Uncertainties and errors

Random errors
A random error  is an error which affects a reading at random.
Sources of random errors include:

  • The observer being less than perfect
  • The readability of the equipment
  • External effects on the observed item

Systematic errors

A systematic error is an error which occurs at each reading.
Sources of systematic errors include:

  • The observer being less than perfect in the same way every time
  • An instrument with a zero offset error
  • An instrument that is improperly calibrated

Precision
A measurement is said to be accurate if it has little systematic errors.

Accuracy
A measurement is said to be precise if it has little random errors.

A measurement can be of great precision but be inaccurate (for example, if the instrument used had a zero offset error.

The effect of random errors on a set of data can be reduced by repeating readings. On the other hand, because systematic errors occur at each reading, repeating readings does not reduce their effect on the data.

The number of significant figures in a result should mirror the precision of the input data. That is to say, when dividing and multiplying, the number of significant figures must not exceed that of the least precise value.

Example:
Find the speed of a car that travels 11.21 meters in 1.23 seconds.

11.21 x 1.13 = 13.7883

The answer contains 6 significant figures. However, since the value for time (1.23 s) is only 3 significant figures. we write the answer as 13.7 m/s.

The number of significant figures in any answer should reflect the number of significant figures in the given data.

Absolute uncertainties
When marking the absolute uncertainty in a piece of data, we simply add ± 1 of the smallest significant figure.

Example:

13.21 m ± 0.01
0.002 g ± 0.001
1.2 s ± 0.1
12 V ± 1

Fractional uncertainties
To calculate the fractional uncertainty of a piece of data we simply divide the uncertainty by the value of the data.

Example:

1.2 s ± 0.1

Fractional uncertainty:

0.1 / 1.2 = 0.0625

Percentage uncertainties
To calculate the percentage uncertainty of a piece of data we simply multiply the fractional uncertainty by 100.

Example:

1.2 s ± 0.1

Percentage uncertainty:

0.1 / 1.2 x 100 = 6.25 %