# I The Normal Distribution - Random Errors

1. Jan 22, 2017

So let's say I do some measurements and obtain a set of measured values. The measurement is characterized by random errors so by making enough measurements, they approach a normal distribution.
In other words, my set of measured values can be approximated by a normal distribution characterized by a mean and standard deviation calculated from the set.
Thus, I have some 95% confidence interval for one standard deviation.

Is it correct to say this: 'By making a new measurement, the probability of the obtained measured value to fall within the 95% confidence interval is 95%.' ?

2. Jan 22, 2017

### Stephen Tashi

That is an ambiguous statement because "confidence interval" has at least two different definitions.

The definition of "confidence interval" is usually taken to mean an interval of specified span (e.g. plus or minus 23.5) about a fixed but unknown population parameter such as the mean $\mu$. In that interpretation, it is correct to say (for a 95% confidence interval for the mean) that a new measurement has a probability of 0.95 of falling within that confidence interval. However , this type of confidence interval has no specific numerical endpoints. For example, the confidence interval $(\mu -23.5, \mu + 23.5)$ does not have specific numerical endpoints because the specific value of $\mu$ is unknown.

In common speech, a different meaning of "confidence interval" is that it signifies an interval of a specific span (e.g. plus or minus 23.5) about an estimated population parameter such as an estimated value 120.8 for the mean of a distribution. If the population parameter is assumed to have a "fixed but unknown" value, it is incorrect to say that there is a 95% probability that a new measurement will be within such a confidence interval. (e.g. It is incorrect to say that there is a 95% chance that a new measurement will be in the interval ( 120.8 - 23.5, 120.8 + 23.5). )

( As a tactic of persuasion, it is common to present intervals by drawing "error bars" around estimated population values, so that these figures indicated specific intervals like (120.8 - 23.5, 120.8 + 23.5). This invites the audience to make the misinterpretation that there is some definite probability that new measurements will be in such a specific interval. )

If we use Bayesian statistics and assume the unknown parameter (e.g. the population mean) has various probabilities of having various values instead having a "fixed, but unknown" value then we can compute the probability that new measurements are within specific intervals about the estimated value of the parameter. Such intervals are called "credible intervals" or "Bayesian confidence intervals".