Confidence interval

In statistics, a **confidence interval**, abbreviated as CI, is a tool that people use when they collect data in order to estimate a certain parameter, such as the mean (average) of a population. A confidence interval gives a range of values that tells others how good we think our estimate is.

Each confidence interval depends on certain properties of the sample(s) that we use to make it. For example, a confidence interval that is made from a sample of 5000 people is going to be much better than a confidence interval made from a sample of only 5 people.

We usually give confidence intervals as a percentage, such as 95%. These percentages are called **confidence levels.**

## Meaning of the term "confidence"

The term _confidence_ has a similar meaning in statistics, as in common use. In common usage, a claim to 95% confidence in something is normally taken as indicating near certainty. In statistics, a claim to 95% confidence simply means that the researcher has seen one possible interval from a large number of possible ones, from which 19 out of 20 intervals contain the true value of the parameter.

## Practical example

A machine fills cups with margarine. It is adjusted so that the content of the cups is 250g of margarine. As the machine cannot fill every cup with exactly 250g, the content added to individual cups shows some variation, and is considered a random variable X.

This variation is assumed to be normally distributed around the desired average of 250g, with a standard deviation of 2.5g. To check if the machine is adequately calibrated, a sample of _n_ = 25 cups of margarine is chosen at random, and the cups are weighed. The weights of margarine are _X_<sub>1</sub>, ..., _X_<sub>25</sub>, a random sample from _X_.

To get an impression of the expectation μ, an estimate is needed. The appropriate estimator is the sample mean:

The sample shows actual weights _x_<sub>1</sub>, ...,_x_<sub>25</sub>, with mean:

If we take another sample of 25 cups, we could easily expect to find values like 250.4 or 251.1 grams. A sample mean value of 280 grams, however, would be extremely rare if the mean content of the cups is in fact close to 250g.

There is a whole interval around the observed value 250.2 of the sample mean within which, if the whole population mean actually takes a value in this range, the observed data would not be considered particularly unusual. Such an interval is called a confidence interval for the parameter _μ_.

To calculate such an interval, the endpoints of the interval have to be calculated from the sample, so they are statistics, functions of the sample _X_<sub>1</sub>, ..., _X_<sub>25</sub>, and hence are random variables themselves.

In our case, we may determine the endpoints by considering that the sample mean  from a normally distributed sample is also normally distributed, with the same expectation _μ_, but with standard error _σ_/√_n_ = 0.5 (grams). By standardizing, we get a random variable

which depends on the parameter μ to be estimated, but with a standard normal distribution independent of the parameter _μ_. Hence it is possible to find numbers −_z_ and _z_, independent of _μ_, where _Z_ lies in between with probability 1 − α, a measure of how confident we want to be. We take 1 - α = 0.95. So we have:

The number _z_ follows from the cumulative distribution function:

and we get:

This might be interpreted as: with probability 0.95, we will find a confidence interval in which we will meet the parameter _μ_ between the stochastic endpoints

and

This does not mean that there is 0.95 probability of meeting the parameter μ in the calculated interval. Every time the measurements are repeated, there will be another value for the mean  of the sample. In 95% of the cases, _μ_ will be between the endpoints calculated from this mean, but in 5% of the cases, it will not be. The actual confidence interval is calculated by entering the measured weights in the formula. Our 0.95 confidence interval becomes:

As the desired value 250 of _μ_ is within the resulted confidence interval, there is no reason to believe the machine is wrongly calibrated.

The calculated interval has fixed endpoints, where μ might be in between (or not). Thus this event has probability either 0 or 1. We _cannot_ say:  "with probability (1 − α), the parameter _μ_ lies in the confidence interval." We only know that by repetition in 100(1 − α) % of the cases, μ will be in the calculated interval. In 100α % of the cases, however, it does not. And unfortunately we do not know in which of the cases this happens. That is why we say: "with confidence level 100(1 − α) %, _μ_ lies in the confidence interval."

The figure on the right shows 50 realisations of a confidence interval for a given population mean μ. If we randomly choose one realisation, the probability that it contains the parameter is 95%. However, it is also possible that we may have unluckily picked the wrong one as well.

## Related pages

- Margin of error
- Prediction interval