to make the value of s useful for other calculations, it is customary to
write it to one more decimal place than the least significant figure of the measurements; we shall do the same for x when it is used in the calculation ofs, but at
no other time. If we did not adopt such a convention, we would frequently find,
for a good set of data, that a standard deviation rounded off to the decimal place
corresponding to the least significant figure of the measurements would have a
value of 0. This would be a useless and misleading result because variations in
the measurements actually exist. A good compromise is to write the average to
the correct number of significant figures as discussed on p 46, but to write the
error statement to one more decimal place than corresponds to the least significant figure in the average. An error statement is the average deviation, the
standard deviation, or one of the various confidence intervals to be discussed on
pp 56-57. Thus, in the example just shown, you would write
x = 10.320 ± 0.0048 m (for average deviation)
x = 10.320 ± 0.0061 m (for standard deviation)
For a very poor set of data, with a very large standard deviation, it is foolish to
show the error statements to one more decimal place than the least significant
figure. If the room measurements just cited actually had a standard deviation of
±0.0756 m, it means that you were silly to think of measuring to the nearest
0.001 m, and in any case you should round the standard deviation to ±0.076 m.
The decision as to when you do, or do not, write the error statement to one
more decimal place than the least significant figure is arbitrary. (Some professors say that you should use the extra decimal place if the standard deviation is
less than 0.4% of the average of the measurements.)
Many calculators permit you to determine x and j directly without the need of
setting out the calculations as described in the last problem. After entering each
measurement through the keyboard, you press the S+ key. After entering all of
the numbers in this fashion, you press the x key (or keys) to get the average and
then the ^ key (or keys) to get the standard deviation. The average—which may
be displayed to, say, eight decimal places—must of course be rounded off to the
proper number of significant figures.
A small complication may arise from the use of calculators that determine x
and s directly. With these calculators, the standard deviation is calculated from
the average deviation that includes all of the digits in the display. The resulting
value probably will differ slightly from the one you would obtain if you previously rounded the average to one more decimal place than the least significant
figure. This difference usually is small and can be ignored. Thus you need not
worry about somewhat different answers obtained with different types of
calculators.
Probability Distribution for Large Numbers of Measurements
In order to appreciate the usefulness of the standard deviation as a measure of
reliability, we must take a closer look at the curves in Figures 5-1 and 5-2. The