Estimating other parameters of a normal population
In normal populations, the mean, µ, is the parameter that is most often estimated. However numerical distributions have other parameters that may be of interest. For example,
These parameters can be estimated using the corresponding summary statistic from a random sample, but the error distribution may be difficult to obtain theoretically.
Simulation
Since the parameters µ and σ are unknown, we cannot perform a simulation with repeated samples from the actual population. However we can conduct a simulation from our best estimate of the population distribution — i.e. replacing µ and σ with the sample mean and standard deviation.
We repeatedly take samples of size n from this approximate population and evaluate the estimation error from using each sample. (Since we know the population from which we are sampling, we know the population parameter and can find the estimation error.)
The standard deviation of these errors is the approximate standard error of the estimator.
Assets-to-liabilities ratios
Researchers in Greece found the assets-to-liabilities ratios of a sample of 68 healthy companies. Low assets-to-liabilities ratios are usually regarded as undesirable for a company.
A quarter of healthy companies have an assets-to-liabilities ratio under what value?
In other words, we want to estimate the lower quartile of the assets-to-liabilities ratio in the wider population of similar Greek companies.
The above dot plot shows the assets-to-liabilities ratios and their lower quartile.
... we estimate that the population lower quartile is 1.217 — i.e. we estimate a quarter of similar companies will have assets-to-liabilities ratios below 1.217.
Approximate population
The assets-to-liabilities ratios have a fairly symmetric distribution so it is reasonable to try simulating random samples from a normal population. The diagram below shows a normal distribution whose mean and standard deviation equal those of our actual data.
We use this normal distribution as a population from which to sample 68 values — simulated assets-to-liabilities ratios for 68 similar companies. For the normal (1.726, 0.639) distribution, the lower quartile is 1.295, so this is the 'target' parameter that is estimated by our simulated samples.
The diagram initially shows a sample of 68 assets-to-liabilities ratios. The error is the difference between the sample lower quartile and the underlying population parameter, 1.295.
Click Take sample a few times and observe that the error varies from sample to sample.
Click Accumulate then take 100 or more samples (assets-to-liabilities ratios of 68 companies). The error distribution is built up as a stacked dot plot at the bottom of the diagram.
The distribution of errors gives an idea of how far the lower quartile of our actual data (1.217) is likely to be from the true unknown population lower quartile.
Click Estimate s.e. and bias to see the standard deviation and mean of the error distribution.
Understanding accuracy of estimate
In the simulation, you should have observed that the bias of the estimator is small, so we will treat it as zero.
The estimated standard error in your simulation was probably around 0.1. We can use the 70-95-100 rule-of-thumb to help interpret its value — the error has approximately 95% chance of being within 2 s.e. of zero and will be almost certainly within 3 s.e. of zero.
The error in our estimate of the population lower quartile, 1.217, is likely to be less than 0.2 and will almost certainly be less than 0.3.