364 15 Combining Information
A normal distribution is characterized by two parameters: the mean and
the variance. Consequently, the measurement of a normal distribution is
a measurement of these two numbers. When these are measured using a
random sample, the mean has thetdistribution, and the variance has the chi-
square distribution. Both of these are approximately normally distributed
for large samples. The mean and variance of the sample mean and sample
variance are well-known:
Sample StatisticsTheorem
LetXbe a random variable from a normally distributed population with meanμ
and varianceσ^2 , For a random sample ofNindependent measurements ofX:- The sample meanXhas meanμand varianceσ^2 /N.
- The sample variances^2 has meanσ^2 and varianceσ^4 /(N−1).
When two populations are compared, one can compare them in a large va-
riety of ways. Their means can be compared with at-test, and their variances
can be compared with either a chi-square test (to determine whether the dif-
ference of the variances is small) or anF-test (to determine whether the ratio
of the variances is close to 1).
It is easy to experiment with these concepts either by using real data or
by generating the data using a random number generator. In the follow-
ing, a random number generator was used to generate two independent
random samples of size 100 from a normal population with mean 10 and
variance 16. For such a large sample size, thetand chi-square distributions
are very close to being normal distributions. The estimates for the distribu-
tions (mean, variance) were (9.31, 13.81) and (10.55, 16.63). Now forget what
these are measuring, and just think of them as two independent measure-
ments. The uncertainty of each measurement is approximately normally dis-
tributed. The mean of the first measurement is the measurement itself, and
the variance matrix is(
0 .138 0
03. 86
)
. The variance matrix of the second
measurement is(
0 .166 0
05. 59
)
. The off-diagonal terms are zero because
the two measurements are independent, and hence uncorrelated. Combin-
ing these two measurements can be done in two ways. The first is to apply
the continuous information combination theorem. The combined distribu-
tion has mean (9.87, 14.97) and variance matrix
(
0 .075 0
02. 28
)
. The second
way to combine the two measurements is to treat them as a single combined