E1C04 09/14/2010
14:7:42 Page 133
So how good is the estimate of the true mean of a variable based on a finite-sized sample? The standard
deviation of the means represents a measure of how well a measured mean value represents the true
mean of the population. The range over which the possible values of the true mean value might lie at
some probability level P based on the information from a sample data set is given as
x Æ t n;P s x ðP%Þ
ð 4:18Þ
where Æt n;P s x expresses a confidence interval about the mean value, with coverage factor t at the
assigned probability, P%, within which one should expect the true value of x to fall. This confidence
interval is a quantified measure of the random error in the estimate of the true value of variable x.
The value s x represents the random standard uncertainty in the mean value, and the value t n;P s x
represents the random uncertainty in the mean value at P% confidence due to variation in the
measured data set. In the absence of systematic error in a measurement, the confidence interval
assigns the true value within a likely range about the sample mean value. The estimate of the true
mean value based on a finite data set is then stated as
x
0
¼ x Æ t n;P s x ðP%Þ
ð 4:19Þ
Equation 4.19 is an important and powerful equation in engineering measurements.
Example 4.4
Consider the data of Table 4.1. (a) Compute the sample statistics for this data set. (b) Estimate the
interval of values over which 95% of the measurements of x should be expected to lie. (c) Estimate
the true mean value of x at 95% probability based on this finite data set.
KNOWN Table 4.1
N ¼ 20
ASSUMPTIONS Data set follows a normal distribution.
No systematic errors.
FIND x; x Æ t n;95 s x ; and x Æ t n;95 s x
p(x)
p(x) –
x'
x
ts x
ts x –
Figure 4.6 Relationships between s x
and the distribution of x and between
s x and the true value x
0 .
4.4 Statistics of Finite-Sized Data Sets 133
14:7:42 Page 133
So how good is the estimate of the true mean of a variable based on a finite-sized sample? The standard
deviation of the means represents a measure of how well a measured mean value represents the true
mean of the population. The range over which the possible values of the true mean value might lie at
some probability level P based on the information from a sample data set is given as
x Æ t n;P s x ðP%Þ
ð 4:18Þ
where Æt n;P s x expresses a confidence interval about the mean value, with coverage factor t at the
assigned probability, P%, within which one should expect the true value of x to fall. This confidence
interval is a quantified measure of the random error in the estimate of the true value of variable x.
The value s x represents the random standard uncertainty in the mean value, and the value t n;P s x
represents the random uncertainty in the mean value at P% confidence due to variation in the
measured data set. In the absence of systematic error in a measurement, the confidence interval
assigns the true value within a likely range about the sample mean value. The estimate of the true
mean value based on a finite data set is then stated as
x
0
¼ x Æ t n;P s x ðP%Þ
ð 4:19Þ
Equation 4.19 is an important and powerful equation in engineering measurements.
Example 4.4
Consider the data of Table 4.1. (a) Compute the sample statistics for this data set. (b) Estimate the
interval of values over which 95% of the measurements of x should be expected to lie. (c) Estimate
the true mean value of x at 95% probability based on this finite data set.
KNOWN Table 4.1
N ¼ 20
ASSUMPTIONS Data set follows a normal distribution.
No systematic errors.
FIND x; x Æ t n;95 s x ; and x Æ t n;95 s x
p(x)
p(x) –
x'
x
ts x
ts x –
Figure 4.6 Relationships between s x
and the distribution of x and between
s x and the true value x
0 .
4.4 Statistics of Finite-Sized Data Sets 133