Consider the case in which a single physical quantity, *y*,
is some function of the 's:
*y* = *y*(_{1}, ...,
_{M}). The "best"
value for *y* is then *y** =
*y*(_{i}*).
For example *y* could be the path
radius of an electron circling in a uniform magnetic field where
the measured quantities are
_{1} =
, the period of revolution,
and
_{2} = *v*, the
electron velocity. Our goal is to find the
error in y given the errors in
. To first order in
(_{i} -
_{i}*) we have

(12) |

A well-known special case of Eq. (12), which holds only when the variables are completely uncorrelated, is

In the example of orbit radius in terms of
and *v* this becomes

in the case of uncorrelated errors. However, if is non-zero as one might expect, then Eq. (12) gives

It is a common problem to be interested in *M* physical parameters,
*y*_{1}, ..., *y*_{M}, which are known
functions of the
_{i}.
In fact the *y*_{i} can be thought of as a new set of
_{i} or a
change of basis from
_{i} to
*y*_{i}. If the error matrix of the
_{i}
is known, then we have

(13) |

In some such cases the ð*y*_{i} /
ð_{a} cannot be
obtained directly, but the
ð_{i} /
ð*y*_{a} are easily
obtainable. Then

__Example 3__

Suppose one wishes to use radius and acceleration to
specify the circular orbit of an electron in a uniform magnetic
field; i.e., *y*_{1} = *r* and *y*_{2} =
*a*. Suppose the original measured quantities are
_{1} =
= (10 ± 1)*µ*s and
_{2} = *v* =
(100 ± 2) km/s. Also
since the velocity measurement depended on the time measurement,
there was a correlated error
= 1.5 × 10^{-3}
m. Find
*r*,*r*, *a*,
*a*.

Since *r* = *v* /
2 = 0.159 m and
*a* = 2*v* /
= 6.28 × 10^{10}
m/s^{2} we have *y*_{1} =
_{1}_{2} /
2 and
*y*_{2} = 2
_{2} /
_{1}. Then
ð*y*_{1} /
ð_{1} =
_{2} /
2,
ð*y*_{1} /
ð_{2} =
_{1} /
2,
ð*y*_{2} /
ð_{1} =
-2_{2} /
_{1}^{2},
ð*y*_{2} /
ð_{2} =
2 /
_{1} . The
measurement errors specify the error matrix as

Eq. 13 gives

Thus *r* = (0.159 ± 0.184) m

For *y*_{2}, Eq. 13 gives

Thus
*a* = (6.28 ± 0.54) × 10^{10} m/s^{2}.