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Error Estimation Experiment
OBJECTIVE
To develop basic working knowledge involving error assessment in experimentation.
BACKGROUND
The experimental determination of any parameter is based upon measurements which by their
nature contain errors. In general, errors fall into two categories: uncertainty or random errors
and systematic errors. Uncertainty errors are due to the inability to read a measurement device
exactly. For example, the finest division on a ruler is normally 1 mm, so that in using a ruler to
measure length one has an uncertainty of ± 0.5 mm. Consider that an experimental
determination will be made for the parameter B. Say that this determination is based upon
measurements x 1 , x 2 ,..., xN. Then mathematically we have
B = fn(x , x ,..., x 1 2 N) (1)
The uncertainty in B, denoted by dB, can then be related to the uncertainty in the measured
values, dxi, by
i
N
i = (^1) i
dx x
B
dB =∑ (^) ⎟⎟ ⎠
(2)
It is useful to utilize a specific example. We are provided with a perfect parallelepiped of
dimensions a x b x c of an unknown material. It is desired to determine the density of the
material and also the uncertainty in this experimental determination of the density. We will
determine the density by measuring the dimensions of the parallelepiped with a ruler,
measuring its mass with a scale, and using the definition
a b c
m
V
m
⋅ ⋅
ρ (3)
Then the uncertainty becomes
dc c
db+ b
da+ a
dm+ m
d = ⎟ ⎠
∂ρ ⎟ ⎠
∂ρ ⎟ ⎠
∂ρ ⎟ ⎠
∂ρ ρ (4)
Evaluating the partial derivatives
a b c
a b c
m
m
m ⋅ ⋅
∂ρ
(5a)
a b c
m =- a b c
m
a
a (^2) ⋅ ⋅
∂ρ
(5b)
and similarly for b and c. Now substituting
dc a b c
m db+ a b c
m da+ a b c
m dm+ a b c
d = 2 2 2 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
ρ (6)
We note that
a b c
m ρ ⋅ ⋅ (7)
and rearrange to get
ρ ρ c
dc
b
db
a
da
m
dm d = (8)
Next, we specify the uncertainty in our measurements. For example
dm = ± 0.5 gm = ± 5 x 10-4^ kg da = db = dc = ± 0.5 mm = ± 5 x 10
Finally, using our measurements, say
m = 100 gm = 0.1 kg a = 10 cm = 0.1 m b = 5 cm = 0.05 m c = 5 cm = 0.05 cm
with
= 400 kg/m (0.1)(.05)(.05)
3 ρ
we determine the numerical value of the uncertainty
ρ
5 x 10
5 x 10
5 x 10
5 x 10 d =(400)
dρ=(400) (.005 +.005 +.01+.01)
3 d ρ=± 12 kg/m
One of the ways in which this uncertainty error appears in our density measurement would
involve running the experiment a number of times and obtaining a number of density values.
To determine the uncertainty in the efficiency, we would now apply Eq.(2) to our expression
given in Eq.(12). In trying to apply Eq.(2) directly, we run into the problem of having eleven
"measurable" quantities with uncertainties which leads to eleven different partial derivatives,
and a whole host of possible algebraic errors. Furthermore, the very long extensive equation
that would result from this can become cumbersome in a single cell of a spreadsheet. An
alternative is to cascade our uncertainties as functions of sets of variables. We begin by letting
the efficiency be a function of the thermal energy and the electrical energy so that
η=fn( Eth ,Eel ) (13)
Then for the uncertainty in the efficiency we have
el el
th th
dE E
dE + E
d = ∂
∂η
∂
∂η η (14)
The two partial derivatives can be evaluated from Eq.(9). The two new differentials, dEth and
dEel, must be evaluated. For the uncertainty in the electrical energy we note that
E el =fn( V,I,τ) (15)
So that via Eq.(2), we have
τ ∂τ
d
E
dI+ I
E
dV+ V
E
dE (^) el= el el el (16)
Once again Eq.(10) can be used to evaluate the partial derivatives. A similar manipulation
may be done for dEth. However, since Eth is calculated from eight measured values it may
prove useful to break it into two parts, one dealing with the beaker energy, Eth,bk, and a second
term dealing with the water energy, Eth,H2O. Then
(17)
E (^) th =Eth,bk + Eth,HO 2
where
(18)
E (^) th, bk=(mcp∆ T)beaker
(19)
E (^) th, HO=(mcp T)water 2
We may then apply Eq.(2) to Eq.(17) to obtain dEth and to Eqs.(18) and (19) to obtain dEth,bk
and dEth,H2O. Then the expressions for dEth,bk and dEth,H2O will only contain uncertainties of
measured parameters.
Each student is responsible for attempting the algebraic evaluation of dη prior to their lab
period. This attempt will be reviewed by your TA. Failure to submit this formulation to
your TA at the beginning of your lab period will result in a 10 point deduction on your
technical memo.
PROCEDURE
A layout of the experimental apparatus is shown below.
Figure 1. Experimental Setup
Holding Beaker
Hot Plate Stirrer
Magnetic Stirrer
X
Water T/C
X
Wall T/C
Current Meter
Immersion Heater
- Measure the mass of the 1000 ml beaker provided. Record the uncertainty in this
measurement.
- Add approximately 1000 ml of water to the beaker. Measure the mass of the water filled
beaker. Record the uncertainty in this measurement and then place this beaker on the magnetic stirrer.
- Fill the other beaker provided with water. This is the holding beaker depicted in Figure 1.
- Measure the wall plug voltage using the digital multi-meter by inserting the probes into the
wall plug. Record the uncertainty in this measurement.
