In any measurement the
numerical
value that you obtain from your instrument is *always* somewhat
different
from the true value of the physical quantity. Your goal is the come to
term
with that fact of life, and characterize the numerical value
accordingly:
estimate the error of your measurement. Here are a few concepts related
to this
issue:

**Statistical (Random) Error:**The statistical uncertainty of a measurement is the uncertainty that reflects the fact that every time you make a measurement, you measure a slightly different quantity each time. The tendency for a measured value to "jump around" from measurement to measurement is the statistical error.**Systematic Error:**This is uncertainty and error in your measurement caused by anything that is*not*statistical uncertainty. This includes instrumental effects, note-taking things into account (will the change in barometric air pressure impact this measurement?) and gross (stupid) errors.**Precision:**This is the extent to which you can specify the exactness of a measurement. Usually it is determined by the quality of the instruments.**Accuracy:**This is the extent to which your measurement is*in fact*close to the true value.

The difference between statistical and systematic errors is illustrated in an example here. Assume you aim arrows at the bull's eye. You may find the magnitude of statistical and systematic errors by looking at the distribution of the shots:

Calculating/estimating
the
statistical error is an important part of your job in the lab.*
*How
do we know if random errors dominate the measurement?
Consider, for
example, measuring the time required for a weight to fall to the
floor. A
random error may occur when an experimenter attempts to push a button
that
starts a timer simultaneously with the release of the weight. To
decide
if the error is truly random, we have to repeat the measurement many
times,
just like in the example above we had to shoot many times (say *N*
= 200
times). Then we look at the measured times, and we find the
smallest and
the largest (say 0.44s and 0.76s). We divide the time in between
to equal
intervals, called "bins". In our example, the first bin, *t _{1}*
is for times between 0.44s and 0.46s, the next one,

The best estimate of
the *true*
fall time *t* is the *mean* value (or average value):

If the experimenter squares each
deviation from the
mean, averages the squares, and takes the square root of that average,
the
result is a quantity called the "root-mean-square" or the
"standard deviation" of
the
distribution. It measures the random error or the statistical
uncertainty of
the individual measurement *t _{i}*:

About two thirds of all the measurements
have a
deviation less than one
from the mean
and 95% of all measurements are within two
of the mean. In accord with our intuition that the *uncertainty of
the mean*
should be smaller than the uncertainty of any single measurement,
measurement
theory shows that in the case of random errors the standard deviation
of the
mean_{ } is given by:

where *N* again is the number of
measurements
used to determine the mean. Then the result of the *N*
measurements of the
fall time would be quoted _{}.

Whenever you make a measurement that is
repeated *N*
> 20 times, you are supposed to calculate the mean value and its
standard
deviation as just described. Needless to say, the procedure can get
very
tedious. Many times we take a shortcut, and use a
simplified
prescription for estimating the random error. Assume you have measured
the fall
time about ten times. In this case it is reasonable to assume that the
largest
measurement *t _{max}* is
approximately
+2
from the mean, and the smallest

is a reasonable estimate of the uncertainty in a single measurement. The above method of determining is a rule of thumb if you make of order ten individual measurements (i.e. more than 4 and less than 20).

Systematic errors result when characteristics of the system we are examining, or the instruments we use are different from what we assume them to be. Your watch being off by five minutes may cause systematic error. If a meter stick we are using expanded over time by 5%, then every reading we record will have a systematic error of 5%. The experimenter causes systematic error if he/she is pushing the timer accurately at the end of the measurement, but he/she is always late at the beginning .

Clearly, taking the average of many readings will not help us to reduce the size of this systematic error. If we knew the size and direction of the systematic error we could correct for it and thus eliminate its effects completely.

To report that the time is about 3 PM is less precise than to say the time is 3:02:45. If your watch displays hours only, you are limited by instrumental precision. But being more precise does not always mean being more accurate - your watch may show seconds, but it could be off by five minutes.

The error due to instrumental precision is a systematic error and cannot be improved by repeating the measurement many times. For example, assume you are supposed to measure the length of an object. The precision will be given by the spacing of the tick marks on the measurement apparatus (the meter stick). You can read off whether the length of the object lines up with a tick mark or falls in between two tick marks, but you could not determine the value to a precision of l/10 of a tick mark distance. Typically, the error of such a measurement is equal to one half of the smallest subdivision given on the measuring device. So, if you have a meter stick with tick marks every mm (millimeter), you can measure a length with it to an accuracy of about 0.5 mm.

While in principle you could repeat the measurement numerous times, this would not improve the accuracy of your measurement. This assumes, of course, that you have not been sloppy in your measurement but made a careful attempt to line up one end of the object with the zero of the meter stick as accurately as you can, and that you read off the other end of the meter stick with the same care. If you want to judge how careful you have been, it would be useful to ask your lab partner to make the same measurements, using the same meter stick, and then compare the results.

If you measure, for example, the gravitational acceleration, and you compare your value to the textbook number, you may conclude that your accuracy was "within 30% of the accepted value". Under no circumstance can this comparison replace a true error estimate. The error you report should be calculated from the errors of the time and length you measured.

Any discrepancy between a generally accepted value and your measurement is NOT AN ERROR but either an indication that you have not fully understood all sources of error in your measurement, or that you made a new discovery! You can always calculate (estimate) the error. But if you do not know the true value, it is impossible to determine to what extent is your measurement accurate.

Even simple experiments usually call for the measurement of more than one quantity. The experimenter inserts these measured values into a formula to compute a desired result. He/she will want to know the uncertainty of the result. Here, we list several common situations in which error propagation is simple, and at the end we indicate a general procedure. If you are faced with a complex situation, ask your lab instructor for help.

When a result *R*
is calculated
from two measurements *x* and *y*, with uncertainties ∆*x*
and ∆*y*,
and two constants *a* and *b* with the additive formula:

*R
= ax + by* ,

and if the errors in *x* and *y*
are
independent, then the error in the result *R* will be:

(∆*R*)^{2} = (*a*
∆*x*)^{2}
+ (*b* ∆*y*)^{2}
.

The reason why we should use this
quadratic form
and not simply add the uncertainties *a*∆*x* and *b*∆*y*,
is that we don't know
whether *x* and *y* were both measured too large or too
small; indeed
the measurement errors on *x* and *y* might cancel each
other in the
result *R*! Independent errors cancel each other with some
probability
(say you have measured *x* somewhat too big and *y*
somewhat too
small; the error in *R* might be small in this case). This
partial
statistical cancellation is correctly accounted for by adding the
uncertainties
quadratically. **Note: **a and *b*
can
be positive or negative, i.e. the equation works for both addition and
subtraction.

When the result *R*
is
calculated by multiplying a constant *a* times a measurement of *x*
times a measurement of *y* (or divided by *y*), i.e.:

*R
= axy*
or *R =
ax/y*,

then the *relative* errors ∆*x/x* and ∆*y/y* add quadratically:

(∆*R/R*)^{2} = (∆*x/x*)^{2} + (∆*y/y*)^{2} .

**Example:** Say quantity *x*
is measured
to be 1.00, with an uncertainty ∆*x*
= 0.10, and quantity *y* is measured to be 1.50 with uncertainty ∆*y* = 0.30, and the constant
*a* =
5.00 . The result *R* is obtained as *R* = 5.00 x1.00 x l.50 = 7.5 . The relative
uncertainty in *x* is ∆*x/x*
= 0.10 or 10%, whereas
the relative uncertainty in *y*
is ∆*y/y* = 0.20 or
20%. Therefore
the relative error in the result is _{}
or 22%,. The absolute uncertainty of the result *R* is
obtained by
multiplying 0.22 with the value of *R*: ∆*R*
= 0.22 x 7.50 = 1.7 .

If your result is obtained using a more complicated formula, as for example:

*R
= a x ^{2}
*sin

there is a very easy way to find out how
your
result *R* is affected by errors ∆*x*
and ∆*y* in *x*
and *y*.
Insert into the equation for *R*, instead of the value of *x*,
the
value *x*+∆*x*,
and find how much *R* changes:

*R*
+ ∆*R _{x}* =

If *y* has no error you are done.
If *y*
has an error as well, do the same as you just did for *x*, i.e.
insert
into the equation for *R* the value for *y+**∆y* instead of *y*,
to obtain
the error contribution ∆*R _{y}*.
The total error of the result

(∆*R*)^{2} = (∆*R _{x}*)

This way to determine the error always
works and
you could use it also for simple additive or multiplicative formulae as
discussed
earlier. Also, if the result *R* depends on yet another variable *z*,
simply extend the formulae above with a third term dependent on D*z*.

Frequently
you perform a series of measurements of a
quantity *y* at different values of *x*, and when you plot
the
measured values of *y* versus *x* you observe a linear
relationship
of the type *y = ax + b*. Your task is now to determine,
from the
errors in *x* and *y*, the uncertainty in the measured
slope *a*
and the intercept *b*. There is a mathematical procedure to do
this,
called "linear regression" or "least-squares fit". Such
fits are typically implemented in spreadsheet programs and can be quite
sophisticated. If you have no access or experience with spreadsheet
programs,
you want to instead use a simple, graphical method, briefly described
in the
following.

Plot the measured
points (*x,y*) and mark for each point
the errors ∆*x* and ∆*y*
as bars that extend from the plotted point in the *x* and *y*
directions. Draw the line that best describes the measured points (i.e.
the
line that minimizes the sum of the squared distances from the line to
the
points to be fitted; the least-squares line). This line will give you
the best
value for slope *a* and intercept *b*. Next, draw the
steepest and
flattest straight lines, see the Figure, still consistent with the
measured
error bars. From these two lines you can obtain the largest and
smallest values
of *a* and *b* still consistent with the data, *a _{min}*
and

In light of the above
discussion of
error analysis, discussions of significant figures (which you should
have had
in previous courses) can be seen to simply imply that an experimenter
should
quote digits which are appropriate to the uncertainty in his result.
The above
result of *R* = 7.5 ±
1.7
illustrates this. It would not be meaningful to quote *R* as
7.53142 since
the error affects already the first figure. On the other hand, to state
that *R*
= 8 ± 2 is somewhat
too casual. It is a
good rule to give one more significant figure after the first figure
affected
by the error.

* http://solidstate physics.sunyb.edu/teach/phy132/lab_instructions

*Information from
http://solidstate.physics.sunysb.edu/teach/phy132/lab_instructions/*