56
Reliability of Measurements
Probability Distributions for Small Numbers of Measurements
In many cases, it is not practical to make large numbers of measurements. In
chemical analyses, there often is only enough material to make a few measurements. The desire usually is to find a true value for the measured quantity, but
the average of a small number of measurements is unlikely to represent the true
value. You can see this if you make several small sets of measurements of the
same quantity; the averages of the various groups are likely to differ somewhat
from one another. Most people have the intuitive feeling that the average will be
closer to the true value as the number of measurements increases. The formal
mathematical statements of probability theory reflect this same viewpoint. It is
possible to make some statements about the results of a small number of measurements, but these statements must be made with less confidence than we
have in statements resulting from large numbers of measurements.
For this discussion we use a distribution curve as before, but this time it is not
represented by an equation as simple as the Gaussian equation. In fact, there is
not just one curve; there are many, one for each size of sample (different
number of measurements). It isn't practical to draw a different curve for each
size of sample, but we can describe the changing nature of the distribution
curves: as the size of the sample gets smaller, the corresponding distribution
curve becomes shorter and broader than the ones shown in Figure 5-1 for the
same value of s. The same statement is true for Figure 5-2, where a larger value
ofs applies. This changing nature of the distribution curve is taken into account
in the t table (Table 5-1), which can be used in place of the curves.
The Precision of a Single Measurement
The calculation of standard deviation (Equation 5-3) is the sanjg whether you
havejnany measurements or only a few; sample size affects only the selection
of the / value. For a single measurement taken at random from a small number
(n) of measurements, the confidence interval for the desired confidence level is
x ± ts
(5-5)
This is a statement of the precision of a single measurement.
Note that you don't know the true value (you didn't take many measurements). You must use the average (x) of your measurements as the best measure available as a substitute for the true value. Note also that the value oft you
choose will depend on the sample size (n) as well as on the confidence level you
desire. The confidence interval will get larger as the number of determinations
gets smaller, or as the confidence level increases.
The Precision of the Mean
The main objective of making a series of measurements usually is to find the
true value, and we would like to indicate the confidence we may have in the
Reliability of Measurements
Probability Distributions for Small Numbers of Measurements
In many cases, it is not practical to make large numbers of measurements. In
chemical analyses, there often is only enough material to make a few measurements. The desire usually is to find a true value for the measured quantity, but
the average of a small number of measurements is unlikely to represent the true
value. You can see this if you make several small sets of measurements of the
same quantity; the averages of the various groups are likely to differ somewhat
from one another. Most people have the intuitive feeling that the average will be
closer to the true value as the number of measurements increases. The formal
mathematical statements of probability theory reflect this same viewpoint. It is
possible to make some statements about the results of a small number of measurements, but these statements must be made with less confidence than we
have in statements resulting from large numbers of measurements.
For this discussion we use a distribution curve as before, but this time it is not
represented by an equation as simple as the Gaussian equation. In fact, there is
not just one curve; there are many, one for each size of sample (different
number of measurements). It isn't practical to draw a different curve for each
size of sample, but we can describe the changing nature of the distribution
curves: as the size of the sample gets smaller, the corresponding distribution
curve becomes shorter and broader than the ones shown in Figure 5-1 for the
same value of s. The same statement is true for Figure 5-2, where a larger value
ofs applies. This changing nature of the distribution curve is taken into account
in the t table (Table 5-1), which can be used in place of the curves.
The Precision of a Single Measurement
The calculation of standard deviation (Equation 5-3) is the sanjg whether you
havejnany measurements or only a few; sample size affects only the selection
of the / value. For a single measurement taken at random from a small number
(n) of measurements, the confidence interval for the desired confidence level is
x ± ts
(5-5)
This is a statement of the precision of a single measurement.
Note that you don't know the true value (you didn't take many measurements). You must use the average (x) of your measurements as the best measure available as a substitute for the true value. Note also that the value oft you
choose will depend on the sample size (n) as well as on the confidence level you
desire. The confidence interval will get larger as the number of determinations
gets smaller, or as the confidence level increases.
The Precision of the Mean
The main objective of making a series of measurements usually is to find the
true value, and we would like to indicate the confidence we may have in the