Standard errors and CIs from formulae
Simple formulae can be found for the standard errors of sample means and proportions. Formulae exist for the standard errors of most commonly used parameter estimates, though some require computer software to evaluate.
For any such estimator, an approximate 95% confidence interval can be found from
estimate - 2 s.e. to estimate + 2 s.e.
(The confidence interval can sometimes be improved by replacing the constant 2 with a slightly different value.)
If no formula for the SE is available...
For some estimators, there is no formula for the standard error, or no available software can evaluate it. In this situation, a different approach is needed.
In this section, we describe how simulations can be used to assess the accuracy of such estimators.
October rainfall in Samaru, Northern Nigeria
In most of Africa, the most important climatic variable is rainfall. A better understanding of the distribution of rainfall can affect the crops that are grown and when they are planted.
What is the October rainfall that is reached in 1 out of 4 years in Samaru, Nigeria?
In other words, we want to estimate the upper quartile of the October rainfall distribution in Samaru.
The diagram below shows October rainfall for the 56 years between 1928 and 1983.
The distribution is very skew — seven years had no rain at all in October and several others had extremely low rainfall, but there was quite high rainfall in a few years. The upper quartile of the rainfall distribution was 57.4 mm.
If the rainfall distribution remains the same as in the past, we would estimate rainfall of 57.4 mm or more in one out of every four Octobers.
There are no convenient formulae for the standard error of this type of estimator, so a simulation is needed to find its standard error.