Induction Tutorial 4 - Experimental Uncertainty

 Contents

Many of the most significant scientific discoveries are the result of meticulous work and record keeping.  They have been accepted because not only are the results reproducible, but have been collected using methods that are considered to be reliable.

You need to do experimental work in which the data can be considered to be reliable.  While there is no practical assessment in the AS syllabus, there will be questions in the examination about certain practical work that you will have to do.  Since you will need to pass  the AS year (depending on your centre) to progress to A-level, there will be practicals that you will have to do to include in your portfolio which is to be submitted to the Board for your A-level exams.

These notes will look at many common terms and concepts that are used in practical work.  A list of the terms used by the AQA is given here.

# Variables

These are the items that are being studied in a scientific investigation.  In physics, the variables are almost always quantitative, i.e. have a value that is a number.

• Categoric variables are those that are described by words, for example, ants, bumble bees, butterflies.  There is no cross-over between the different classes.  You can’t have a half-way version of an ant and a bumble bee.

• Ordered variables are not numerical but are in a ranked order, for example, thin, medium, thick.

• Discrete variables are those that are in whole numbers.  You can have 1, 2, 3, …, but you cannot have 1.2 or 3.5.

• Continuous variables have a numerical value of any value, e.g. 3.64 W.

These variables are used in three ways in experiments:

• The independent variable is the one that we change (e.g. by changing the length of resistance wire);

• The dependent variable changes in response (e.g. the resistance increases).

• The control variable, which is kept constant to make sure that it’s a fair test.  If you were measuring temperature rise with the power of a heater, you would keep the start temperature the same, use the same volume of water, etc.

Any type of variable can be the independent or dependent variable.

# Errors in measurement

Errors affect the reliability of your data, and it’s important that you are aware of the sources of these errors.  Errors arise from:

• Wrong technique, for example holding a ruler in your hand to measure something that you are holding in the other hand.  The ruler will wobble…

• Positioning your eye to read the scale.  Parallax errors can occur.

• Reaction time in stopping a timer.

• Uncertainty in a reference point.  It is worth having a clear reference point when something is moving.  This is often called a fiducial or fiduciary mark.

Parallax errors can be minimised by use of a mirror behind the pointer.  When you cannot see the image of the pointer, you can be sure that you are reading the correct result.  A parallax mirror can be seen on the meter in the picture below:

 Image from Physics for You Write down the reading shown by positions 1, 2, and 3 in the picture above.

All of these errors are random.  Random errors arise from faulty technique or faults in the equipment.

 Sometimes random errors are called human error.  Do not use this term in the examination as no credit is given.  This is because the phrase human error is too vague and covers not only faulty technique, but also a whole range of other human frailties, e.g. the pilot who stalls his aeroplane on take-off, or “having one’s fingers in the till”, or “playing away from home”.   To err is human, to forgive divine.

Random errors can be reduced by taking repeat readings.

Systematic errors result in readings all being shifted too high, or too low.  This can be the result of the wrong calibration of the instrument, or a change in the zero point of the instrument.

In this case, the zero has shifted.

 How would you take into account the error from this newtonmeter?   Image from Physics for You

The zero point on many instruments can be adjusted, for example, this voltmeter.

Image from Physics for You

 How would you fix the zero error on this meter?

Improving Accuracy of Measurements

A fiducial mark is a reference point.  Suppose you are timing a swinging pendulum.  You want to know when one swing finishes and the next starts, so you place a pointer on the bob and one on the clamp stand, like this:

You count each time the pointer passes the fiducial mark from, say, left to right.  You start timing when the pointer first passes the fiducial mark from left to right.  A whole swing is when the pointer passes the fiducial mark the next time from left to right.  This fiducial mark is in the middle of the swing, so that it doesn't matter if the amplitude of the swings gets less.  You need to count multiple swings (say 10) to reduce the uncertainty in the timing.  The period of the swing is the time for 10 swings ÷ 10.

When a fiducial mark is used to indicate a zero point, it is called a datum point.

Using a set square is a valuable way to check that you are reading a scale correctly:

You place a pointer onto the slotted mass.  When there is no force from the slotted mass, you set up the zero mark using the set square to make sure it is exactly opposite the pointer.   As you load this spring, you can measure the stretch using the set square to make sure you are reading from the pointer.  This reduces uncertainty.

A plumb-line is a thin piece of cotton with a small weight on it.  It always hangs vertically.  You will use one when finding the centre of mass of an irregular sheet of material.

Decorators use plumb-lines for making sure that wallpaper hangs vertically.

# Accuracy and Precision

When you take a measurement, you want the measurement to be accurate and precise.  However it is important to understand the difference between the two words:

• Accurate measurements are close to the true value;

• Precise measurements arise from the smallest scale reading that the instrument can give.  A digital voltmeter can give a precision of 0.001 V (the minimum reading that it can take).

If the voltmeter is not properly calibrated, it may be precise, but not accurate.  Similarly a watch with a precision of 10-6 s per day is not accurate if it's telling the wrong time.  The diagram shows the idea by using airgun shots around a target:

If the shots are on the bulls-eye, they are accurate.  If the shots are close together, they show high precision.  A tight group of shots may be precise, but are not accurate if they are not on target.

When measuring length with a metre ruler, you can measure to a precision of ± 1 mm.

The AQA definition for precision that is used in the examinations is the smallest division that can be read.  Yes, I know you can read it to about half a division, but that is what the AQA have decided.

An electronic thermometer that can read to ± 0.1 oC is more precise than an alcohol thermometer that reads to ± 0.5 oC.

Look at this voltmeter, which is reading 244 V.

 (a) What is the precision of this voltmeter? (b) What is reading on this voltmeter including the precision?

However there is more to precision than the above:

• A precise instrument gives consistent readings when taking the same measurements.  For example:

A beaker is weighed 3 times on balance A.  The readings are 73 g, 77 g, and 71 g.  The range is 77 – 71 = 6 g.

The same beaker is weighed on balance B.  The readings are 75, 73, and 74 g.

What is the range for balance B? Which instrument is more precise?

Three students are asked to determine the capacity of a box for storing ball bearings. Each of them uses a different ruler and only one takes care doing it. Here are their results:

 Name Width (cm) Height (cm) Depth (cm) Capacity (cm³) Andy 10 9 7 Bill 11 8 6.5 Cass 10.8 7.9 6.4

(a)  Calculate the capacity that each student works out and complete the table.

(b)  The true value is 545 cm3. Which is the most accurate result?

# Taking Measurements

Note that the word data is a plural word.  The singular, datum, is used in the context of a reference point from which measurements are made.   A single measurement is best referred to as a data item.

When you read your instruments, you are recording data.  The data that you record are then processed in a certain way to get:

• A graph;

• Processed data items, (e.g. resistance from voltage and current);

• Or both.

Data are made reliable by taking repeat readings.  Whenever you take a set of repeat readings, there is always going to be a certain amount of variation.  A free-fall experiment involving dropping a ball bearing through a viscous (gooey) liquid may give these results:

10.1 s, 10.2 s, 9.9 s, 10.0 s; 10.3 s; 10.6 s

There is variation because there are reaction times in operating the stopwatch.

The range of the results can be worked out:

Range = maximum value – minimum value.

 What is the range of the data above?

To get a value that we can plot on a graph, we need to take an average (or mean):

Average = (Sum of the readings) ÷ total number

 What is the average of the data above?

Notice from your answers that the average is not necessarily the midpoint of the range.

You can also test for the reliability of your data by checking the results obtained by other students in your class.  If they are close, you can be confident that your data are reliable.

# Processing Data and Quantification of errors

We have looked at how errors can arise.  Now we need to see how we can quantify the errors, i.e. turn the errors into numbers.

Suppose several students measured the diameter of the same ball bearing with a micrometer.  The results were:

1.21 mm; 1.20 mm; 1.18 mm; 1.25 mm; 1.24 mm; 1.19 mm

The average of these readings is:

1.21 + 1.20 + 1.18 + 1.25 + 1.24 + 1.19 = 1.21 mm

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The range of these readings can be worked out using:

Range = maximum value – minimum value

Range = 1.25 – 1.18 = 0.07 mm

The error (or uncertainty) is found by taking half the range:

Uncertainty = 0.5 × range = 0.5 × 0.07 = 0.035 mm

So we can write:

Diameter = 1.21 ± 0.04 mm

The diameter of the ball bearing can be confidently measured as:

1.17 mm to 1.25 mm

When you write down such an error remember to write in the units.

If you have repeat readings that are all identical, then you need to work out the uncertainty from the precision of the meter.  The same applies if you have not done repeats.

If we have several different variables, we need to combine the errors for each one.  Clearly it does not make sense to combine quantities of different units, e.g. millimetres, amps, ohms.

However we can work out the fractional error for all quantities.  These are numbers, so they can easily be combined.

We can work out the fractional error as:

Fractional error = error in the measurement ÷ average value

In this case:

Fractional error = 0.035 ÷ 1.21 = 0.029

You can convert that to a percentage.  In this case, it’s 2.9 %.

In AQA exams, the fractional uncertainties are always expressed as percentages.  If an overall or absolute error is asked for, you need to write:

Resistance is 6.3 ohms +/- 0.2 ohms

When combining errors, the following rules apply:

• When quantities are added or subtracted, the absolute errors are added together.

• When quantities are multiplied or divided, the fractional (percentage) errors are added together.

• If a quantity is squared, the error is multiplied by 2.  If it’s cubed, the error is multiplied by 3.

• If a square root is taken, the error is halved.

Here are two resistors in series.  We know that:

In this case we simply add up the resistances and add the absolute uncertainties.

 What is the total resistance and what is the total uncertainty of these two resistors?

 Do not add the percentage uncertainties.    The percentage uncertainty of the 25 W resistor is (1 ÷ 25) × 100 = 4.00 %   The percentage uncertainty of the 45 W resistor is (1 ÷ 45) × 100 = 2.22 %   The total percentage uncertainty would be 6.22 %.  This translates into an absolute uncertainty of:   0.0622 × 70 = 4.44 W   This is over twice what it actually is.

# Multiplying or Dividing

The majority of calculations involve the multiplication and/or the division of several numbers.  In this case the percentage uncertainties add together.

Let’s look at what happens in a calculation.  Suppose the voltage has an error of 2.5 %, and the current has an error of 3.6 %.  If we calculate the resistance using R = V/I, the error will be:

Error = 2.5 + 3.6 = 6.1 %

And if we calculate the power, we use P = IV:

Error = 2.5 + 3.6 = 6.1 %

 Do not put the error into a formula in this way: Error in the resistance = V/I = 2.5 ÷ 3.6 = 0.64 % Wrong Error in the power = VI = 2.5 × 3.6 = 9 % Wrong

Here are some data for a resistivity calculation.  The formula for resistivity is:

(The symbol r is rho, the Physics code for resistivity).

The uncertainty in the area is 8.5 %;

The uncertainty in the Resistance is 6.3 %;

The uncertainty in the length is 0.86 %.

 (a)  Calculate the uncertainty in the resistivity. (b)  What are the units for resistivity?

If we have a lot of quantities with different uncertainties, it can be helpful to use an equation to sum them:

Where:

• DW is the overall fractional uncertainty;

• DX is the uncertainty in quantity X;

• DY is the uncertainty in quantity Y;

• DZ is the uncertainty in quantity Z.

We can also write for a similar equation for the absolute uncertainty:

# Reliable and valid evidence

Evidence is data that are considered to be relevant to the investigation.  When investigating resistivity, the lengths of the resistance wire are relevant data; they are evidence.  The colours of the connecting wire or the manufacturers of the meters are not.  They are not evidence.

We want our evidence to be reliable:

• Reliable evidence stems from data that you can trust.

• If other students did the same experiment using the same equipment and method, they would get the same result.

The evidence must be valid as well.  Valid evidence is reliable and relevant.  For example:

• Measuring the extension of a spring to find the force pulling on it.  This is relevant, so the evidence is valid, provided that the data are reliable.

• Measuring the volume of a resistor to investigate the resistance.  This is not valid evidence, as the volume is not relevant to the resistance.

You can check the validity of your data by using secondary evidence, e.g. someone who has done the experiment before, and observed the same things.