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 determine 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"1, ..., "X"25, 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"1, ...,"x"25, 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"1, ..., "X"25, 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.