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Dr. Robin Rehagen, LPC Physics
Understanding Experimental Error
Table of Contents
- Overview
- Accuracy vs. Precision
- Measuring Accuracy
- Measuring Precision
4.1 Instrumental Resolution
4.2 Random Error
4.3 Errors in Parameters of Fitted Functions
4.4 Error Propagation
4.5 Error in Trigonometric Functions
4.6 Percent Error
- Types of Error
1. Overview
It is essential in lab class to understand the limitations of your experiments. Scientists keep
track of their experimental limitations in several ways:
- Know the assumptions made when using a certain physical model. For example, we
often assume that we can ignore air resistance when using kinematics equations.
- Be aware of bias: instances in which your sample of data may be inherently different
than the population you wish to study. For example, say I want to study the average
height of a human. If I chose to only ask professional basketball players to give their
heights and no one else, then my final answer will be biased to very high values because
basketball players do not consist of a representative sample of the world population.
- Know that every measurement you make has some uncertainty associated with it.
This uncertainty may come from limitations of your measuring device (like the size of
the tick marks on a ruler) or the random variations that are associated with everyday life
(if you want to measure human body temperature, every healthy hum will have a
slightly different temperature).
A more detailed list of the types of error that may come up in your experiments may be found
in Section 5. As you write the discussion section of your lab report, you may use this list to help
you evaluate the types of error present in your experiment.
Note that the words error and uncertainty are equivalent, and completely interchangeable.
2. Accuracy vs. Precision
We will start with some definitions:
- Accuracy indicates how close your experiment is to the “right answer”. If you knew in
advance that your internal body temperature was 98.4°F, then you would say a
thermometer is accurate if it could reproduce that known value.
- Precision indicates how well your experiment can reproduce the same result. If you
took your temperature ten times, and the thermometer always read 98.9° on each
measurement, it would be very precise – but not accurate.
It is important that our experiments are both precise and accurate. Often, accuracy is hard to
determine in real experiments – we don’t always know what the answer “should be”. In fact, in
scientific research we almost never know what the answer “should be”. Therefore, when
scientists talk about “uncertainty” or “error” on a measured value, they are almost always
talking about the precision of their measurement, not the accuracy.
3. Measuring Accuracy
In order to measure accuracy, we must know the “answer” to our scientific question ahead of
time. Sometimes this will be relevant in lab class. For instance, if I am trying to measure g , the
acceleration due to gravity, I know from previous experiments that the average value of g has
been found to me 9.8 m/s
2
. Therefore, I can measure the accuracy of my experiment. Let’s say
that I measured g = 9.7 m/s
2
. In order to determine how accurate my experiment was, I can
calculate a percent difference from the accepted vale of g :
Percent Difference = ,
My Value − True Value
True Value
m
s
;
m
s
;
m
s
;
For this experiment, I would say that my percent difference from the accepted value for g is 1%.
This statement gives the accuracy of my experiment.
In lab, it is not always possible to determine a percent difference. If you have no “known value”
to compare your answer to, you cannot calculate a percent difference. If you are able to
calculate it, then you should use it to test the accuracy of your experiment. If you find that your
percent difference is more than 10%, there is likely something wrong with your experiment and
you should figure out what the problem is and take new data.
4. Measuring Precision
Precision is measured using two different methods, depending on the type of measurement
you are making. These methods are described in Sections 4.1 and 4.2. There are special
circumstances when more complicated methods are necessary, such as finding the errors in
parameters of a fit (discussed in Section 4.3) or finding the error in a trigonometric function
(discussed in Section 4.5). Depending on the details of your experiment, you may need to
propagate your measurement errors as described in Section 4.4. At the end of your
Acceleration (m/s
2
(± 0.01 m/s
2
Trial 1 9.
Trial 2 9.
Trial 3 9.
Trial 4 10.
Trial 5 9.
Trial 6 9.
Trial 7 9.
Trial 8 9.
Trial 9 9.
Trial 10 9.
Mean
9. 82 ± 0.04 m/s
2
Table 1 – Here we show the acceleration of the falling ball as calculated in each
of the eight trials, along with the mean acceleration of all trials. The error in
individual acceleration measurements is 0. 01 m/s
2
. Our final result is a
measurement of g = 9.82 ± 0.04 m/s
2
This student measured g , the acceleration due to gravity, ten separate times. Even though we
know that the true value of g does not change from trial to trial, the student’s measurements
do. The student correctly quotes the instrumental uncertainty on his measurement device to
be ± 0.01 m/s
2
. However, we can see that the actual numbers vary much more widely that ±
0.01 m/s
2
. How should he quantify his uncertainty?
The first thing to note is that in cases where your measurements fluctuate randomly, you must
take multiple trials (10 minimum) in order to get a good idea of the amount of fluctuation.
Once you have taken multiple trials, you should then find the mean of your measurements.
The mean tells you your “best guess” of what the correct answer is, and this is the final
numerical result that you should quote in your lab report.
There is a simple formula to determine the error on the mean value
Error on the Mean =
where 𝜎 is the standard deviation of your data and N is the total number of data points (10, in
this case). Standard deviation 𝜎 is defined as
E
I
;
L
IMN
where 𝑥̅ is the mean of your measurements and each 𝑥 I
represents an individual measurement.
Don’t worry about the details of this formula, you can quickly use the Excel function =STDEV()
to calculate 𝜎 for you. But it is worth pausing now to take a look at what Excel is actually
calculating when you use =STDEV(). The term (𝑥
I
− 𝑥̅ ) is the difference between an individual
measurement and the mean. We then square that term so positive and negative deviations
from the mean are treated the same. We add up the
I
;
deviations for each individual
measurement, and then divide by the total number of measurements to get an “average”
deviation. (Note that we divide by 𝑁 − 1 instead of 𝑁 for complicated reasons that are best
discussed in a statistics class.) Finally, we take the square root to undo the fact that we squared
the
I
term. Thus, the standard deviation is a measurement of the average amount
that the data deviates from the mean value.
To recap: If your data fluctuates between different trials, take the mean of the data and
calculate the error on the mean. For data that fluctuates, the error on the mean gives you a
better estimate of your precision than the instrumental errors on individual data points. For
fluctuating data, the error on the mean is almost always larger than the instrumental error, and
is thus a more conservative way to measure your experimental uncertainty.
Note: If you take significantly fewer than 10 trials, the error on the mean becomes less
meaningful. Ideally, you should always perform at least 10 trials in any experiment to quantify
your random error. If for some reason you only took a very small data sample (fewer than 10
trials), then a “quick and dirty” way to estimate your error is to use the half-range formula:
maximum data value
− (minimum data value)
The numerator represents the full range of your data. When you divide by 2, you get “half the
range”. This will give you an idea of the uncertainty in your measurements; however, running
10 or more trials and calculating the error on the mean is much more rigorous.
4.3 Errors in Parameters of Fitted Functions
If you have created a scatter plot with your data and are fitting a function to it, the parameters
of the fitted function will all have errors. In some cases, LoggerPro is able to calculate the
errors in the fit parameters. Please consult your instructor during lab to verify whether or not
LoggerPro is properly estimating fit parameter errors.
In the cases where LoggerPro does not accurately estimate fit parameter errors, you can use
Excel to calculate the errors for you. Excel can only do this for a linear function (i.e., a straight-
line function). The process of using Excel to calculate errors in a linear fit is described below.
The Excel function LINEST (“line statistics”) is able to calculate the errors in the slope and y-
intercept of a linear function of the form 𝑦 = 𝑚𝑥 + 𝑏. To do so, follow the directions below:
- The empty 2x2 square should populate with numbers. The top line of numbers shows
your slope and y-intercept. These 2 numbers should match the equation for your fit.
The bottom line shows error in slope and error in y-intercept.
4.4 Error Propagation
Often in a laboratory setting, we have to take measurements of many different variables in
order to calculate the quantity which will give us the answer to our scientific question. How do
the errors in all of our separate variables combine together? The error propagation equation
tells us how these errors combine into the error in our final answer.
For example, say you wanted to calculate the time it would take for a ball to fall a very small
distance (for example, the length of your index finger). Using a stopwatch to measure the time
directly would be very inaccurate because of poor human reaction time, so you decide to use
kinematic equations to model an equation for the time of fall. You find that the time of fall
should be
𝑡 = E
You then choose to measure h and g and use those measurements to determine the time t.
You measure the length your index finger, because you know that this is the height h you are
dropping the ball. You measure h = 5.2 ± 0.05 cm, or in SI units: h = 0.05 2 ± 0.0005 m. You
chose to use an instrumental uncertainty on this measurement because the length of your
finger does not fluctuate randomly. Thus the uncertainty of 0.05 cm is half of the smallest tick
mark on your ruler.
Next, you measure g. We will assume that you took the data shown earlier in Table 1, and you
have calculated a mean value of g to be 9.82 ± 0.04 m/s
2
. The uncertainty of 0.04 m/s
2
is the
error on your mean , which is the correct way to calculate uncertainties for fluctuating data.
Now – how do you calculate t? And what is the uncertainty in t? To calculate t , you simply plug
in your measured values into the equation:
E
E
- 052 m
m
s
;
= 0. 102 s
To find the uncertainty in t , we need to use the Error Propagation Formula. For any quantity F
that is a function of multiple variables F(x,y,z,…) , the error in F (which we call 𝑒 \
) is defined as
\
E
]𝑒
^
a
;
+ ]𝑒
b
a
;
+ ]𝑒
c
a
;
where 𝑒 ^
b
, and 𝑒
c
are the errors in individual measurements. If you are unfamiliar with
partial derivatives (such as the term
f\
f^
), they are simply normal derivatives where all other
quantities in the equation are treated as constants except the variable with respect to which
you are taking the partial derivative.
Let’s write out the error propagation formula for the student’s experiment. The quantity he
wants to calculate is t , and t is a function of two variables: t(h,g) each of which have
uncertainties 𝑒 g
and 𝑒
h
. So the formula for the error in t is:
i
E
]𝑒
g
a
;
+ ]𝑒
h
a
;
Let’s evaluate the partial derivatives one by one. To evaluate
fi
fg
, we would treat the number 2
and the variable g as constants, so:
jE
k = jE
k
lℎ
N/;
n = jE
k ]
oN/;
a = E
To evaluate
fi
fh
, we would treat the number 2 and the variable h as constants, so:
jE
k
l √2ℎ
n
l 𝑔
oN/;
n
l √2ℎ
n ]−
op/;
a = −
E
p
Now we plug in these expressions to the equation for 𝑒
i
Here, 𝑒
yz{ (|)
represents the error in the function tan
, and the 𝑒
|
is the error in 𝜃 that you
determined during your experiment. The equation above can also be applied to sine and
cosine:
~{ (|)
sin(𝜃 + 𝑒
|
) − sin (𝜃 − 𝑒
|
~ (|)
cos(𝜃 + 𝑒
|
) − cos (𝜃 − 𝑒
|
Here is an example. If your expression looks like
sin (𝜃)
the best thing to do is replace sin (𝜃) with
to obtain the expression
O
If that is not possible in your particular experiment, then the next best thing to do is replace
sin (𝜃) with another variable (let 𝑧 = sin (𝜃), for example) and rewrite the expression as
Then you can find the error in 𝑧 using Eq. 2 above, and use the error propagation formula to
propagate the errors in 𝑧 and 𝑥 into an error in your final answer, 𝑦.
4.6 Percent Error
After you calculate your final answer and the error on that answer, you should quote a percent
error. This puts your error value in context, by indicating what percent of your measurement is
uncertain. Percent error is defined as
Percent Error =
measurement error
measurement
so in the case described above, the percent error is
Percent Error =
i
0005 s
102 s
Eq (2)
Eq (3)
and the student may report that he has a 0.5% error in his calculated time of fall. Note that the
student should report both the actual numerical value and error (0.102 ± 0.0005 s) and the
percent error (0.5%). If the experiment was one in which a percent difference from the true
value can be calculated, the percent difference should be reported as well.
5. Types of Error
A measurement of a physical quantity is always an approximation. The uncertainty in a
measurement arises, in general, from three types of errors.
Personal errors - Carelessness, poor technique, or bias on the part of the experimenter. The
experimenter may measure incorrectly, or may use poor technique in taking a measurement, or
may introduce a bias into measurements by expecting (and inadvertently forcing) the results to
agree with the expected outcome. Gross personal errors, sometimes called mistakes or
blunders, should be avoided and corrected if discovered. As a rule, gross personal errors are
excluded from the error analysis discussion because it is assumed that the experimental result
was obtained by following correct procedures. The term "human error" should also be avoided
in error analysis discussions because it is too vague to be useful.
Systematic errors : These are errors which can be traced to an imperfectly made instrument, a
badly calibrated instrument, or to the personal technique and bias of the observer. Systematic
errors consistently skew measured values in the same direction (too high or too low).
Systematic errors cannot be detected or reduced by increasing the number of observations,
and can be reduced by applying a correction or correction factor to compensate for the effect.
Random errors : These are errors for which the causes are unknown or indeterminate, but are
usually small and follow the laws of chance. Random errors can be reduced by averaging over a
large number of observations, as described in Section 4.2.
The following are some examples of systematic and random errors to consider when writing
your error analysis.
- Incomplete definition (may be systematic or random) - One reason that it is impossible
to make exact measurements is that the measurement is not always clearly defined. For
example, if two different people measure the length of the same rope, they would
probably get different results because each person may stretch the rope with a different
- Physical variations (random) - It is always wise to obtain multiple measurements over
the entire range being investigated. Doing so often reveals variations that might
otherwise go undetected. These variations may call for closer examination, or they may
be combined to find an average value.
- Lag time (systematic) - Some measuring devices require time to reach equilibrium, and
taking a measurement before the instrument is stable will result in a measurement that
is generally too low. The most common example is taking temperature readings with a
thermometer that has not reached thermal equilibrium with its environment.
