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Confidence Interval of a Mean Normal approximation methodDefinition & Properties Assumptions & Requirements
Definition and Properties
The confidence interval provides a measure of the reliability of our estimate of a statistic, whether the mean or any other statistic that we calculate from our data. It can be defined as that range which when attached to a sample statistic would enclose the true parametric value on a given proportion (1−α) of occasions when it is calculated from randomly selected
The confidence interval of the mean of a measurement variable is commonly estimated on the assumption that the statistic follows a normal distribution, and that the variance is therefore independent of the mean. This is known as a normal approximation confidence interval. Providing the distribution is not too skewed, central limit theorem means this assumption should be valid if your sample size is large. If the distribution is only moderately skewed, sample sizes of greater than 30 should be sufficient. The assumption will not be valid for small samples from a skewed distribution.
For a large sample (n > 30) your estimate of the standard error will be relatively unbiased. Hence you can say that 1.96 'standard errors' either side of the mean will enclose 95% of the means in that population. A common approximation is to give the arithmetic mean ± twice the standard error of the mean.
Small sample size
For means of smaller samples, your estimate of the standard error will be biased. This will result in the confidence interval being too small. You correct for this error by using t as the multiplier rather than z.
Finite population correction
If your sample is comprised of more than 10% of the total population, you need to include the finite population correction. For a large sample the confidence interval is given by :
If the data are drawn from a skewed population (especially if your sample size is less than 30), or if the variance is dependent on the mean (as with proportions), alternative methods should be used. The 'traditional' solution to this is to use an appropriate transformation. The commonest are the logarithmic, square root and arcsine square root transformations. Full details can be found in the More Information Page on
Assumptions and Requirements