# Error Bars for Peaks in Noisy Data

I'm doing an experiment where peaks in amplitude $A$ (the dependent variable) are expected as one varies the frequency $f$ (the independent variable).

Based on our theoretical model,

• Away from the peaks, the value for the amplitude is approximately zero.
• Close to the peaks, the amplitude as a function of frequency approximates a Gaussian distribution.

Currently, my method for identifying the peak is to take the mean and standard deviation of the amplitude, and defining peaks as data points where the amplitude is more than three standard deviations (above) the mean. This is possible as the data consists mostly of data where the amplitude is at zero.

However, what I'm interested in is a good way to define a meaningful error bar for the value of the peak frequency at each peak. Close to the $i^{th}$ peak, what I'm currently doing (inspired by an idea of the full-width half maximum of a signal) is to take half the value of amplitude $A_{i,max}/2$ and find the range of values of $f$ for which the amplitude is higher than this value, taking the minimum and maximum of this range as the error bar. This error bar would then indicate the range of values where there is a 98% level of confidence that the peak frequency $f_i$ is indeed in that range.

Is there any other meaningful means of handling the data, in particular in defining a meaningful error bar?

• Can you get multiple measurements at each frequency? Or if not, how fine grained can your variations on frequency be (so you have multiple measurements close to each peak, if not on it). – Peter Ellis Feb 6 '13 at 20:12
• Hi @PeterEllis, I can make multiple measurements of amplitude at each frequency. Of course, each measurement of the amplitude will have its own error bar. Regarding the variations of frequency, the minimum change in frequency for my experiment is 5 Hz. The range of value of $f$ for which the amplitude is higher than half the maximum amplitude is currently around 30Hz, but I have not taken multiple measurements yet. – Vincent Tjeng Feb 7 '13 at 11:49