Intensity measurement
mean intensity
Typically when measuring intensity, we make a series of
measurements. From this we determine the mean intensity and its standard
deviation. But how many measurements should we make? Time is often too limited
or too expensive to make many long measurements. Therefore should we make many
short measurements or only a few long ones?
The mean `x and standard deviation s of a series of measurements are given by
where xi are the individual measurements
and n is the number of measurements. The standard deviation is an
estimate of the spread in the measurements (population). Strictly this is valid only if the
variations in xi about the mean follow a normal distribution.
Often it is not recognised that this estimate of standard
deviation is not precise, especially when only a small numbers of measurements
are made. The real standard deviation s is given by
where c 2[a;b] is an inverse form of the c 2 distribution, called the percentage points of the c 2 distribution, and is the value of c 2 which would give a c 2 distribution value of a for b degrees of freedom; a is the confidence level. If a =0.9 then there is a 90% confidence that s is inside the two limits given above.
The length of an intensity measurement is commonly called the integration time. The total measurement time therefore is the product of the integration time and the number of measurements.
Number of independant intensity measurements
In Intensity I, the conclusion appears to be that the best results are achieved with a small number of measurements (3 or 4) with a long integration time. But perhaps there is a better way. To illustrate, let's choose a particular total measurement time, say 5 s. We collect a measurement every 0.1 s. We now have 50 individual measurements.
To estimate the mean we add the 50 values together and then divide by 50. Or we could put them into 5 groups of 10, take the mean of each group and then average the 5 groups. This is equivalent to changing the integration time from 0.1 s to 1 s. But we find that the two means are the same, even though the standard deviations are different. So perhaps the standard deviation is telling us something about how the intensity is varying rather than telling us how good the mean is. We can determine the confidence in the real mean m from
We can determine the confidence in the real mean m from where t[1-a/2;n-1] are the percentage points of Student’s t distribution.
Spikes
In Intensity II - contrary to Intensity I - the conclusion appears to be that the best results are achieved with a large number of measurements with a short integration time. Statistically this makes sense, the more measurements the better defined is the distribution of values and hence the better defined is the mean. This, after all, is the basis of Student's t distribution. Conversely, part of the uncertainty in a small number of measurements is that the distribution is not well defined. But there are other advantages to using a large number of measurements. Sometimes a spike occurs in intensity measurements. There are many possible causes of occasional spikes. The problem is that without care they can affect the mean.
Two spikes are shown here, the other points were generated as normal noise about a mean of 100. Such spikes can be detected in software and eliminated.