Course 1 Part 1

Question 1

Error vs uncertainty

Answer 1

Experimental uncertainty is the correct terminology
Error reinforces the misconception that a mistake was made

Every measurement has uncertainty

Question 2

Random error

Answer 2

Seen in scattered repeated measurements

• Some variation can be reduced by careful experimentation, e.g. controlling the temperature when measuring reaction rates.
• But always a fundamental limit due to “noise”
• Repeated measurements allow effects of random noise to be reduced

Question 3

What is systematic error

Answer 3

• Characteristic is systematic deviation from “true measurement e.g. measuring 10 when the true measurement is 24.
• Imagine Charles obtained 1620.183 mm compared to 1571 and 1573 mm from Adam and Bernadette. Either a mistake or some problem with the equipment.
• CAN be measured by “% agreement”.
• CANNOT be reduced by repeating measurements. Identify and eliminate!

Mistakes can be ignored - do it again! An eg would be not zeroing a top pan balance (so can usually see systematic error through repeats)_

Question 4

Calculations are subject to what type of error?

Answer 4

• Calculations are not subject to random errors
• The same calculation will always return the same result e.g. 12.7689543 J. (ignoring minor differences due to rounding / machine precession OR coding / operator errors)

• Calculations are subject to systematic errors in any approximations used
• Calculations involving electrons in molecules are not usually exact.
• If systematic error is present, agreement with experimental data is NOT a reliable measure of calculation quality; Method B may be “getting the right answer for wrong reason”.

Question 5

Accuracy vs precision

Answer 5

High precision implies a low spread of results (low random error)
High accuracy means that the average result is close to “true” answer (low systematic error)
High precision and high accuracy are always desirable, but not always essential.
Note can only really access accuracy and precision from multiple data points.

Question 6

In what case is low accuracy acceptable?

Answer 6

• To comply with supranational regulations, car speedometers generally read systematically higher than the true speed: ( eg due to differences in overall tire diameter with different tire manufacturers and wheel sizes, a factor is designed into the speedometer function that
increases the displayed speed. )
• Speedometer can only measure wheel rotations per second (νr) but actual road speed depends on tyre circumference (some uncertainty in radius r).
• Hence speedometer calibration is deliberately inaccurate!
• High accuracy may require careful calibration (verification against known standards).
• If we are only interested in differences, accuracy may not be a priority, e.g. in bomb calorimetry, we only care about the change in temperature.

Question 7

Intro to distribution

Answer 7

• If we measure a value of x many times, we can plot how often we obtain different values of x as a histogram*
• If x is a continuous variable (rather than an integer), then the “bins” correspond to a range of values e.g. 8 values of x were between 9.5 and 10.5
• Note that a large number of samples is needed to obtain a smooth curve
• The histogram in this limit is termed a distribution, and is characteristic for different statistical (random) processes.

Question 8

What would the distribution of a dice look like?

Answer 8

• Rolling a 6-sided die should result in a uniform distribution of outcomes
• The distribution proves the die is fair
• Distributions are generally normalised (integral is 1); vertical scale (for discrete outcomes) is then probability of
obtaining given outcome.

Question 9

What is the probability density?

Answer 9

Formal term for the vertical axis when dealing with a continuous distribution (contrast to discrete values when rolling a die)
An example of continuous data would be results in a titration

Question 10

The normal distribution

Answer 10

As number of measurements increases, histograms of experimental results always towards a normal or Gaussian distribution (classic curve check slide 21)

Center of distribution is called X
If symmetric the center (mean) is also likely the mode
Sigma determines the width of the distribution e.g. at half height

Probability of observing a result between two x values is the area under the curve (therefore total area = 1)

Question 11

Poisson distribution

Answer 11

• applicable for counting infrequent, random “independent” events eg radioactive decay, photon counts etc and is known as shot noise
• characterised by 1 parameter eg av. count rate, av. number of counts in fixed interval (Ñ (pretend tilda is straight)

As Ñ increases, tends to Gaussian distribution