# Why do the confidence intervals increase

## Confidence intervals

A confidence interval or confidence interval denotes an interval of possible parameter values in which an examined population parameter is located with the probability of the confidence level (1-α). The confidence interval provides information on the accuracy of the point estimate of an examined population parameter, such as the arithmetic mean.

The size of the confidence level can be freely selected by the user and depends on the respective question. Usually confidence levels of 0.95 or 0.99 are chosen. A confidence level of 0.95 describes, for example, that 95% of the 95% confidence intervals of the samples taken from a population cover the expected value μ of the population if a large number of samples are drawn from the same population. This fact is shown in Figure 1 with the expected value μ = 100, a sample size n = 15 and the standard deviation in the population σ = 15.

The green and dark blue markings below the distributions of the interval boundaries indicate those cases in which the confidence interval of the sample does not include the expected value of the population. In about 2.5% of the cases the upper limit of the confidence interval is below the expected value (blue) and in 2.5% of the cases the lower limit of the confidence interval is above the expected value μ = 100 (green). The lower limit G_{u} and the upper limit G_{O} the confidence interval can be estimated using the following formulas:

X̅ is the point estimate for the population mean, e.g._{1-α / 2} the quantile of the standard normal distribution and σ_{X̅} is the standard error of the point estimate of the population mean. The standard deviation of the population is required to calculate the standard error. Since this is often not known, the standard error of the sample mean s_{X̅} as an estimate for σ_{X̅} used (see standard error). Instead of using z-values of the standard normal distribution, the interval boundaries would be determined using quantiles of the t-distribution. For a deeper understanding, reference is made to Rudolf & Kuhlisch (2008). The width of the confidence interval depends on various parameters. A higher specified confidence level (1-α), i.e. 99% instead of 95%, for example, leads to a wider confidence interval. The higher the standard deviation, the wider the confidence interval. If the sample size is increased, the confidence interval becomes narrower.

kkk The video explains the creation of a confidence interval in more detail.

kkk The extent to which the confidence interval depends on various parameters can be seen graphically in the interactive simulation.

**further reading**

Rudolf, M., & Kuhlisch, W. (2008). *Biostatistics: An Introduction for Life Scientists* (Chapter 4.2). Munich: Pearson studies.

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