54
Reliability of Measurements
3. Relationship between cr and area under the curve. If we express the
values of x in terms of a, we find that the normal distribution curve
always has the same shape, regardless of the size of cr. That is, the area
under the curve for a particular multiple of cr on either side of /u, is the
same for any normal distribution curve.
Values of x between
Area under this portion of the curve
(j. -
/LI - 2cr and /A + 2cr
0.954 (shaded area in Figure 5-2)
/u- 3crand|ii+ 3cr
0.997
Thus, whenever you have a very large number of measurements, the
probability is that 68.3% of them (about two-thirds) will have values
within the range /A ± cr (that is, within one standard deviation of the
average value). Similarly, 95.4% of the measurements (about 19 out of
20) probably will have values within the range /u. ± 2cr, and only 0.3%
of them (3 in 1000) are likely to have values outside the range /u, ± 3cr.
This also means that the probability is only 0.3% (3 times out of 1000
measurements) that any single measurement will yield a value differing
by more than 3 cr from the value p.
4. Confidence interval and confidence level. We have seen that 3 out of 1000
measurements probably will have values outside the range /u, ± 3cr.
The range of values obtained in a particular large set of measurements
will depend on the particular extreme values that happen to be obtained. Because the normal distribution curve is so regular, it is useful
to express the average result in a form that reflects some particular
percentage of the measurements, rather than listing the particular extremes obtained. For example, you might wish to report the result as
fj, ± to; where t is chosen so that the range will include some particular
percentage of the measurements. For example, we have seen that a
choice of t = 2 will yield a range that includes 95.4% of the measurements. Suppose you wish to report a range that includes 80% of the
measurements; in this case, you will need to consult a t table such as
Table 5-1 to find the appropriate value oft. We wish to find the value of
/ corresponding to a confidence level of 80%; the confidence level is the
probability that any measurement picked at random will fall within the
range /i ± t cr. Using the bottom line of the table (representing an infinite number of measurements), we see that the desired value of/ is
1.282. Therefore, we can say that there is a probability of 80% that any
random measurement will fall within the range /j. ± 1.282cr; this range
is called the confidence interval. When a result is reported with a confidence interval, the corresponding confidence level should be stated to
make the range meaningful. For example, a result might be reported as
25.342 ± 0.003 with a confidence level of 80%.
Reliability of Measurements
3. Relationship between cr and area under the curve. If we express the
values of x in terms of a, we find that the normal distribution curve
always has the same shape, regardless of the size of cr. That is, the area
under the curve for a particular multiple of cr on either side of /u, is the
same for any normal distribution curve.
Values of x between
Area under this portion of the curve
(j. -
/LI - 2cr and /A + 2cr
0.954 (shaded area in Figure 5-2)
/u- 3crand|ii+ 3cr
0.997
Thus, whenever you have a very large number of measurements, the
probability is that 68.3% of them (about two-thirds) will have values
within the range /A ± cr (that is, within one standard deviation of the
average value). Similarly, 95.4% of the measurements (about 19 out of
20) probably will have values within the range /u. ± 2cr, and only 0.3%
of them (3 in 1000) are likely to have values outside the range /u, ± 3cr.
This also means that the probability is only 0.3% (3 times out of 1000
measurements) that any single measurement will yield a value differing
by more than 3 cr from the value p.
4. Confidence interval and confidence level. We have seen that 3 out of 1000
measurements probably will have values outside the range /u, ± 3cr.
The range of values obtained in a particular large set of measurements
will depend on the particular extreme values that happen to be obtained. Because the normal distribution curve is so regular, it is useful
to express the average result in a form that reflects some particular
percentage of the measurements, rather than listing the particular extremes obtained. For example, you might wish to report the result as
fj, ± to; where t is chosen so that the range will include some particular
percentage of the measurements. For example, we have seen that a
choice of t = 2 will yield a range that includes 95.4% of the measurements. Suppose you wish to report a range that includes 80% of the
measurements; in this case, you will need to consult a t table such as
Table 5-1 to find the appropriate value oft. We wish to find the value of
/ corresponding to a confidence level of 80%; the confidence level is the
probability that any measurement picked at random will fall within the
range /i ± t cr. Using the bottom line of the table (representing an infinite number of measurements), we see that the desired value of/ is
1.282. Therefore, we can say that there is a probability of 80% that any
random measurement will fall within the range /j. ± 1.282cr; this range
is called the confidence interval. When a result is reported with a confidence interval, the corresponding confidence level should be stated to
make the range meaningful. For example, a result might be reported as
25.342 ± 0.003 with a confidence level of 80%.