IB Physics/Measurements and Uncertainties (2016)/Uncertainties and Errors (2016)

Errors
Physics is an experimental science, and not all measurements are made in an infinitely precise way. As such, physicists have developed mathematical methods to deal with the errors that crop up from doing experiments in the real world - a field of study called Error Analysis. In IB Physics, this is lightly tested but critical for your IA, where you do an experiment. In the most formal sense, think of an error as a deviation, or difference, of a measured value from an accepted (or real) value.

Random Errors
A random error is an error affecting a measured value that an experimenter has found which is unknown and unpredictable. It is usually caused by:


 * 1) Variations in the experimental situation, like random fluctuations in temperature. In these cases, the maximum random error, (or uncertainty), is of unpredictable and usually unknown size.
 * 2) Estimating when reading a measured value off of a measuring instrument, like taking the mass of a body from a digital scale. In these cases, you can usually know the size of the maximum random error, (or uncertainty), from the precision of the measuring instrument.

In doing practical experiments in Physics, it is important to reduce the random error of your measurements in order to get the most accurate possible result.

The easiest way to reduce random error in experiment is using equipment that is more precise. (Precision here refers to how closely measured values agree with each other. A measurement being precise might not imply it's close to the true value- just that it's very reliably consistent.) For instance, one could use an electronic balance that is precise to 0.01 g over a kitchen balance that is precise to 1 g.

Systematic Errors
Systematic errors are errors that cause the measured value of a quantity to be different from the real value by a reasonably consistent amount. They are usually associated with a particular measuring instrument or experimental technique.

For example, a 30 cm plastic ruler with margins of 0.3 cm on either side of the 0 cm and 30 cm markings would have a systematic error if you used it incorrectly, by assuming the end of the ruler was the real 0 marking, and lining that up against, say, the height of a block to measure it. This kind of systematic error is called a zero error, so named because measuring a 0 cm actual value would leave you with a systematic error of 0.3 cm.

You can reduce systematic errors in an experiment by identifying them from your equipment and method, followed by accounting for them in your calculations, usually by subtraction. You can also reduce systematic errors by using equipment that has been tuned to have none, and by using existing equipment in a way that will not lead to the creation of systematic errors, like measuring from the actual 0 cm mark of a ruler.

Uncertainties
Uncertainties are extra information added to the accepted value of a quantity that signify how precise our knowledge of a quantity is.

For example, one might measure a length of a copper cable as $$0.600\,\mathrm{m}$$, with a yardstick accurate to the nearest $$0.001\,\mathrm{m}$$. Measuring a length on an analogue ruler is taken to be the act of finding where two points are on the ruler, and finding the difference. Each point on the ruler is a millimetre apart, so the real start and end of the wire can be recorded by a point on the ruler that is at most half a millimetre away. Subtracting these two points creates a total uncertainty of two half-millimetres, or one millimetre.

We would therefore write the quantity signifying our knowledge of the length of copper wire as $$0.6\,\mathrm{m} \pm 0.001\,\mathrm{m}$$. The symbol in the centre is pronounced "plus or minus", and it shows that the value could be up to $$0.6 \,\mathrm{m} + 0.01 \,\mathrm{m} = 0.601 \,\mathrm{m}$$, or as low as $$0.6 \,\mathrm{m} - 0.001 \,\mathrm{m} = 0.599 \,\mathrm{m}$$. That is, from the accepted value minus the uncertainty up to the accepted value plus the uncertainty.

Note: this use of the plus-or-minus symbol means that uncertainties are always positive numbers and should be treated as such. This is why there are $$|\mathrm{absolute\, value\, signs}|$$ on all of the formulas about uncertainties. These signs mean that you must change whatever is within them, positive or negative, into a positive number of equal magnitude. The signs have other meanings in maths, but this is beyond the scope of this Wikibook.

Absolute Uncertainties
An absolute uncertainty is an uncertainty attached to a quantity, which is of a certain known value. It answers the question - how wrong exactly are we? It is not a proportion of the accepted value. The absolute uncertainty of a given quantity $$x$$ is written as $$\Delta x$$. For instance, if we knew that $$x = 300 \pm 6$$, we could say that $$\Delta x = 6$$.

You should write them to a maximum of 1 or 2 (based on your judgement) significant figures. The reasoning behind this is that the extra significant figures add no real meaning to our understanding of the main value.

Fractional Uncertainties
A fractional uncertainty is an uncertainty attached to a quantity which represents the magnitude of the absolute uncertainty involved, as a proportion of the accepted value for the quantity. Basically speaking, it answers the question - how much does our wrongness matter? For instance, in our example of the wire, we could calculate the uncertainty like this:

$$\frac{\Delta x}{x} = \frac{0.001\,\mathrm{m}}{0.600\,\mathrm{m}} = 0.001666\ldots$$

There is usually no need to round a fractional uncertainty because, very often, it is an intermediate step in calculating something more complex. The symbol $$\frac{\Delta x}{x}$$ represents a fractional uncertainty.

Percentage Uncertainties
Percentage Uncertainties of quantities are much like fractional uncertainties, except for the fact that they are expressed as a percentage rather than a fraction or decimal. The mathematical skill of converting a fraction or decimal into a percentage is below the scope of the IB Physics Syllabus, and so will not be detailed here.

For our example of the yardstick, the percentage uncertainty is easily found by converting the fractional uncertainty to a percentage. The symbol which represents a percentage uncertainty, $$\frac{\Delta x}{x}$$, is the same. See here:

$$\frac{\Delta x}{x} = \frac{0.001\,\mathrm{m}}{0.600\,\mathrm{m}} = 0.001666\ldots = 0.17\ldots\% \approx 0.2\%$$

Percentage Uncertainties are the usual way to express uncertainties by the proportion of a value. Exactly the same as absolute uncertainties, you should write them to a maximum of 1 or 2 (based on your judgement) significant figures.

Uncertainties from Averages
When you have taken multiple readings of a value to find an average and increase your accuracy, there are two ways to determine the uncertainty of your final average value.


 * 1) If the range of the data is more than the range of possible values of the mean when the same absolute uncertainty is kept, then you must account for this in expressing your final value. For instance, if your experimental data for a value are $$5.5 \pm 0.2, 6.5 \pm 0.2, 7.5 \pm 0.2$$, then you have underestimated the random error, because the ranges of possible values do not overlap on one single value. You need to correct this. The rule of thumb for doing so is for the absolute uncertainty of the mean to be half of the range of the data; the reasoning behind this is to include all of the data points as possible values. In our example, the range of the data is $$2.0$$, so we would express our mean as $$6.5 \pm 1$$.
 * 2) If the uncertainty of each value means that the possible range of real values from each reading is smaller than the range of the data, then you simply retain the same absolute uncertainty when expressing the data's mean. For instance, if your data were $$6.6 \pm 0.2, 6.5 \pm 0.2, 6.7 \pm 0.2$$, then, because every value is within the uncertainty of the mean ($$6.6 \pm 0.2$$), there is no need to increase the uncertainty to account for the spread of the data.

A note on the reasoning behind keeping the same absolute uncertainty for the mean even though taking an average is used to make a measurement more accurate is made a few sections below.

Graphical Analysis
The skills involved with graphing a relationship between two quantities based on experimental data are generally prerequisite to the IB Physics course. However, here's a quick refresher on how you should do your graphs.


 * 1) Linearise the axes to test for a straight-line graph if necessary for analysis: if $$A^3 = kB^2$$, then plot $$A^3$$ and $$B^2$$ on the axes rather than leaving the two quantities as they are.
 * 2) Include a descriptive title on the graph, and turn on all major and minor grid lines.
 * 3) Label the axes in the format "description symbol /unit"; for instance, a label like "distance $$s\,/\mathrm{m}$$" is appropriate.
 * 4) Ensure that the numbers on the axes are labelled and spaced out in a logical manner - without the unit, as you have already divided the quantity by the unit in your labelling scheme.
 * 5) Generally, a physicist would analyse an experiment by plotting the thing they are changing (the independent variable) on the horizontal x-axis, and plot the thing they are measuring (the dependent variable) on the vertical y-axis.
 * 6) Make sure you use "X" markers or "+" markers rather than circular blobs to indicate a plotted point, so that it is visually more precise.

That's pretty much the sum of physics graphing skills that you would be expected to have up to the point of the IB course. For the rest of this section of the article, assume I am talking about Microsoft Excel (2020) (and not Google Sheets - Sheets is incapable of proper error bars and uncertainty analysis).

Error Bars
Error bars are a physicist's most useful visual interpretation of uncertainty. They are extensions of the plotted marker that show how far up, down, left, or right, the real value indicated by the marker could possibly be plotted on the graph; that knowledge is based on the uncertainties of the value you are plotting. Search: "error bars physics" on Google Images for a visual idea of them.

Error bars for an experiment might all be of equal size, or different points may have different uncertainties - in this case, you should record, or generate, a column of uncertainty values (e.g. a column for $$\Delta x$$ and $$\Delta y$$ alongside the typical columns for $$x$$ and $$y$$) for the spreadsheet software to use as values for error bars.

Generally speaking, the error bars up and down in the y-axis and the error bars left and right in the x-axis will be the same size, unless you know for sure that the uncertainty in one direction is far less or far greater than the uncertainty in another direction.

Sometimes, error bars are not appropriate for a graph simply because they are so small. For instance, if you are working in lengths of a couple of metres, but your uncertainties in the length are just a millimetre - that being the precision of the ruler - then your error bars with respect to length are likely to be so small on the plotted graph that they can be ignored.

A useful exercise, after this section, would be to see if you can use drawing and intuition based on error bars to demonstrate the rules for adding uncertain quantities together, multiplying an uncertain quantity by a constant, and, as a challenge, multiplying or dividing two uncertain quantities together.

Uncertainty of Gradient and Intercepts
Finding the uncertainty of a gradient or an intercept in physics is very useful because it allows uncertainty values to be "calculated" for quantities that we are unable to directly measure.

For instance, most experiments involving the acceleration of free-fall, $$g$$, do not measure it directly - they derive it from another dependent and independent variable, perhaps as the gradient of the graph that links the two variables together. Another example is how experiments determine the value of absolute zero - physicists back in the 19th century absolutely would not have had access to any equipment that might let them bring the temperature of something down even close to absolute zero, and they determined it using the x-intercept of a graph linking two quantities about the behaviour of a gas together.

The technique to use here is called the use of "max-min lines". Essentially, you try to plot a "max line", which is as steep as possible but still passes through all the error bars, and then a "min line", which is as shallow as possible, but, again, passes through all the error bars. The logic behind this is that you are pushing the error bars to their limits, using your knowledge of how the real point may be anywhere inside the rectangle created by the error bars.

After you have created these max-min lines, you can take the steeper gradient to be the maximum possible gradient, and then you can take the shallower gradient to be the minimum possible gradient. As a rule of thumb, you can generally take the arithmetic mean (add then divide by 2) of the two gradients to get your "measured value" of the gradient. Naturally, it then follows that the gap between your measured gradient and each of the maximum and minimum possible gradients is the uncertainty in the gradient.

You can analyse the intercepts of the graph (intercepts with another line, with the axes, etc) with a similar technique. Note the extreme possible values of the intercept, taking them to be the maximum and minimum values as appropriate, and, again, take the arithmetic mean of the values to be your "measured value". The uncertainty in the intercept is calculated from the difference between the measured and the extreme values.

Uncertainties Expressed as Ranges
There are, of course, some exceptions. For instance, you might realise that the difference between a max line and a min line in some pattern that is close to vertical might end up massive - think a minimum of 10 and a maximum of 5,000. Here, taking the arithmetic mean is not appropriate, and so by necessity you would have to express the gradient as a range (i.e. $$10 \leq m \leq 5,000$$).

Expressing uncertainties in any form as a range is an easy skill. Although it is not always necessary, it can be helpful. Generally, given an uncertain quantity $$x \pm \Delta x$$, knowing that the extremes are such that $$m - \Delta m \leq m_{real} \leq m + \Delta m$$, for instance, $$300 \pm 6$$, we would express our knowledge of the quantity with the range statement "$$x$$ is between 294 and 306"; or, simply with the interval/inequality $$294 \leq x \leq 306$$.

Calculation: Propagating Uncertainties
When determining the uncertainty of a value which has been itself calculated from other uncertain values, the following rules apply.

Adding and Subtracting Values
When adding and subtracting values from each other, you will need to add the absolute uncertainties together.

For instance, $$6.0 \pm 0.2 + 4.2 \pm 0.1 = 10.2 \pm 0.3$$.

The reasoning behind this is that, when adding two uncertain values, $$a \pm \Delta a$$ and $$ b \pm \Delta b$$, the lowest possible value is $$(a+b) - \Delta a - \Delta b$$, and the highest possible value is $$(a+b) + \Delta a + \Delta b$$. The total range of possible values is equal to the sum of the ranges of possible values for the two quantities you added together. In other words,


 * If $$a + b = c$$,
 * then $$\Delta a + \Delta b = \Delta c$$
 * because $$a \pm \Delta a + b \pm \Delta b = (a+b)\pm (\Delta a + \Delta b)$$.

Multiplying or Dividing an Uncertain Value by an Exact Value
The rules regarding propagating an uncertainty through multiplication and division are perhaps some of the most important.

If you have some uncertain value $$a \pm \Delta a$$, then it can take a lowest value of $$a - \Delta a$$ and a highest value of $$a + \Delta a$$. If you multiply the lowest value and the highest value each by some exact factor $$k$$, then you'll get a lowest value of $$ka - k\Delta a$$ and a highest value of $$ka + k\Delta a$$.

The main value has been scaled up by a factor of $$k$$, becoming $$ka$$. At the same time, we can see that the minimum and maximum possible values have scaled up by the same factor - the amount by which they are lower or higher than $$ka$$, that amount being the uncertainty in $$ka$$, has correspondingly been increased by that factor of $$k$$. Mathematically speaking, we can write this in a really simple form - $$\Delta ka = k \Delta a$$. In the IB formula booklet, this isn't actually included - you may have noticed that this is in fact just an application of the rule regarding adding uncertain quantities together.

Dividing a quantity by a constant, which we'll call $$j$$, can be thought of as multiplying by $$\frac{1}{j}$$. So, likewise to the previous paragraph, $$\Delta \frac{a}{j} = \frac{1}{j}\Delta a$$.

As an example - say you have the uncertain quantity $$100 \pm 4$$. If you divided this quantity by 4, you'd get $$25 \pm 1$$; the absolute uncertainty is divided by 4, and the percentage uncertainty remains the same - a very important fact to bear in mind. If you multiplied this uncertainty by 4, you'd get $$400 \pm 16$$. Notice that I've kept the second significant figure; that's because rounding it to an absolute uncertainty of 20 would make it significantly more uncertain than it really was (the uncertainty increases in size by a whole 25%! That is a meaningful difference, and so I have concluded with my own judgement that 2 significant figures is appropriate here).

Multiplying or Dividing two Uncertain Values
The rules get a little more complicated when we move into the realm of multiplying and dividing uncertain values by one another. Let's try and find minimum and maximum values when we multiply two uncertain quantities together.

Suppose you have two quantities, $$a$$ and $$b$$, that multiply to the quantity $$c$$. Suppose that each of the data are uncertain by an amount $$\Delta a$$ and $$\Delta b$$ respectively. In other words, $$ab = c$$.

At a minimum, $$c = (a - \Delta a)(b - \Delta b) = ab - a \Delta b - b \Delta a + \Delta a \Delta b$$, taking the smallest values of the initial data possible. At a maximum, $$c = (a + \Delta a)(b + \Delta b) = ab + a \Delta b + b \Delta a + \Delta a \Delta b$$.

Try to do this with your own pencil and paper - if you subtract the minimum from the maximum, you find a range of values $$2(a \Delta b + b \Delta a)$$ wide; the uncertainty is $$a \Delta b + b \Delta a$$. You will also find that the measured value of the quantity $$c$$ is $$ab + \Delta a \Delta b$$, because that's the centre-point between the minimum and maximum values. Consider that, usually, $$\Delta a \Delta b$$ will be too small to affect the measured value. You can basically cancel it to 0. As such, our final value for $$c$$ is $$ab \pm (a \Delta b + b \Delta a)$$.

Generally, the rule for multiplying two uncertain values is that the percentage uncertainty of the result will be the sum of the percentage uncertainties that you are multiplying together. This is a property that can be seen in the calculation above. If we try to calculate the overall percentage uncertainty in $$ab$$, we get the below.

$$\frac{a \Delta b + b \Delta a}{ab} = |\frac{\Delta a}{a}| + |\frac{\Delta b}{b}|$$

This explains the rule for multiplying two uncertain quantities together. Now we ought to consider what happens if we divide one uncertain quantity by another. Say we have a quantity $$\frac{a}{b} = c$$. $$a$$ and $$b$$ are both uncertain by an amount $$\Delta a$$ and $$\Delta b$$ as before.

At a maximum, $$c = \frac{a + \Delta a}{b - \Delta b}$$. At a minimum, $$c = \frac{a - \Delta a}{b + \Delta b}$$. The difference between these two quantities is $$\frac{a + \Delta a}{b - \Delta b} - \frac{a - \Delta a}{b + \Delta b} = \frac{ab + \Delta a \Delta b + a \Delta b + b \Delta a - (ab - a \Delta b - b \Delta a + \Delta a \Delta b)}{b^2 - (\Delta b)^2} = 2\frac{a \Delta b + b \Delta a}{b^2 - (\Delta b)^2}$$. As before, $$(\Delta b) ^2$$ is a quantity that would generally be insignificant to the final result, so we cancel it to 0. Keep in mind that the algebra here is quite hard, and I wouldn't expect every reader of this textbook to follow it. The general rule for division is stated later.

We find that, dividing by two, the absolute uncertainty is $$\frac{a \Delta b + b \Delta a}{b^2} = \frac{a}{b}(\frac{\Delta b}{b} + \frac{\Delta a}{a})$$. You may realise that this absolute uncertainty is the new value multiplied by the sum of the percentage uncertainties - the percentage uncertainty in the new value is the initial percentage uncertainties added up. You may want to come back to this section later and attempt the algebra yourself. Until then, the rule to be remembered is quite simple, below.

To summarise - when multiplying or dividing two uncertain quantities by one another, the percentage uncertainty of the answer can be found by adding up the percentage uncertainties of the two inputs. When $$a = bc = \frac{d}{f}$$:

$$|\frac{\Delta a}{a}| = |\frac{\Delta b}{b}| + |\frac{\Delta c}{c}| = |\frac{\Delta d}{d}| + |\frac{\Delta f}{f}|$$

Raising Uncertain Values to a Power
When raising an uncertain value, which we'll call $$x$$, to a certain power, which we'll call $$n$$, the rule is actually quite simple. You take your original percentage uncertainty in $$x$$ and multiply it by the absolute value of the power. This is a pretty much direct logical conclusion that can be drawn from our knowledge of the rules for multiplying and dividing uncertain quantities.

For instance, if we want to calculate the percentage uncertainty in $$x^4$$, we just need to note that $$x^4 = x \times x \times x \times x$$. If we want to calculate the percentage uncertainty in $$\frac{1}{x^3}$$, we just need to note that $$x^{-3} = 1/x/x/x$$. In the first case, we are multiplying four $$x$$'es together, and so the percentage uncertainty must be $$\frac{\Delta x^4}{x^4} = |\frac{\Delta x}{x}| + |\frac{\Delta x}{x}| + |\frac{\Delta x}{x}| + |\frac{\Delta x}{x}| = 4|\frac{\Delta x}{x}|$$. In the second case, we are dividing an exact quantity by $$x$$ 3 times, and so the percentage uncertainty must be $$\frac{\Delta x^{-3}}{x^{-3}} = |\frac{\Delta x}{x}| + |\frac{\Delta x}{x}| + |\frac{\Delta x}{x}| = 3|\frac{\Delta x}{x}| = |-3\frac{\Delta x}{x}|$$.

This uncertainty principle needs a little more mathematical proving if we are to use it for fractional and decimal powers. However, I'm just going to leave it - the math is beyond the scope of this course. To summarise what is learned in this section:

$$\frac{\Delta x^n}{x^n} = |n\frac{\Delta x}{x}|$$

Special Functions
For IB SL and HL Physics, calculating the uncertainty of a value after it's been passed through a special function, like $$\mathrm{sin}$$ or $$\mathrm{ln}$$, is not necessary. In pursuing analytical work that needs this, the fractional uncertainty of the function's input is taken to be that of its output. For instance, if I am $$1\%$$ uncertain in a quantity $$A$$, then I would also say that I am $$1\%$$ uncertain in $$\sin A$$ or $$\ln A$$.

This paragraph is only relevant for some very difficult IA topics that may need it. For some functions $$f(x)$$ where a shift in $$x$$ of $$\Delta x$$ causes changes of wildly different size to $$f(x)$$, it is more appropriate to manually calculate the highest and lowest possible values of $$f(x)$$ across the domain of possible real values of the quantity from $$x + \Delta x$$ to $$x - \Delta x$$, which are often (but not necessarily) equal to $$f(x + \Delta x)$$ and $$f(x - \Delta x)$$.