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.