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.