Definition
A confidence interval for a population parameter is an interval of numbers calculated from a data sample and a confidence level, C, such that if the interval were calculated from repeated samples, C% of the samples would contain the true value of the parameter. Confidence intervals provide an interval estimate for a population parameter.
Theoretical example
We are interested in estimating the mean of a population, µ. The population has a standard deviation of s. We have a random sample of 100 observations from this population. For this sample, the sample mean is denoted by 
. The sample mean has a normal distribution with mean µ and standard deviation &sigma/10. The 95% confidence interval for µ is [ - 1.96 s/10, + 1.96 s/10]. We obtain the number 1.96 from a table giving values for a standard normal distribution, using the fact that the probability that a standard normal random variable lies in the interval [ -1.96,1.96 ] is 0.95.
Practical example
We are interested in estimating the average amount, µ, spent by teenage shoppers at an online music store in a one-month period with 90% confidence. We know that the standard deviation of purchases is $16. We have data on 64 teens, who purchased an average of $56 worth of music online. The $56 represents the sample average for this particular sample. A different sample of 64 teens would have a different sample average purchase. The sample average is a random variable. The sample average, , has a normal distribution with a mean of µ and standard devation of $16/8 = $2.
A 90% confidence interval for %mu; is given by [ $56 - 1.645 x $2, $56 + 1.645 x $2 ] = [ $52.71, $58.29 ]. The number 1.645 is obtained from a standard normal table: the probability that a standard normal random variable is in the interval [ -1.645, 1.645 ] is 90%. We are 90% confident that the true value of µ is contained in this interval.