|Question: How do I make a matching
Answer: Examples are given here
. The table template is here.
|Question: How do I write an equation of fit?
- Replace the generic x- and y-variables with the symbols that
represent the relevant physical quantities.
- Replace the coefficients with the numerical values obtained
by the fit. Round these to the proper number of significant figures
and include units.
|Question: Why would I use a graph of residuals, and
how do I create such a graph in Logger Pro?
Answer: The process is described in L113.
Residuals are used to assess goodness of fit.
|Question: How do I assess
the reproducibility of a measurement?
This involves taking repeated measurements, finding mean deviations,
and finding the percent mean deviation. The process is described
|Question: How do I know whether to calculate a percentage
difference or a percentage error between two values?
When you have no reason to expect that one value is any better than
the other, find the percentage difference. When one of the values
is taken as an accepted value because it is trusted by the scientific
community (for example, the value of g), find the percentage error.
|Question: How do I calculate percentage difference?
Answer: Simply divide the difference of two
values by the sum of the values and multiply the quotient by 100.
% Difference = 100 ∙ (Value 1 - Value 2) ÷
(Value 1 + Value 2)
Note that the sign of the result tells you by inspection which
value is the larger. This is helpful when looking for systematic
errors in a measurement technique.
|Question: How do I calculate
Answer: Divide the difference
of two values by the accepted (or expected) value and multiply the
quotient by 100.
% Error = 100 ∙ |Measured Value - Accepted Value| ÷
Finding the absolute value of the difference is an accepted practice,
although it's not essential.
|Question: Why do I always get
the wrong number of significant figures in calculations of errors?
Answer: The experimental error and percentage difference formulas involve both subtraction and division. You must
use both the addition/subtraction and multiplication/division rules
for significant figures. Apply the subtraction rule first and retain
the smaller number of decimal digits. Then apply the division rule
and retain the smaller number of significant figures. Most people
use only the division rule. This almost always gives the incorrect
number of significant figures.
|Question: How do I estimate
percentage uncertainties (relative error) in measurements, and why would I do this?
Answer: Measurements are inherently uncertain.
The uncertainty is affected by such things as the method of measurement,
the measuring instrument, and variations in the measured characteristic.
Suppose you were measuring the diameter of a cylinder. Some ways
of doing this are to lay a ruler across the diameter and sight the
edges of the cylinder, to roll the cylinder without slipping through
one complete rotation on a piece of paper and divide the result
by π, or to place the cylinder between the jaws of vernier calipers.
Each method has a different uncertainty, and different measuring
instruments can be read with different precisions. If the cylinder
is out of round, that influences the results as well.
See L101 and L103
to review how to calculate uncertainties.
Question: How do I determine the number of significant figures in calculations of mean deviation and % mean deviation?
The same rules for significant figures apply here as to all situations. However, you can save time by noting the following:
- The mean of a set of measurements, the deviations from the mean, and the mean deviation should all have the same number of decimal digits.
- The % mean deviation is the ratio of the mean deviation to the mean. Apply the multiplication rule to determine the number of significant figures in this final calculation.
Important note about rounding error: The determination of % mean deviation requires a number of arithmetic operations. There is the potential for rounding error in each step. In order to minimize rounding error, carry extra digits in intermediate calculations and round in the final step. Here's a shortcut method.
|Question: Why is the phrase human error unacceptable
in a discussion of errors?
Answer: The phrase human
error is non-descriptive. It doesn't show a critical examination
of the process of measurement in the experiment in question.
|Question: What determines the number of significant
figures in a measurement?
Answer: The significant
figures include those digits that are certain as well as the first
uncertain digit. Leading 0's are not considered significant, because
they don't result from a measurement. They're simply placeholders.
|Question: How do I know what digits are certain? Stated
differently, how do I meet a criterion for a particular number of
Answer: A method that
isn't correct is to assume that every digit in the readout of an
instrument is significant. For example, if you measure one oscillation
of a spring and obtain a reading of 1.32 s, don't assume that all
three digits are significant. But how do you know which of those
digits are significant? That is, which ones can you be sure of and
what is the first digit of which you are unsure? You can take repeated
measurements. Let's suppose your measurements are 1.32, 1.19, 1.27,
1.34, 1.21 s. You can be certain the first digit is 1, but there's
variation in the second digit. That's the first uncertain digit,
so that makes this a 2 significant figure measurement. The third
digit of the readings is meaningless.
|Question: How can I increase
the number of significant figures in a measurement?
Generally you find a way to make the measurement larger. For example,
if you're measuring the diameter of a penny, you place several pennies
side-by-side along a ruler as
shown here. This method works because all pennies have the same
diameter or nearly so. If you're measuring a repeated time interval
such as the oscillation of a spring or pendulum, you time several
cycles instead of just one. In these methods, you reduce the percentage uncertainty--also called relative uncertainty--in the measurement as follows:
Measurements of distance and time intervals have endpoint errors.
These are inherent errors in judging the position of the ends of
an object on a ruler or in judging when to start and stop a stopwatch.
The errors have about the same size whether you're measuring a small
interval or a large one. However, the error becomes a smaller percentage
of the measurement itself the larger the measurement becomes. The
goal is generally to reduce the percentage (or relative) errors. Suppose,
for example, that you wanted to achieve a precision of 1 part in
a 100 (1%) in a measurement of the period of a pendulum. How many
consecutive cycles would you have to time. First, you need to estimate
your endpoint timing uncertainty. This is a personal judgment. For
sake of this example, suppose the uncertainty is 0.1 s. In order
to achieve the 1% criterion, your time intervals would need to be
10 s. (0.1 is 1% of 10.) If the period of the pendulum was about
half a second, then you'd need to measure the time for 20 consecutive
cycles. You would make this measurement several times and calculate
the percentage mean deviation in order to verify that you did indeed
meet the 1% criterion.
Question: How can I minimize rounding error in calculating % mean deviation?
Answer: Click here