95% Confidence Interval: 4.446 to 5.554

This interval calculator helps you determine the confidence interval for the population mean based on a sample of observations from that population. The confidence interval is the range of possible values that we believe the population mean could be, based on a desired confidence level for the statement. You have the option of setting the confidence level when making this claim: if you want to be highly confident the population mean is within that range, you will set a very wide confidence interval for that value to ensure the value is present within that range. On the other hand, if you can tolerate some level of error, you can set a lower desired confidence level and interpret the results more narrowly. If you are working with a population proportion, you should consider using our Confidence Interval calculator for proportions.

By convention, the 95% confidence level is commonly used in statistical analysis so we have set the default value for the calculator at that level. It is not uncommon for engineering applications to be developed to an even higher confidence level such as 99%, often with additional tests included.

The confidence interval will be centered around the sample mean. In the absence of other claims, such as used in Bayesian statistics, this is our best estimate of the population mean. From there we look at the standard deviation of the sample. This provides our best estimate of the variance within the underlying population and influences the width of the confidence interval. Along the same lines, the sample size is incorporated into the analysis. The standard deviation and sample size give us some perspective on how much natural variation we should expect to see in the data. Finally, we use the desired confidence level (usually 95%) to establish how "wide" a "slice" we should use as the confidence interval, based on the acceptable odds of failure (population mean located outside the confidence interval).

For those of you using the calculator to check statistics homework, we present a breakout of the relevant calculations. This includes both the formula and a version populated with numbers from the calculator. This is intended to help you check your calculations.

In the real world, confidence intervals are a powerful tool but their insights should be balanced with other perspective. They are highly sensitive to any changes in the properties of the underlying statistical population from which they are drawn, including the possibility of shifts in statistical parameters caused by a non-stationary process (where values change over time). These are surprisingly common in manufacturing quality and social sciences applications. It is useful to supplement your conclusions through techniques such as holdout sample validation (data mine one sample, check another to ensure results replicate) or replicating your findings across similar subgroups of a data set. Insights that replicate well across samples, regardless of statistical precision, should be taken seriously. Along the same lines, be wary of putting too much stock in sampling conducted with a very large sample size - while you "technically" may have a narrow confidence interval, it is very possible you are commingling data from multiple operating periods or are dealing with non-stationary operating behaviors that change over time. Replication is a powerful antidote to this sort of mess.