Statistics of measurements

In all scientific studies, the precision with which a quantity is measured is all important. For example, consider the detection or discovery of a weak source. If the uncertainty in the measured intensity is large, one might not be convinced the source was even detected; it could have been a perfectly normal fluctuation in the background noise. Another example is the comparison of two intensities. Did that quasar change its optical brightness since the measurement of 6 months ago? If the quasar in fact changed its intensity by 3% but the two measurements were each made with only 5% accuracy, true variability could not be claimed. To detect a 3% difference with confidence, one should require each measurement to be accurate to significantly less than "±1%".

In measurements governed by counting statistics, one must define what one means by an error or accuracy because there is no absolute limit to the possible fluctuations in a measured number. The measured number can, in principle, always be found to be bigger than some quoted limit, such as ± 1%, if one is willing to wait long enough, possibly many lifetimes. Usually, one quotes a one standard deviation limit ±a within which about 2/3 of the measurements will fall.

A quantity that varies with time or frequency can be compared to theoretical models such as a Gaussian peak or a blackbody spectrum. Here again the uncertainties in the data must be understood. Statistical fluctuations could either mask a possible agreement or they could make the data falsely appear to agree with the model.

The understanding of a few basic principles of statistics is often sufficient. If one must resort to complex statistical arguments to "prove" that a measured effect is real, it may, in fact, not be real. In this case, it is often wiser to return to the telescope for additional measurements so that one can make (or disprove) the case with the basic statistical arguments.

Here we present some of the concepts underlying the assignment of error bars to measured numbers and to values derived from them. We then present the least squares method of fitting data to a model and also the x2 (chi-square) method of evaluating such a fit.

Instrumental noise

Every instrument has its own characteristic background noise. In the absence of any photons impinging on it, apparent spurious signals will be produced. For example, as noted above, cosmic ray particles will pass through a CCD leaving an image in one or more pixels that could be mistaken for a stellar image. Also, the readout process produces a noise of its own. Cosmic rays passing through proportional counters produce pulses much like those due to x rays, and radio amplifiers exhibit quantum shot noise.

Instrument designers incorporate features to reduce or eliminate these effects as much as possible. For example, anticoincidence schemes may be incorporated to identify signals from the most penetrating cosmic ray particles, and the noise in many solid state detectors is reduced by cooling the detectors to liquid gas temperatures. However, residual instrument background always remains at some level. The observer copes with this by measuring the background as carefully as possible and then by subtracting it from the on-source data. However, the measurement of the instrument noise has its own error, and the result of the subtraction will still carry an uncertainty (error) associated with the background.

Some residual instrumental noise is statistical in nature (e.g., that due to randomly arriving cosmic rays) and it can be measured as precisely as desired, given sufficient time. Other noise is systematic in nature; it does not behave in a predictable manner and hence can not easily be eliminated by better measurement. Examples are the gain change in an amplifier due to unmeasured temperature changes, the aging of detector or electronic components, and the varying transparency of thin clouds passing in front of the celestial source under study. It is very important that the sources of, and estimates of, both the statistical and systematic errors in a given measurement be stated when presenting a result.

Statistical fluctuations - "noise"

Statistical noise is a term applied to the inherent randomness of certain types of events. Consider the detection of photons from a steady source, one that is not pulsing or flaring. Although there is an average rate of arrival of photons, the actual number detected in a limited time interval will fluctuate from interval to interval. The nature of these fluctuations can be quantified.

Poisson distribution

Consider a steady celestial source of constant luminosity, that produces, on average, 100 counts in a detector every second. The 100 photons do not arrive with equal intervals, 10 ms between each count, like the ticks of a clock. Instead, they arrive randomly; each photon arrives at a time completely uncorrelated with the others. (This is not the case for a pulsing source.) The average rate of arrival is governed by a fixed probability of an event occurring in some fixed interval of time. In our case, there is a 10% chance of an event occurring in every millisecond, but no certainty that the event will indeed arrive in any specific millisecond. This randomness leads to a variation in the number of counts detected in successive 1-s intervals, e.g.,... 105, 98,87, 96, 103,97,101,...

A distribution of counts N (x) can be obtained from many such measurements; one measures the number of times N a given value x occurs. For example, N (95) is the number of times the value 95 is measured. For a random process such as photon arrival times, this distribution is well known theoretically as the Poisson distribution. Our experimental distribution would not match this function exactly because each value N (x) would itself have statistical fluctuations. We will find that longer observations with more accumulated counts improve the agreement with the theoretical function.

The Poisson distribution (not derived here) gives the probability Px of detecting an integer number x of events, mx e-m

when the average (mean) number of events over a large number of tries is m. For example, if the mean (average) value of the measurements is m = 10.3, the probability of detecting x = 6 photons in a single observation is Px = 0.056. If, instead, the mean is m = 6.0, the probability of detecting x = 6 is greater, Px = 0.161, but still smaller than one might have guessed. This tells us that it is not particularly likely that one will actually detect the mean number. In these cases, the counts were accumulated over some fixed time interval.

The Poisson distribution is valid for discrete independent events that occur randomly (equal probability of occurrence per unit time) with a low probability of occurrence in a differential time interval dt. The distributions Px vs. x for m = 3.0, 6.0 and 10.3 are shown in Fig. 7.

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