Random Errors

• Random (accidental) errors are those errors remaining after systematic errors have been eliminated.

• Characteristics: Positive and negative with same occurrence probability for same absolute value of the signal.

• Less probable as magnitude of the absolute value increases.

Random Errors

• Approaches zero as the number of measurements increase.

• For a given measurement method, random errors do not exceed a fixed value. If random errors do exceed a fixed value, that experiment should be repeated and studied separately.

Random Errors

• Random errors imply one measures n times to have a set of x (datum) that can be averaged (xn) . If the set of values is finite then each average is different. The averages follow a Gaussian Distribution

• Having a variance of σ2/n and σ2 is the variance of x.

• Confidence interval for x is

xn – uncertainty ≤ x ≤ xn + uncertainty

Random Errors

• And the uncertainty is k times the square root of the variance or :

kσ/√nOne can obtain k from the tables of the

normal Gaussian Distribution tables. The confidence interval has a probability of Conf. Interval Probability = 1 – α where α

is also found from the Gaussian Distribution tables.

Random Errors

• The normalized Gaussian Distribution is shown at the following site:

http://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm

In normalized functions, the total probability is 1. So 1-α is the total probability minus the “tail” of the Gaussian Distribution.

Random Errors

• See Example 1.2 page 19 of the text.