Understanding Accuracy, Precision and Uncertainties in Measurements, Summaries of Elementary Mathematics

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Accuracy, Precision and

Uncertainties

Precision

• How close your results are to each other
• How repeatable they are
• The number of decimal places or significant figures that an instrument can measure

Example 1

• The actual temperature of a classroom is 18.5°C
• A digital thermometer measures it to be 12.231°C
  • is this accurate or precise?

It is very precise because it gives a value to 5 significant figures, but it is not very accurate because it is a long way from the true temperature

Example 2

• The actual temperature of a classroom is 18.5°C
• A digital thermometer measures it to be 19°C
  • is this accurate or precise?

It is accurate because it gives a value close to the actual value, but it is not very precise because it only reads to two significant figures

Systematic errors

• Equipment errors e.g. digital scales being incorrectly zeroed or a thermometer that consistently reads 3°C too high
• Systematic errors can be reduced by replacing equipment, calibrating equipment or using multiple sets of equipment to compare readings
• Systematic errors mainly affect accuracy

Summary questions 1

Please write your answers in your books and attempt these questions without looking back at the previous slides. Spend 5 minutes on this.

1. Describe the difference between accuracy and precision
2. Describe the difference between random and systematic errors and give an example of each

Percentage difference

• The percentage difference is a simple calculation to compare the value obtained in an experiment to its true value
• It gives an idea of how accurate your result is

% 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 =

𝑑𝑑𝑒𝑒𝑒𝑒𝑑𝑑𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑𝑑𝑑𝑒𝑒𝑒𝑒𝑒𝑒 𝑣𝑣𝑒𝑒𝑒𝑒𝑣𝑣𝑑𝑑 − 𝑒𝑒𝑑𝑑𝑣𝑣𝑑𝑑 𝑣𝑣𝑒𝑒𝑒𝑒𝑣𝑣𝑑𝑑 𝑒𝑒𝑑𝑑𝑣𝑣𝑑𝑑 𝑣𝑣𝑒𝑒𝑒𝑒𝑣𝑣𝑑𝑑 × 100 %

• The vertical bars mean take the ‘modulus’ (absolute value) – so you ignore the minus sign if the numerator is negative
• A rule of thumb is that if your % difference is less than 10% then your experimental result is fairly accurate

Example 1

• An experiment to measure the time taken to travel a certain distance gives a reading of 11.8 s. The true time taken is 10.3 s. Calculate the percentage difference and comment on the accuracy of the experiment.

% 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 11 .8−10 10. 3.^3 × 100 % = 14.6 %

It is not a very accurate experiment because the % difference is greater than 10 %.

Summary questions 2

Please write your answers in your book and attempt these questions without looking back at the previous slides. Spend 5 minutes on this.

1. An experiment to measure the temperature of an aluminium block gives a result of 94.2°C. The true value is 81.8°C. Calculate the percentage difference and comment on the accuracy of the experiment.
2. An experiment to measure the mass of a copper block gives a result of 1.01 kg. The true value is 1.06 kg. Calculate the percentage difference and comment on the accuracy of the experiment.

Summary questions 2 - answers

Question 1

% 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 94 .2−81 81. 8.^8 × 100 % = 15.2 %

It is not a very accurate experiment because the % difference is greater than 10 %.

Absolute uncertainties

• These are expressed as ± the smallest increment that your instrument or measuring device can measure.
• For example a stopwatch that can measure to 0.1 s has an absolute uncertainty of ± 0.1 s.
• So, if this stopwatch gives a reading of 5.0 s it should really be quoted as 5.0 s ± 0.1 s.
• This means that the range of true values is 4.9 s to 5.1 s.

Percentage uncertainties

• These are the absolute uncertainties expressed as a percentage of the measured value
• So, for the stopwatch that produces a reading of 5.0 s ± 0.1 s, the percentage uncertainty is:

× 100 = 2 %

Absolute and Percentage uncertainties

• As seen previously, if the stopwatch is used to measure 5 s, the percentage uncertainty is:

× 100 = 2 %

• If the same stopwatch is used to measure 50 s, the percentage uncertainty is: 0. 50

× 100 = 0.2 %

Absolute and Percentage uncertainties

• This is why in an experiment you want your measured range to be as large as possible – because it reduces the % uncertainty in your result.
• For example measuring the period of a pendulum oscillation by timing over multiple swings rather than just one swing