Introduction to Gaussian Errors Department of Physics 1st Year Laboratory Dr S P Tear

Histograms of repeated measurements Repeated measurements of the same quantity x1, x2, x3, … xn Histrogram of the frequency of values of xi: Measurements in B are less precise than in A –wider distribution of values x f(x) A x f(x) B Gaussian Errors

Gaussian or Normal distribution A very large number of measurements having random errors will approximate to a continuous bell shaped curve.  Gaussian or normal distribution is the standard deviation & is the mean Gaussian Errors

Standard deviations & probability Probability of the value of a single measurement lying: a) within x=- and x=+ is 1  b) within 2 s.d. of the mean is ~95%. c) within 1 s.d. is ~68%. Gaussian Errors

Estimating standard deviation We can estimate the standard deviation from a set of n measurements: Taking an infinite number of measurements would allow us to calculate the precise value of the standard deviation for the particular measurement we are making. Gaussian Errors

Precision of n measurements How precise is the mean of n measurements? We could take another set of n measurements, and calculate a new mean We could repeat this several times. It turns out the means themselves form a Gaussian distribution, with a width, the standard error, which is given by where  is the standard deviation of a set of n measurements. Gaussian Errors

Single measurement… If we make a single measurement of a quantity subject to random error we have no idea how reliable this quantity is. The best we can do is make an estimate based on the precision of the instruments we’re using. Gaussian Errors

Two measurements… If we make just two measurements we have a mean, an estimate of the standard deviation, and an estimate of the error in the mean (the standard error). Gaussian Errors

More measurements… If we make more measurements of the same quantity, we have a mean, a more precise estimate of the standard deviation, and a reduced standard error in the mean because taking more measurements leads to a more precise estimate of the true mean Gaussian Errors

Standard error reduces as n The standard error will reduce as the square root of the number of measurements, e.g. doubling the number of measurements reduces the standard error by 2 Gaussian Errors

How many measurements? the time we take in repeating measurements increases linearly with the number of measurements; but the precision of the measurements only increases as the square root of the number of measurements. Don’t spend too much time reducing the standard error – judgement based on the size of other errors in your experiment. Gaussian Errors

Example If Z = AB Then If the error in A is 3% and in B 1%, then = 0.03 to 1 s.f. i.e. 3%. Thus the 1% error in B was negligible in the presence of 3% error in A. Gaussian Errors