Inferential statistics gets its name from what happens in this branch of statistics. Rather than simply describe a set of data, inferential statistics seeks to infer something about a population on the basis of a statistical sample. One specific goal in inferential statistics involves the determination of the value of an unknown population parameter. The range of values that we use to estimate this parameter is called a confidence interval.

### The Form of a Confidence Interval

A confidence interval consists of two parts. The first part is the estimate of the population parameter. We obtain this estimate by using a simple random sample. From this sample, we calculate the statistic that corresponds to the parameter that we wish to estimate. For example, if we were interested in the mean height of all first-grade students in the United States, we would use a simple random sample of U.S. first graders, measure all of them and then compute the mean height of our sample.

The second part of a confidence interval is the margin of error. This is necessary because our estimate alone may be different from the true value of the population parameter. In order to allow for other potential values of the parameter, we need to produce a range of numbers. The margin of error does this.

Thus every confidence interval is of the following form:

Estimate ± Margin of Error

The estimate is in the center of the interval, and then we subtract and add the margin of error from this estimate to obtain a range of values for the parameter.

### Confidence Level

Attached to every confidence interval is a level of confidence. This is a probability or percent that indicates how much certainty we should be attributed to our confidence interval.

If all other aspects of a situation are identical, the higher the confidence level the wider the confidence interval.

This level of confidence can lead to some confusion. It is not a statement about the sampling procedure or population. Instead it is giving an indication of the success of the process of construction of a confidence interval. For example, confidence intervals with confidence of 80% will, in the long run, miss the true population parameter one out of every five times.

Any number from zero to one could, in theory, be used for a confidence level. In practice 90%, 95% and 99% are all common confidence levels.

### Margin of Error

The margin of error of a confidence level is determined by a couple of factors. We can see this by examining the formula for margin of error. A margin of error is of the form:

Margin of Error = (Statistic for Confidence Level)(Standard Deviation/Error)

The statistic for the confidence level depends upon what probability distribution is being used and what level of confidence we have chosen. For example, if *C*is our confidence level and we are working with a normal distribution, then *C* is the area under the curve between -*z*^{*} to *z*^{*}. This number *z*^{*} is the number in our margin of error formula.

### Standard Deviation or Standard Error

The other term necessary in our margin of error is the standard deviation or standard error. The standard deviation of the distribution that we are working with is preferred here. However, typically parameters from the population are unknown. This number is not usually available when forming confidence intervals in practice.

To deal with this uncertainty in knowing the standard deviation we instead use the standard error. The standard error that corresponds to a standard deviation is an estimate of this standard deviation. What makes the standard error so powerful is that it is calculated from the simple random sample that is used to calculate our estimate. No extra information is necessary as the sample does all of the estimation for us.

### Different Confidence Intervals

There are a variety of different situations that call for confidence intervals.

These confidence intervals are used to estimate a number of different parameters. Although these aspects are different, all of these confidence intervals are united by the same overall format. Some common confidence intervals are those for a population mean, population variance, population proportion, the difference of two population means and the difference of two population proportions.