5
Reliability of Measurements
In pure mathematics or in counting, every number has an exact meaning. The
figure 2, for example, means precisely two units (not approximately two units).
In using numbers to express the results of measurements, however, we use
numbers with inexact meanings, because no measurement is perfectly accurate.
If we say that an object has a length of 2 m, we mean that it is approximately 2
m long. We would not be surprised to find that the actual length differs from
2.000000 m.
In expressing the results of measurements, we should use numbers in a way
that indicates the reliability of the result. Suppose two persons measure the
diameter of a dime with a centimeter scale. One person reports a result of 1.79
cm; the other finds the diameter to be 1.80 cm. Both would agree that the
desired reading is near the 1.8 cm mark and just slightly toward the 1.7 cm side
of that mark. However, one person has estimated the value as 1.79 cm, whereas
the other feels that the edge of the dime is close enough to the mark to report a
value of 1.80 cm. How should the uncertainty be expressed in reporting this
result? It would be correct to report the result either as 1.79 cm or as 1.80 cm,
but it would not be correct to give the average value of 1.795 cm. This last figure
implies that the true value is known to lie near the middle of the range between
1.79 cm and 1.80 cm, which is not the case.
Suppose the measurements are repeated with a caliper whose vernier scale
permits careful measurement to the nearest 0.01 cm and estimation to the
Reliability of Measurements
In pure mathematics or in counting, every number has an exact meaning. The
figure 2, for example, means precisely two units (not approximately two units).
In using numbers to express the results of measurements, however, we use
numbers with inexact meanings, because no measurement is perfectly accurate.
If we say that an object has a length of 2 m, we mean that it is approximately 2
m long. We would not be surprised to find that the actual length differs from
2.000000 m.
In expressing the results of measurements, we should use numbers in a way
that indicates the reliability of the result. Suppose two persons measure the
diameter of a dime with a centimeter scale. One person reports a result of 1.79
cm; the other finds the diameter to be 1.80 cm. Both would agree that the
desired reading is near the 1.8 cm mark and just slightly toward the 1.7 cm side
of that mark. However, one person has estimated the value as 1.79 cm, whereas
the other feels that the edge of the dime is close enough to the mark to report a
value of 1.80 cm. How should the uncertainty be expressed in reporting this
result? It would be correct to report the result either as 1.79 cm or as 1.80 cm,
but it would not be correct to give the average value of 1.795 cm. This last figure
implies that the true value is known to lie near the middle of the range between
1.79 cm and 1.80 cm, which is not the case.
Suppose the measurements are repeated with a caliper whose vernier scale
permits careful measurement to the nearest 0.01 cm and estimation to the