We have seen in the preceding sections how to calculate the errors on directly measured quantities. Very often, however, it is necessary to calculate other quantities from these data. Clearly, the calculated result will then contain an uncertainty which is carried over from the measured data.

To see how the errors are propagated, consider a quantity *u* = *f(x,
y)* where *x* and *y* are quantities having errors
_{x} and
_{y},
respectively. To simplify the algebra, we only consider a function of
two variables here; however, the extension to more variables will be
obvious. We would like then to calculate the standard deviation
_{u} as
a function of _{x} and
_{y}. The variance
_{u}^{2} can be
defined as

To first order, the mean may be
approximated by *f*(,
). This can be
shown by expanding *f(x, y)* about
(,
) Now, to express the
deviation
of *u* in terms of the deviations in *x* and *y*, let us
expand (*u* - ) to
first order

where the partial derivatives are evaluated at the mean values. Squaring (62) and substituting into (61) then yields

Now taking the expectation value of each term separately and making use of the definitions (8, 9) and (10), we find

The errors therefore are added quadratically with a modifying term due to the covariance. Depending on its sign and magnitude, the covariance can increase or decrease the errors by dramatic amounts. In general most measurements in physics experiments are independent or should be arranged so that the covariance will be zero. Equation (64) then reduces to a simple sum of squares. Where correlations can arise, however, is when two or more parameters are extracted from the same set of measured data. While the raw data points are independent, the parameters will generally be correlated. One common example are parameters resulting from a fit. The correlations can be calculated in the fitting procedure and all good computer fitting programs should supply this information. An example is given in Section 7.2. If these parameters are used in a calculation, the correlation must be taken into account. A second example of this type which might have occurred to the reader is the estimation of the mean and variance from a set of data. Fortunately, it can be proved that the estimators (49) and (52) are statistically independent so that = 0!