Phy 1002: Exercise on Error Estimations
September 15, 2019
1. How many significant numbers are there in the following numbers:
1234
123400
123.4
1001
1000.
10.10
1010
1.01 × 103
1.010 × 103
0.015
1.5 × 10−2
1.50 × 10−2
0.0001010
100.0
2. Suppose we want to determine a quantity x that is a function of two measured variables, u and
v. We shall determine the characteristics of x from those of u and v from the fundamental dependence:
x = f (u, v), where u, v are uncorrelated.
Although it may not always be exact, we shall assume that the most probable value for x is given
by x̄ = f (ū, v̄), where q̄ denotes the average or mean of q.
The error in the resulting value for x can be found by considering the spread of the values of x
resulting from combining the individual measurements ui ,vi , . . . , where i=1,2,...N, into individual
results xi : In the limit of an infinite number of measurements (N), the mean of the distribution will
coincide with the average x given above and we can find the variance σx2 (which is the square of the
standard deviation σx ) from:
N
1 X
σx2 = lim [
(xi − x)2 ]
(1)
N →∞ N
i=1
2 ∂f
Show that: σx2 ≈ σu2 ( ∂f
)2 + σv2 ( ∂f
)2 + 2σuv
), where the covariance is
( ∂u )( ∂f
∂u
∂v
∂v
2
σuv
N
1 X
= lim [
(ui − u)(vi − v)]
N →∞ N
i=1
(2)
and the partial derivatives are all evaluated at the mean values of the measured quantities.
If the fluctuations in the measured quantities u and v are uncorrelated, then, on the average, we
should expect to find equal distributions of positive and negative values for this term, and we should
expect the term to vanish in the limit of a large random selection of observations.
Note that similar formulas apply to standard errors, as well, why?
3. Find the standard error σx in x as a function of the errors in σu and σv in u and v for the
following functions:
a) x = 1/[2(u + v)]
b) x = 1/[2(u − v)]
c) x = 1/u2
d) x = uv 2
e) x = u2 + v 2
f)x = a sin(bu/v)
g)x = a ln(bu)
h)x = ebu+cv
i) x = t2 + p2
4. Snell’s law relates the angle of refraction θ2 of a light ray light ray traveling in a medium of index
of refraction n2 to the angle of incidence θ1 of a ray traveling in a medium of index n1 through the
equation n2 sin θ2 = n1 sin θ1 . Find n2 and its uncertainty from the following measurements:
θ1 = (22.03 ± 0.2)◦
θ2 = (14.45 ± 0.2)◦
n1 = 1.0000
(3)
5. Suppose you determine the acceleration of gravity g by measuring the oscillation period T of a
pendulum with length t. The value of g follows from:
4π 2 t
g= 2
T
(4)
You measure T = 2.007 ± 0.002 s and t = 1.000 ± 0.002 m. Determine g and its standard error.
6. The angular speed of a disk may be determined by measurements of angle and time t, made
by an angular sensor and a digital clock. In an experiment, the disk rotates by 4θi = θi − θi−1 = 1◦
during the time interval 4ti = ti − ti−1 = 1ms from ti−1 to ti , what is the mean speed ωi of the disk
for this interval?
If the resolutions of angular sensor is 0.09◦ and that of the clock is 10 µs, what is the estimated
error in ωi ?
The speed of a point on the disk is given by v = Rω, where R is its radius. If the radius is measured
with a Vernier caliper, of resolution 0.1 mm, to be 20 mm, what is the estimated error of the point
when the disk is rotating at the angular speed ωi ?